Morfismos, Vol 9, No 2, 2005

VOLUMEN 9 NÚMERO 2 JULIO A DICIEMBRE DE 2005 ISSN: 1870-6525

Morfismos Comunicaciones Estudiantiles Departamento de Matem´aticas Cinvestav

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VOLUMEN 9 NÚMERO 2 JULIO A DICIEMBRE DE 2005 ISSN: 1870-6525

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Morfismos

Editorial Guidelines “Morfismos” is the journal of the students of the Mathematics Department of CINVESTAV. One of its main objectives is for students to acquire experience in writing mathematics. Morfismos appears twice a year.

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If you have any suggestions about the journal, let the Editors know and we will gladly study the possibility of implementing them. We expect this journal to foster, as a preliminary experience, the development of a correct style of writing mathematics.

Morfismos

Contenido Riemann and his zeta function Elena A. Kudryavtseva, Filip Saidak, and Peter Zvengrowski . . . . . . . . . . . . . . 1

Secondary operations in K-theory and the generalized vector field problem (revisited) Jes´ us Gonz´ alez and Maurilio Velasco-Fuentes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

Complete intersection toric ideals of oriented graphs Enrique Reyes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

Morfismos, Vol. 9, No. 2, 2005, pp. 1–48

Riemann and his zeta function Elena A. Kudryavtseva

1

Filip Saidak

∗

Peter Zvengrowski

Abstract An exposition is given, partly historical and partly mathematical, of the Riemann zeta function ζ(s) and the associated Riemann hypothesis. Using techniques similar to those of Riemann, it is shown how to locate and count non-trivial zeros of ζ(s). Relevance of these investigations to the theory of the distribution of prime numbers is discussed.

2000 Mathematics Subject Classification: 11M06, 11M26, 11A41, 11N05.

Keywords and phrases: meromorphic functions, Riemann zeta function, gamma function, Riemann hypothesis.

1

Introduction

The aim of this note is to give a straightforward introduction to some of the mysteries associated with the Riemann zeta function ζ(s) of a complex variable s and the Riemann hypothesis (usually written RH) about the location of its zeros, both from an historical and a mathematical perspective. The mathematical development will be largely self contained, and understandable to readers having a basic acquaintance with real and complex analysis. We hope to elucidate the answers to the following questions: (a) What is the Riemann zeta function? (b) What is the RH? (c) Why is this conjecture considered so important? ∗ 1

Invited Article.

Partially supported by a PIMS Visiting Research Fellowship.

1

2

E. A. Kudryavtseva, F. Saidak, and P. Zvengrowski

(d) Using techniques available to Riemann, how can one actually locate zeros of ζ? (e) How much did Riemann know about RH (did he even consider it relevant)? (f) What is the history of this problem since Riemann? (g) What is some of the current research being done on RH? (In particular, recent work of the authors will be briefly mentioned in this context.)

The zeta function is intimately connected with the distribution of the primes. Indeed one of Riemann’s primary motivations for studying it was to prove the Prime Number Theorem, cf. (13). Discussion about the distribution of primes will therefore be included (cf. §4). Another extremely important aspect of the Riemann zeta function is its very significant generalizations, however we only give the briefest of introductions to this.

The outline of the paper is as follows. Section 2 clarifies the notations used. In §3 meromorphic functions f and their zeros are introduced, including results for the functions f = sin, Γ, ζ that will be used in the sequel. The RH is stated here. In §4 the history of the zeta function and the distribution of primes, from Euclid [27] through Riemann [76], is sketched. The next two sections develop the mathematical theory of ζ and its zeros starting with basic results such as the intermediate value theorem from real analysis and the argument principle from complex analysis, and leading to the location of the first three zeros of ζ along the critical line. As mentioned in (d), this will be done using techniques that Riemann himself may well have used. In §7 we return to the historical perspective, discuss Weierstrass’ contributions, and address questions (c), (e), (f), and (g), including Siegel’s very important 1932 study [80] of Riemann’s Nachlass. Appendices A, B give short proofs, respectively, of the Prime Number Theorem and von Mangoldt’s Theorem. Details of the proof of the Riemann–von Mangoldt explicit formula 4.6 appear in Appendix C, as well as further discussion, partly speculative, of question (e).

2

Notation

All notations used in this paper are standard, however we list some of them here for completeness and convenience.

3

Riemann and his zeta function

log z = ln z

the natural logarithm of a complex number z

⌊x⌋

the greatest integer ≤ x, also floor of x

the nearest integer to a real number x, x ̸∈

⌈x⌋

{x}

f (x) ∼ g(x)

≈ " " $ , , n

p

" " ,

n≤x

" ρ

p≤x

+Z

x − ⌊x⌋, the fractional part of x f (x) f is asymptotic to g, i.e. lim =1 x→∞ g(x) approximately equal, for two complex numbers # !

N " 1 − log N n n=1

Euler’s constant, γ := lim

γ

1 2

respectively

p

respectively

∞ "

n=1 ⌊x⌋

"

n=1

,

N →∞

,

"

,

p prime

≈ 0.5772

$

p prime

"

p prime, p≤x

the sum taken over all zeros ρ of ζ (or ξ) in the entire critical strip, with their multiplicities, and in order of increasing |Im(ρ)|

For functions f, g of a complex (or real) variable z, where g is positive real valued, we define (a) f (z) = o(g(z)) if lim|z|→∞ |f (z)|/g(z) = 0,

(b) f (z) = O(g(z)) if there exists a constant C > 0 such that |f (z)| ≤ Cg(z) as |z| → ∞.

For any piecewise continuous function f of a real variable x, with only jump discontinuities, we define f˜(x) := 21 lim (f (x + ε) + f (x − ε)). ε→0

3

Meromorphic functions

Recall that an entire function f : C → C is one that is complex differentiable (i.e. holomorphic, equivalently analytic) at each s ∈ C. One calls a function meromorphic on a subset A ⊆ C if it is defined on some open neighbourhood U of A, except at a discrete (possibly empty) subset S ⊆ A, and is holomorphic everywhere on U \ S with poles at the points of S. The remainder of this section gives examples of meromorphic functions (as well as their poles and zeros) that will be important in the subsequent development of the Riemann zeta function ζ(s). Throughout this note we write s = σ + it for a complex variable,

4

E. A. Kudryavtseva, F. Saidak, and P. Zvengrowski

σ = Re(s), t = Im(s) being respectively the real and imaginary parts of s, a tradition that goes back (at least) to Landau [57] in 1909.

Example 3.1 Let f (s) = p(s) q(s) be a rational function, where p, q are relatively prime polynomials of degree m, n respectively. Then f has m zeros and n poles, where zeros are always counted with their multiplicities, and poles with their orders. A zero of multiplicity 1, or a pole of order 1, is called simple. As a specific illustration of this type, consider √ 3 f (s) = 2 s +1 . Here f has the three (simple) zeros s = −1, 21 ± 23 i, s −4is−4

and the pole of order two s = 2i. In this way, any two disjoint finite sets with assigned multiplicities (respectively orders) can be obtained for the zeros and poles of a meromorphic function, which can be taken to be a rational function.

As the following examples show, for more general meromorphic functions the number of zeros or poles can well be infinite.

Remark 3.2 For any non-constant meromorphic function the numbers of zeros and poles are at most countably infinite, since it is standard that the sets of zeros and poles must be topologically discrete subsets of C (cf. [58], III, §1 and V, §3). The latter property will be useful in the sequel.

Example 3.3 Let f (s) = sin(s). Then f is an entire function (hence meromorphic with no poles), and all zeros lie on the real axis, namely s = nπ, n ∈ Z. To verify the statement about the zeros, recall that sin(s) = sin(σ + it) = sin(σ) cosh(t) + i cos(σ) sinh(t).

An easy calculation now shows that | sin(s)|2 = sin2 (σ) + sinh2 (t), hence sin(s) = 0 implies both sinh(t) = 0 and sin(σ) = 0, i.e. t = 0 and σ = nπ.

We remark that the function sin satisfies the well known functional equations sin(s + π) = − sin(s), sin(−s) = − sin(s).

Example 3.4 f (s) = Γ(s). The gamma function is usually introduced in real analysis courses via the integral due to Euler [28]:

(1)

Γ(s) =

!

0

∞

xs−1 dx = ex

!

∞ 0

xs dx , ex x

s ∈ R, s > 0.

5

Riemann and his zeta function

The second form for this integral is called the Mellin transform [70] of e1x . (The integral (1) can also be viewed as a Laplace transform.)

The requirement s > 0 guarantees convergence. An easy exercise in integration by parts shows that Γ(s) = (s − 1)Γ(s − 1) for s > 1. Also Γ(1) = 1 is clear, hence Γ(n) = (n − 1)! for n ∈ N. Exactly the same arguments work if s = σ + it ∈ C, σ > 0, and thus the above integral defines a holomorphic function Γ(s) for σ > 0, satisfying the functional equation Γ(s) = (s − 1)Γ(s − 1), s ∈ C and σ > 1. Using the functional equation and the principle of analytic continuation extends Γ to a meromorphic function on C with simple poles at s = 0, −1, −2, . . . .

Other basic properties of the gamma function are the reflection formula of Euler (2)

Γ(s)Γ(1 − s) =

π , sin(πs)

s ̸∈ Z (i.e. sin(πs) ̸= 0),

and the duplication formula of Legendre (3)

√ 1 Γ(s)Γ(s + ) = π 21−2s Γ(2s), 2

2s ̸∈ Z≤0 .

From (2) it is easy to see that Γ(s) ̸= 0 for all s in its domain. An interesting historical discussion of the gamma function is given in [21], and proofs of (2), (3) can be found in the classic work of Artin [3] as well as many other texts. For example, in [58], XV, §2, proofs are given based on the Weierstrass product formula (30).

Example 3.5 f (s) = ζ(s). The Riemann zeta function will be described in much more detail in §4 and thereafter. Here we introduce it by three equivalent formulae, the usual Dirichlet series, the Euler product, and a Mellin transform expression similar to (1): (4)

# ∞ ∞ " !

1 1 xs dx 1 , = = ζ(s) = ns 1 − p−s Γ(s) 0 ex − 1 x p

σ > 1.

n=1

The integral representation of ζ(s) in (4), at least for s ∈ R, s > 1, is due to Abel [1] in 1823. Riemann [76] obtained it by making the change of variable x = tn in the definition (1), and then summing for all n ≥ 1, as shown by the following sequence of formulae, for σ > 1: # ∞ s−1 # ∞ s−1 x t Γ(s) Γ(s) = dx, = dt, and x s e n etn 0 0

6

E. A. Kudryavtseva, F. Saidak, and P. Zvengrowski

Γ(s)ζ(s) =

!

∞ 0

ts−1 dt. et − 1

In all three cases in (4) the condition σ > 1 is necessary for convergence. Again, by analytic continuation (cf. (10), or (16), (17)), ζ(s) can be extended to a meromorphic function on C with a single (simple) pole at s = 1, and satisfying a functional equation relating ζ(1 − s) and ζ(s), cf. (18). Using the functional equation we shall see in §5 that ζ has (simple) zeros at s = −2, −4, −6, . . . . These are called the trivial zeros, and we shall also see that the functional equation implies that all other zeros, the non-trivial zeros, lie in the critical strip 0 ≤ σ ≤ 1. The line σ = 12 is called the critical line.

The Riemann Hypothesis asserts that, for any non-trivial zero s = σ + it of ζ, σ = 1/2, i.e. all non-trivial zeros of ζ(s) lie on the critical line.

Remark 3.6 All functions f considered in 3.3, 3.4, 3.5 satisfy f (s) ∈ R for all real s in their domain, which is equivalent to f (¯ s) = f (s) for any meromorphic function on C. This means that all their zeros are real or occur in conjugate pairs σ ± it.

4

History from Euclid to Riemann

Let us now look at some of the fundamental ideas and theorems that played an important rˆole in the historical development of the theory of the Riemann zeta function, up to and including Riemann’s monumental 1859 paper [76], which is also quite remarkable since it is only eight pages long (see Appendix of [25] for an English translation). It should be mentioned that while much of this work (including Riemann’s), coming before modern standards of mathematical rigour were introduced to analysis by Weierstrass and his successors (notably Hardy), falls short of what would be considered acceptable proofs today, this in no way detracts from the originality and significance of this pioneering work.

The Fundamental Theorem of Arithmetic, originating in Book IX of Euclid’s Elements [27] (Proposition 14), states that every n ∈ N has a unique representation, up to order, as a product of prime numbers " m mk 1 m2 (5) n = pm pi i , mi ≥ 1.

1 p2 · · · pk = pi |n

7

Riemann and his zeta function

Here the existence of a prime factorization easily follows by induction.

Uniqueness, likely first proved by Gauss in 1801 [37] (although tacitly assumed by many prior authors), also can be proved by induction and Euclid’s lemma (Proposition 30 of [27]), i.e. p|ab =⇒ p|a or p|b. Euclid used the existence of a prime decomposition to show that there are infinitely many primes [27] (Proposition 20).

The Fundamental Theorem of Algebra, proved by Gauss in his Thesis [36] of 1799, states that if P (z) is a polynomial of degree n > 0 with complex coefficients, then P (z) has a unique (again up to order) factorization into n monic factors of degree 1 and a constant non-zero factor, over the complex numbers. In other words, P (z) has n zeros (or roots) in C, counted with multiplicities, and factors as (6)

P (z) = a(z − z1 )(z − z2 ) · · · (z − zn ) = a

n ! (z − zi ), i=1

for zi ∈ C and a ∈ C \ {0}. Partial results (for situations in R) had been obtained by Girard [39] in 1629 and Descartes (his Rule of Signs) [23] in 1637, and again this theorem was tacitly assumed by various authors prior to Gauss.

In the early 1730s, Euler found new, ingenious ways to combine (5) and (6) with theorems from analysis in order to prove new results in number theory. In 1737, in his Variae observationes [32] he used (5) to prove that the function ζ(s) has the product representation (4) for all real s > 1. Its significance comes from the fact that, for the very first time, one has an explicit link between prime numbers, natural numbers, and analysis, that can be used to study the distribution of primes (Euler mainly considered the special case where s is an integer, s > 1). As an immediate application " of1 this product representation, the divergence of the harmonic series n n = ζ(1) gives a new proof of Euclid’s theorem on the infinitude of primes. As a second application, taking logarithms of both sides of the product representation in (4), Euler himself was able to obtain, for s > 1, $ % 1 −1 log 1 − s p p ∞ # 1 ## # 1 1 = + = + R(s), ps mpms ps p p p

log ζ(s) = (7)

#

m=2

8

E. A. Kudryavtseva, F. Saidak, and P. Zvengrowski

! !

1 1 1 1!

1 !∞ where 0 < R(s) < p ∞ m=2 2pm = 2 n=2 n(n−1) = p p(p−1) < 2 1 + that the divergence of the harmonic 2 . Letting ! 1 s → 1 , we now see! series n n = ζ(1) implies that p p1 = ∞, a non-trivial statement of Euler [32] about the density of the primes.

Remark 4.1 Using Euler-Maclaurin summation (cf. §6), discovered by Euler [29] in 1732, and Maclaurin [67] in 1742, Euler found: " 1 # x dt # x {t} {x} = − ∈ (log x, 1 + log x), dt + 1 − (8) 2 n t x 1 t 1 n≤x

and then guessed the 1874 theorem of Mertens [71]: "1 p≤x

p

∼ log log x.

! ! !

1 The inequality (similar to (7)) log n≤x n1 < p≤x p1 + 12 p≤x p(p−1) and (8) imply a weaker, but still very interesting lower bound ⎛ ⎞ (−1 &' "1 " 1 1 1 ⎠ − 1 > log ⎝ > log log x − 1− p p 2 p(p − 1) 2 p≤x

p≤x

p≤x

for all x ≥ 2. This gives a second non-trivial statement about the density ! of the primes, strengthening the conclusion p p1 = ∞ from (7).

2

In 1734 (see [30] and [31]), Euler showed that ζ(2) = π6 , a difficult question proposed by Cavalieri’s student Mengoli [97] as early as 1650.

Factoring the function sin x in terms of its zeros: 0, ±π, ±2π, . . . , as if it were a polynomial in (6), Euler found its product representation (see also 7.3), and he equated it with the Taylor series of sin x, ( ∞ +, ∞ ' & & x -. x -, x2 sin x = x 1+ =x 1− 1− 2 2 πn πn π n n=1

= x−

x3 3!

n=1

+

x5 5!

−

x7 7!

+ ... .

!

Comparing the coefficients at x3 immediately gives us − n 21 2 = − 3!1 . π n Another proof of the formula for ζ(2) can be obtained using the Fourier series expansion of f (x) = x2 , −π ≤ x ≤ π, evaluated at x = π. See also 5.2 for another method.

9

Riemann and his zeta function

Remark 4.2 In 1837, Dirichlet [24] generalized parts of Euler’s work on the zeta function in two significant ways. First, in (4), he now thought of s ∈ R, s > 1, whereas Euler had mainly considered cases where s ∈ Q, s > 1, see also (18). Second, Dirichlet introduced the generalization of (4) and of (17) $ ! χ(n) " # χ(p) −1 (9) L(s, χ) = = , s > 1, 1− s ns p n p

where χ is a Dirichlet character modulo a prime q (we do not define this here, it is not necessary for the further discussion). Using this he generalized Euler’s argument and proved his celebrated theorem [24] that, for any coprime a and b, we have !

1 = ∞. p p≡b (mod a)

He thereby proved a famous conjecture of Legendre [60], that any arithmetical progression {an + b | n ∈ N}, where a, b are relatively prime integers, contains an infinitude of prime numbers.

From the work of Euler and Dirichlet, it became clear that analytical methods were a powerful tool in number theory.

The main reason Riemann, who was a student of Dirichlet, was able to make tremendous advances in the theory of the zeta function, was the growth of the new field of complex analysis, created by Fourier, Cauchy, Gauss, and others in the period 1800-1830. In his thesis, submitted to the University of G¨ottingen in 1851, Riemann himself vastly enlarged this new branch of analysis. Such basic notions as the Cauchy-Riemann equations, the Riemann mapping theorem, and Riemann surfaces are among his many contributions to the subject, especially to that part now called geometric function theory. Probably no mathematician, for at least the 50 years following Riemann’s death (at age 39, in 1866), came close to his mastery of geometric function theory, which he used to good advantage in his work on the zeta function.

In 1859, Riemann defined ζ(s) as a function of a complex variable s.

The first step was to extend (or to analytically continue) the definition (4) of ζ(s) to all of C \ {1}. This can be accomplished by noticing that, %∞ −s for σ > 0, n = s n x−s−1 dx, and so $ ∞ ∞ # & ∞ ∞ & ∞ ! ! !

1 dx dx = s = s (10) ζ(s) = s s+1 s+1 n n x n x n=1

n=1

n=1

10

E. A. Kudryavtseva, F. Saidak, and P. Zvengrowski

=s

!

∞

1

(

"

n≤x

=

dx 1) s+1 = s x

s − s s−1

!

!

∞

1

∞

1

⌊x⌋ dx = s xs+1

{x} dx xs+1

!

1

∞

x − {x} dx xs+1

for σ > 1 .

Since {x} ∈ [0, 1), it follows that the last integral converges for σ > 0, and defines a continuation of ζ(s) to the half-plane σ = Re(s) > 0. If one continues this process, one can extend ζ(s) to a holomorphic function on all of C \ {1}. One sees from (10) that s = 1 is a simple pole with residue 1. See also (16) for Riemann’s original technique, or (17).

Remark 4.3 Note that for s real, s > 0, the last integral in (10) is always positive real. It follows at once from (10) that ζ(s) < 0, s ∈ (0, 1), and clearly ζ(s) > 1 for s ∈ (1, ∞).

Remark 4.4 The next step of this continuation process gives ! s 1 s(s + 1) ∞ {x}2 − {x} ζ(s) = − − dx, σ > −1. s−1 2 2 xs+2 1 It follows at once that ζ(0) = − 12 . Applying Stirling’s formula (22) and the definition of γ, one also shows ζ ′ (0) = − 12 log(2π).

In order to see a deeper connection between ζ(s) and prime numbers, let us now follow Riemann’s ideas [76] and employ the logarithmic derivative of the Euler product (4), using (7): (11)

−

∞ ∞ "" " log p Λ(n) ζ ′ (s) = = ms ζ(s) p ns p m=1

for σ > 1,

n=2

where Λ(n) is von Mangoldt’s function [68], defined as Λ(pm ) = log p for a prime power pm , and 0 otherwise. We also define the Chebyshev function (cf. [13]) " " Λ(n) = log p. ψ(x) := n≤x

pm ≤x

Note that (11) can also be written, for σ > 1, as (12)

! ∞ ! ∞ ∞ "" ψ(x) log p ζ ′ (s) −s = − = x dψ(x) = s dx. ms ζ(s) p xs+1 1 1 p m=1

Riemann and his zeta function

11

We now turn to the early developments related to the Prime Number Theorem (PNT), stated in (13) below, a much stronger statement ! 1 about the asymptotic distribution of the primes (e.g. compared to p p = ∞).

In a letter written to the astronomer (his former student) Encke in 1849 (cf. [38] or Appendix B of [40]), Gauss stated that he observed as early as 1792 or 1793 (when he was only 16) that the density of primes around a number x appears to be on the average inversely" proportional to log x, x dt and therefore the logarithmic integral Li(x) := 2 log provide t should ! a good approximation to the prime counting function π(x) := p≤x 1.

Gauss’ work on this question, both as a youth and in his 1848 letter, was empirical in nature and unpublished.

Prime Number Theorem (PNT): (13)

π(x) ∼ Li(x).

Remark 4.5 The PNT was proved in 1896, cf. Appendix A. The RH is closely related to refining the PNT further by estimating the error in the approximation (13), indeed, it is equivalent to this error being √ O( x log x), cf. [54] and §5.5 of [25]. See also A.2 and A.3.

The first published account is probably due to Legendre [60] in 1798, again based on empirical observations. Legendre conjectured that, for x large, π(x) ≈ x/(log x − 1.08366). Let us mention here that Legendre’s formula clearly implies π(x) ∼ logx x , and this in turn is equivalent to PNT, due to (15). Gauss, in his 1849 letter, compared Legendre’s formula to Li(x) for values of x = 5 × 105 , 106 , 1.5 × 106 , . . . , 3 × 106 .

He noted that while the Legendre formula seemed to have smaller deviations from π(x), these deviations seemed to be growing more rapidly than for Li(x), and therefore it was “quite possible they may surpass them” (i.e. the deviations of the Legendre formula would eventually become larger than for Li(x)).

The two memoirs published by Chebyshev [12], [13] in 1848 and 1850 comprised the first mathematical attack on the PNT. ! In [13] he defined the function ψ(x) (above), also defined θ(x) := p≤x log p, realized that ψ(x) is nearly equal to x (e.g. for n = 102 , 103 , 104 , 105 , ψ(n)/n = 0.94045, 0.99668, 1.00134, 1.00051 respectively), and can be estimated more easily than π(x). He was able to prove that, for x sufficiently large, it satisfies the inequality Ax < ψ(x) < Bx with A = 0.92129, B = 1.10555. He used this to show that, for x large, x x (14) 0.89 < π(x) < 1.11 log x log x

12

E. A. Kudryavtseva, F. Saidak, and P. Zvengrowski

(see also [25], §1.1).2 As an application, Chebyshev [13] gave the first proof of Bertrand’s postulate3 , as well as obtained the following interesting result: ! for any positive ! non-increasing function F = F (n), n ∈ N, the series n F (n) and p F (p) log p either both converge or both diπ(x) , x→∞ x/ log x

verge. Chebyshev [12] also proved that lim

if it exists (possibly

x − log x), if it exists (possibly infinite), infinite), equals 1, and lim ( π(x) x→∞ equals −1. As a consequence, based on the asymptotic expansion of Li(x): " # x x x x + + . . . + (n − 1)! n + o , (15) Li(x) = log x log2 x log x logn x

for any fixed n, he showed (by taking n = 1 and n = 2) that log xx−1 provides the best approximation to π(x) among all formulae of the form x x A log x−B (in particular, both log x and Legendre’s empirical formula), provided that the above two limits exist. Validity of the latter assumption was finally confirmed in 1899, see §7 and A.2.

Riemann’s original motivation for his study of the zeta function was to obtain an explicit formula such as (49) for π(x), similar to 4.6 below, and to prove the PNT. Being aware of Chebyshev’s work (indeed Chebyshev had met Riemann’s mathematical mentor Dirichlet in 18524 ), Rie2

Chebyshev’s method was used later to give an elementary proof of PNT, see Appendix A. 3 In his group-theoretical investigation in 1845, Bertrand used the following proposition which he only verified within the limits of tables of primes: for each integer x ≥ 7 there exists a prime p ∈ ( x2 , x − 2], cf. [57], §4. In fact Chebyshev [13] strengthx ened this by proving π(2x) − π(x) > 35 log(2x) for x sufficiently large.

4 “In the summer of 1852, however, he [Chebyshev] was sent on an official mission, lasting six months, to visit the cities of Berlin, London and Paris. The main purpose of this was the inspection of factories and workshops, in order to learn about the use of steam engines and other types of machinery. ...

However while studying new technologies in the daytime he found opportunities in the evenings to meet the foremost mathematicians in the places he was visiting. For example in Berlin he spent a considerable time with Dirichlet, in London with Cayley and Sylvester, and in Paris he was warmly received by Liouville, who introduced him to other French mathematicians. ...” Excerpted from [51], Ch. 4.

“It was of great interest for me to become acquainted with the celebrated geometer Lejeune-Dirichlet. ... [I] found an occasion each day to talk with this geometer concerning [applications of calculus to number theory] as well as other questions on pure and applied analysis. ... [I attended] with particular pleasure one of his lectures on theoretical mechanics.” Excerpted from Chebyshev’s report on his trip to Western Europe [14], p. XVII.

13

Riemann and his zeta function

mann came up with the revolutionary idea of applying Fourier analysis (of which he was a master) in order to get more precise information about π(x) via a function Π(x), cf. (46), which is analogous to Chebyshev’s ψ(x). For the purpose of Fourier analysis we also modify ˜ as given in §2. ψ slightly (by a standard procedure) to ψ, A fundamental link between the functions ζ(s) and ψ(x) can be ˜ obtained by inverting (12) to get an analytic expression for ψ(x).

In 5 fact, starting from (12), the classical Fourier inversion formula implies, for any fixed a > 1, 1 T →∞ 2πi

˜ ψ(x) = lim

!

a+iT

a−iT

" ′ # ds ζ (s) − xs , ζ(s) s

x > 0.

Now consider the closed rectangular contour C with vertices a ± iT , −(2n + 1) ± iT , with counterclockwise orientaion, where T → ∞ is suitably chosen (cf. C.1 (a)) and n ∈ N, n ≥ T log T . With careful estimations of the modulus of the integrand on the horizontal edges of C, as well as the left hand edge, one shows that the contribution of these three edges approaches 0 with T → ∞ as above, and hence (cf. C.1) ˜ ψ(x) = lim

T →∞

1 2πi

! " ′ # ds ζ (s) − xs , ζ(s) s C

x > 1.

The latter integral is easily evaluated using the residue formula (cf. [58], VI, §1). The poles of the integrand inside C occur at s = 1, at the non-trivial zeros ρ of ζ, at the trivial zeros −2n of ζ, and at s = 0. The residues are, respectively, x (since s = 1 is a simple pole ρ −2n of ζ, cf. (10)), −m(ρ) xρ , m(ρ) ∈ N being the multiplicity of ρ, x2n ′

(0) (since all trivial zeros of ζ are simple, cf. 5.1), and − ζζ(0) = − log(2π), see 4.4. This leads directly to formula (38), which in turn leads to the important “explicit formula”, stated by Riemann [76] in slightly different form (47), and proved in both Riemann’s form and the following form by von Mangoldt [68]: 5

When σ = a > 1, we can rewrite (12) as ∞

f (t) = 0

t∈

,y∈

e−ity g(y) dy,

where

f (t) := −

ζ ′ (a + it) 1 · , ζ(a + it) a + it

. By the Fourier inversion formula, g(y) = lim

T →∞

1 2π

g(y) := T −T

˜ y) ψ(e , eay

eity f (t) dt.

14

E. A. Kudryavtseva, F. Saidak, and P. Zvengrowski

4.6 Explicit formula (Riemann–von Mangoldt [68], 1895) For x > 1, " # ! xρ x2 1 ˜ + log ψ(x) = x − − log(2π), ρ 2 x2 − 1 ρ the sum being extended over all zeros ρ (with multiplicities) of ζ in the entire critical strip, in order of increasing |ρ| (compare §2). !

Evidently, the explicit formula 4.6 gives a very precise description of the error committed in the approximation ψ(x) ∼ x, and more importantly, it relates (e.g. in Appendix A) the estimation of this error to the location of the non-trivial zeros. Since the vertical distribution is reasonably well known, see (33), 7.5, the horizontal location of the zeros becomes of paramount importance, see also Remark A.3.

We close this historical discussion by appending Riemann’s formula in [76] to obtain the analytic continuation of ζ to all of C \ {1}: $ (−z)s dz Γ(1 − s) H(s), where H(s) := , (16) ζ(s) = z 2πi C e −1 z s ∈ C \ N, and C = C1 ∪ C2 ∪ C3 is the contour from +∞ to +∞ shown in Figure 1 with C2 a circle of radius ε, 0 < ε < 2π. ✬✩ C2 ✛ ■ ❛❵ ❛❵

0

C3 ❘ ✫✪ ε

✲

C1

Figure 1: The Contour C for the Hankel Integral Nowadays H(s) is known as the Hankel integral ([58], XV, §4), and the exponential function (−z)s = es log(−z) is defined by taking log(−z) to be the principal value of log on C with the negative real axis deleted.

It follows that Im(log(−z)) varies from −π to +π on C2 , and hence one defines log(−z) = log |z| − πi on C1 , log(−z) = log |z| + πi on C3 . The radius ε of C2 is taken less than 2π so that z = 0 is the only zero of the denominator ez − 1 inside or on C. The exponential decay of the integrand and Lemma 1.1 of [58], XV (which allows one to differentiate under the integral sign) show that H is an entire function.

15

Riemann and his zeta function

Formula (16) for σ > 1 follows from (2) and the Mellin transform expression (4) of ζ, by showing that ! ! ! ! ∞ xs dx H(s) = , σ > 1, + + = 2i sin(πs) ex − 1 x C1 C2 C3 0 " " " where lim ( C1 + C3 ) equals the right hand expression, while C2 equals ε→0+

s

s−1 −z on C , and hence 2πi times the average value of (−z) 2 ez −1 = (−z) ez −1 has limit 0 as ε → 0+ . Thus (16) gives the analytic continuation of ζ to C \ {1}.

5

Symmetry and the associated ξ function

In 1749, Euler returned to the subject in his paper [35], see also his 1748 book [33]. This time he considered the closely related function6 (17)

φ(s) :=

∞ # (−1)n+1

n=1

ns

= (1 − 21−s ) ζ(s) .

n+1 $ xn , which is abUsing the associated power series φ(s, x) = n (−1) ns solutely convergent for |x| < 1, s ∈ R, and taking limits as x → 1− (cf.

Hardy [46], §2.3, for details), he proved that, for any integer m ≥ 2, we have % m −1 (m − 1)! if m is even, (−1)m/2+1 m 2 m−1 φ(1 − m, x) π (2 −1) = lim φ(m, x) x→1− 0 if m is odd.

He then formally replaced lim x→1− φ(s, x) by φ(s) and, with the help of the cosine function, rewrote this in the simple form φ(1 − m) 2m − 1 πm = − m m−1 (m − 1)! cos φ(m) π (2 − 1) 2

for all m ∈ N \ {1}.

At this point Euler states his belief that the same should remain true for all real numbers, i.e. φ(1 − s) 2s − 1 πs = − s s−1 Γ(s) cos φ(s) π (2 − 1) 2

for s ∈ R \ {1, 0, −1, −2, . . . } .

6 The series in (17) uniformly converges in the half plane σ ≥ ε for any ε > 0 [57], §42, and thereby determines an analytic continuation of ζ(s) to σ > 0.

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E. A. Kudryavtseva, F. Saidak, and P. Zvengrowski

Due to (2), this is equivalent to saying that, for all s ∈ R \ Z≥0 , we have ! πs " ζ(1 − s), (18) ζ(s) = 2s π s−1 Γ(1 − s) sin 2

the so called functional equation of ζ. Euler did not know how to prove this intriguing assertion, but he verified it for several non-integer values of s, e.g. s = 12 , 32 , and 52 .

In 1859 Riemann [76] was the first to indicate that (18) is true, indeed for all s ∈ C \ Z≥0 . Today, many proofs of this important result exist (see [25], [55], [57], [58], or [85]). In [76] Riemann first expressed ζ(s) in terms of the Hankel integral H(s) for all s ∈ C \ N, cf. (16).

He then evaluated H(s), for σ < 0, by reversing the orientation of the contour C shown in Figure 1, and applying the residue formula (cf. [58], VI, §1) to the domain D exterior to C (taking account of the poles of the integrand in this domain at z = 2πki, k ∈ Z \ {0}, where ez − 1 = 0).

This yields the functional equation (18) for σ < 0, and therefore for all s ∈ C\Z≥0 , since the difference of the two sides of (18) is a meromorphic function on C having a non-discrete set of zeros, see 3.2. Since D is not bounded, to make Riemann’s argument rigorous one can replace D by its intersection with a large square |Re(z)| < (2n+1)π, |Im(z)| < (2n+1)π, and take the limit of the integral as n → ∞. Notice that |ez − 1| > 1/2 # (−z)s dz = 0, σ < 0 on the boundary Q of the square, hence lim n→∞ Q ez − 1 z (see also [25], §1.6).

Remark 5.1 Strictly speaking (18) does not hold when Γ(1 − s) is undefined, which as we saw in 3.4 is true precisely for the poles at s = 1, 2, 3, . . . . However, for s = 3, 5, 7, . . . , ζ(s) is some positive real number and | sin( πs 2 )| = 1, so an easy continuity argument shows ζ(1 − s) then must equal 0, i.e. 0 = ζ(−2) = ζ(−4) = . . .. As mentioned in 3.5, these are called the “trivial” zeros of the zeta function. We now see that they are the only possible zeros on the real axis, and are simple zeros. Since any convergent infinite product with non-zero factors cannot equal 0, the Euler product formula (4) already shows ζ(s) ̸= 0, σ > 1. Then (18) and the observations about the zeros of the functions sin, Γ in Examples 3.3, 3.4, show that ζ(s) ̸= 0, σ < 0, apart from the trivial zeros on the negative real axis. The assertion in Example 3.5, that all non-trivial zeros lie in the critical strip 0 ≤ σ ≤ 1, is thus proved.

17

Riemann and his zeta function

Remark 5.2 Alternatively, the result about the trivial zeros of ζ in Remark 5.1 can be thought of as a special case of the following explicit formula, which can be easily derived from the Hankel integral (16), cf. [25], §1.5: ζ(−n) = (−1)n

Bn+1 , n+1

n = 0, 1, 2, . . . .

Here Bn is the nth Bernoulli number [7], defined by ∞

! Bn z n z = . ez − 1 n! n=0

For example, B0 = 1, B2 = 1/6, B4 = −1/30, B6 = 1/42, B8 = −1/30, B10 = 5/66, B1 = −1/2, B3 = B5 = B7 = . . . = 0. In view of the functional equation (18) with s = −2k + 1, the above formula for ζ(−n), with n = 2k − 1, is equivalent to Euler’s famous formula in [34]: ζ(2k) = (−1)k+1

(2π)2k B2k , 2 · (2k)!

k ∈ N.

Having derived the functional equation (18), Riemann proceeded at once to obtain a more symmetric form by defining (19)

ξ(s) :=

s 1 s s(s − 1)ζ(s)Γ( )π −s/2 = (s − 1)ζ(s)Γ( + 1)π −s/2 .

2 2 2

Proposition 5.3 (Riemann [76], 1859) The function ξ satisfies (a) ξ(s) = ξ(1 − s),

(b) ξ is an entire function, and ξ(s) = ξ(s), (c) ξ( 12 + it) ∈ R,

(d) If ξ(s) = 0, then 0 ≤ σ ≤ 1, (e)7 ξ(0) = ξ(1) = 1/2,

(f ) ξ(s) > 0 for all s ∈ R. Outline of proof: Using the properties (2), (3) of the gamma function, deriving the functional equation (a) for ξ from that of ζ, i.e. (18), is a straightforward exercise. The second expression in the definition (19) shows at once that ξ is holomorphic for σ ≥ 0, since the simple pole of ζ at 1 is removed by the factor s − 1, and there are no other poles for σ > 0. But then (a) implies ξ holomorphic on all of C. The second

18

E. A. Kudryavtseva, F. Saidak, and P. Zvengrowski

part of (b) follows trivially from (19) and 3.6. Combining (a) with (b) and 3.6 gives (c), and similarly for (d) by first noting ξ(s) ̸= 0, σ > 1.

The known values Γ(1) = 1, ζ(0) = −1/2 (see 3.4, 4.4) imply (e) for ξ(0), and the functional equation (a) then gives the result for ξ(1).

Finally, to prove (f), first note from (1) that Γ(s) > 0 for all s ∈ R, s > 0. Combining this with Remark 4.3 and the definition (19) of ξ proves (f) for s > 0, s ̸= 0, 1. Combining this with (e) then proves (f) for all s ≥ 0, whence the functional equation (a) shows that (f) holds for all s ∈ R. !

Corollary 5.4 The zeros of the function ξ are identical to the nontrivial zeros of the function ζ. !

It is now possible to understand what Riemann meant when he stated (his version of) the RH, which we quote in the original German, followed by an English translation ([25], Appendix): “Man findet nun in der That etwa so viele reele Nullstellen innerhalb dieser Grenzen, und es ist sehr wahrscheinlich, dass alle Wurzeln reel sind” (One finds in fact about this many real roots within these bounds, and it is very likely that all of the roots are real). At this stage (the third page) of his paper [76], Riemann is referring to the function ξ(1/2 + iu) of the complex variable u. The fact that all zeros of this function are real (i.e. u ∈ R) is equivalent to the fact that all zeros of ξ(s) have real part Re(s) = σ = 1/2, which by Corollary 5.4 is equivalent to RH. Remark 5.5 Riemann used the letter t for the complex variable that we have denoted by 1/2 + iu above (so as to avoid any confusion with the previous use of t throughout this paper). In fact Riemann’s choice of the letter t was somewhat unfortunate and has led to some confusion in the literature, as well as a minor error in Riemann’s paper, see the footnotes to 5.3 (e) and (47).

It is also now possible to anticipate Riemann’s strategy in [76] for locating zeros of ζ (equivalently of ξ) in the critical strip, which we will carry out in detail in the next section. Estimating the real number ξ(1/2 + it) for various real values of t in an interval 0 ≤ t ≤ T , at least closely enough to determine its sign, will guarantee the existence of at least N zeros (along this portion of the critical line) when the sign changes N times, by the intermediate value theorem. Further, the argument principle (a standard result in complex analysis, stated

Riemann and his zeta function

19

immediately below as Theorem 5.6), and some further estimation of a suitable contour integral, will allow us to count the number ! !" #!

(20) N (T ) := ! s ∈ C ! 0 ≤ Re(s) ≤ 1, 0 ≤ Im(s) ≤ T, ζ(s) = 0 !

where each zero is counted with its multiplicity. When N ≥ N (T ), it follows that there are exactly N zeros in this portion of the critical strip, all lying on the critical line and simple.

Theorem 5.6 (Cauchy’s principle of the argument) Let f be meromorphic on a simple closed curve C and in its interior. Further assume f has no zero or pole on C. Then 1 ∆C arg(f (z)) = Z − P, 2π

where Z equals the number of zeros (with multiplicities counted), and P the number of poles (with orders counted), of f in the interior of C, and ∆C arg(f (z)) equals the net change in the argument arg(f (z)) as z makes one counterclockwise circuit of C. !

In our application of 5.6 we will have f = ξ, thus P = 0. It is important to also note that & $% ′ % ′ f (z) f (z) 1 dz = dz.

(21) ∆C arg(f (z)) = Im i C f (z) C f (z) Furthermore, the first equality in (21) holds more generally for any path C, not necessarily closed.

6

Location of the first three zeros of ζ

Following the strategy outlined before 5.6, let us choose T = 28. We shall show that N ≥ 3 and N (28) = 3.

6.1

Demonstration that N ≥ 3

We already know (Proposition 5.3 (f)) that ξ(1/2 + it) is positive for t = 0, and now outline a method that will show ξ(1/2 + 18i) < 0, ξ(1/2 + 23i) > 0, ξ(1/2 + 27i) < 0. Thus there must be at least three zeros on the portion of the critical line s = 1/2 + it, 0 < t < 28.

Our technique to approximate the ξ values, at least accurately enough

20

E. A. Kudryavtseva, F. Saidak, and P. Zvengrowski

to determine the signs, is based on the Euler-Maclaurin summation method and simple computations that can be done by hand. It is clear that with modern computers similar computations can easily be carried out for much larger values of T .

The Euler-Maclaurin summation formula arises from the approximation of a discrete sum by a definite integral, and can be found in many references (cf. [25], §6.2, or [77], §3). The theory is more or less elementary and involves Bernoulli numbers as well as their generalization to Bernoulli polynomials. A simple example, familiar ! from elementary calculus, is the approximation of the harmonic sum nj=1 1/j by the "n definite integral 1 (1/x)dx = log n, see (8), or the last equality in (10) for another example. We will content ourselves in this section with a couple of further examples which illustrate the method and apply to our proposed computations. A nice feature of the method is that it enables one to estimate partial sums of potentially divergent series, with a strict control of the error term.

Example 6.2 The sharp Stirling series for log Γ(s). The formula, derived by Stirling [84] in 1730 (for s ∈ R, s > 0), states that, if s = reiθ , r > 0, −π < θ < π, then n

# B2k 1 1 log Γ(s) = (s − ) log s − s + log(2π) + + R2n (s), 2 2 (2k − 1)2k s2k−1 k=1

where one has the strong upper bound (due to Stieltjes [83], see also [25], §6.3) for the error term |R2n (s)| ≤

$

1 cos(θ/2)

& %2n+2 & & & B2n+2 & & & (2n + 1)(2n + 2)s2n+1 & .

It may not be obvious that this infinite series is actually divergent.

The divergence is due to the fact that the Bernoulli numbers actually grow very rapidly, for example B26 = 8 5536 103 ≈ 1 425 517.17, or more generally8 √ ' n (2n |B2n | ∼ 4 πn . πe 8

This asymptotic formula for the Bernoulli numbers is very accurate. For example, for n = 13, it gives B26 ≈ 1 420 956, compare 6.2. It does not seem to appear in the literature, but can be deduced from [59] or [20].

Riemann and his zeta function

21

Nevertheless, one can use the first few terms of the series to estimate log Γ(s) very accurately, i.e. with very small remainder. As a consequence, we also obtain the “classical” Stirling formula (22)

Γ(s) ∼

!

2π ss , s es

σ ≥ 0, |s| → ∞.

As a specific example (that will be used later), take s0 = 5/4 + 9i and n = 1. Then log Γ(s0 ) = (s0 − 1/2) log(s0 ) − s0 +

1/6 1 log(2π) + + R2 (s0 ), 2 1 · 2 · s0

where the inequality |s0 | > 9 and the Stieltjes remainder formula give |R2 (s0 )| < 4 · (1/30)/(3 · 4 · 93 ) ≈ 1.52416 × 10−5 . Evaluating the above then gives log Γ(5/4+9i) ≈ −11.5698+11.9265i, where the magnitude of the remainder shows that the accuracy is to about six significant digits.

Exponentiating this gives Γ(5/4 + 9i) ≈ 10−6 (7.57806 − 5.64057i), again with about six digit accuracy.

We remark that several calculations with complex numbers are involved in the above evaluation, and also the use of the well known formula log(r · eiθ ) = log r + i θ. This must be applied carefully since θ is only unique mod(2π); we take the branch of the logarithm function (for log( 54 + 9i)) where 0 ≤ θ < π/2. Mathematical software can differ on the choice of branch, so an answer differing by 2mπi, for some integer m, can easily occur. For example MAPLE gives log Γ(5/4 + 9i) ≈ −11.56982768 − 0.6398651938i, to ten digit accuracy. Of course, this difference of 4πi becomes irrelevant once the exponential is taken.

Before turning to our next example, we state an Euler-Maclaurin summation formula for Γ′ (s)/Γ(s), essentially the derivative of the first formula in 6.2, with n = 0, that will be of use in other parts of this paper: (23)

1 Γ′ (s) = log s − + R0′ (s), Γ(s) 2s

" " " B2 " iθ where |R0′ (s)| ≤ sec3 (θ/2) · " 2s 2 ", and s = re , r > 0, −π < θ < π, cf. [25], §6.3. The corresponding estimations for all n ≥ 0 are given in [77], §8.

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E. A. Kudryavtseva, F. Saidak, and P. Zvengrowski

Example 6.3 Estimating ζ(s). In somewhat similar fashion to the previous example, Euler-Maclaurin summation can be applied to the tail of the Dirichlet series to obtain an accurate estimation of ζ(s), even for s in the critical strip (where the Dirichlet series diverges). It gives us N −1 !

1 N 1−s B2 · s 1 + + + + ... ζ(s) = s s j s−1 2N 2N s+1 j=1

+

B2n · s(s + 1) . . . (s + 2n − 2) + R2n,N (s), (2n)!N s+2n−1

where the error (due to Backlund [6], see also [25], §6.4) is controlled by " " " s + 2n + 1 B2n+2 · s(s + 1) . . . (s + 2n) " " , σ > −2n. " · |R2n,N (s)| ≤ " " σ + 2n + 1 (2n + 2)!N s+2n+1

Example 6.4 Computation of ζ(1/2 + 18i). Specializing the previous example, with N = 6, n = 4, we have 1 1 1 1 1 1 + s + s + s + + s s−1 2 3 4 5 (s − 1)6 2 · 6s 1 1 1 6 ·s 30 · s(s + 1)(s + 2) 42 · s(s + 1) . . . (s + 4) − + 2! · 6s+1 4! · 6s+3 6! · 6s+5 1 30 · s(s + 1) . . . (s + 6) + R8,6 (s), 8! · 6s+7

ζ(s) = 1 + + − where

" "s+9 " |R8,6 (s)| ≤ " 1 · "2 +9

5 66

" · s(s + 1) . . . (s + 8) "" ", " 10! · 6s+9

σ > −8.

Evaluating this at s = s1 := 1/2 + 18i with modern computational tools is easily done, but it is worthwile at least thinking about how much work it would have been for Riemann, Backlund, or Gram to do this by hand.

In particular, evaluating the exponentials such as 1/6s1 +1 = 6−3/2−18i involves using the well known identity mx+iy = mx · (cos(y log m) + i sin(y log m)). The estimation of the remainder R8,6 (s1 ) is somewhat simpler, e.g. one can use |s1 (s1 + 1) . . . (s1 + 8)| < |s1 + 8|9 < 209 .

The outcome of the calculations is ζ(1/2 + 18i) ≈ 2.32922 − 0.18865i, with error less than 10−3 . Thus the value is accurate to about three significant digits.

23

Riemann and his zeta function

We now return to the original goal of calculating ξ(1/2 + 18i) = (−1/2 + 18i)ζ(1/2 + 18i)Γ(5/4 + 9i)π −1/4−8i . The difficult parts are already done in Examples 5.1 and 5.3, and the calculation π −1/4−8i ≈ −0.4798582 + 0.5778631i is routine (with seven significant digits accuracy). One then finds ξ(1/2+18i) ≈ −10−4 ×2.986 with about three significant digits accuracy. This proves ξ(1/2+18i) < 0. With calculations quite similar to those above, and again about three digits accuracy, one finds ξ(1/2+23i) ≈ 10−6 ×5.622 > 0, ξ(1/2+27i) ≈ −10−7 ×5.656 < 0.

The first goal of this section, showing that N ≥ 3, is thus accomplished.

6.5

Demonstration that N (28) = 3

To commence the second objective of this section let us apply the Principle of the Argument 5.6 to ξ(s) using the simple closed rectangular curve D = D(T ) with vertices −1, 2, 2 + T i, −1 + T i, traversed in that order. Let us also write C = C(T ) for the contour consisting of the portion of D from 2, to 2 + T i, to 1/2 + T i. Finally, define ! ! "" 1 it t + (24) ϑ(t) := Im log Γ − log π.

4 2 2 This function has the following estimation, due to Stirling’s formula (6.2) with n = 0: ! " ! " T 1 T π T (25) ϑ(T ) = log − − +O , T → ∞.

2 2π 2 8 T This approximation can be further refined up to O(1/T 2n+1 ) by using (6.2), for any n (see also [25], §6.5, or [53], III, §4).

Riemann had the asymptotic estimate N (T ) ∼ (T /2π) log(T /2π) − T /2π (without proof, however see also Theorem 7.5 and B.2), but the following exact formula of Backlund [5] is a substantial improvement: Proposition 6.6 (Backlund [5], 1914) that ζ(s) ̸= 0 on C, (26)

N (T ) =

1 1 ϑ(T ) + 1 + Im π π

(b) If also Re(ζ(s)) ̸= 0 on C then

(a) For any T > 0 such !#

C

" ζ ′ (s) ds . ζ(s)

1 N (T ) = ⌈ ϑ(T ) + 1⌋. π

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E. A. Kudryavtseva, F. Saidak, and P. Zvengrowski

Remark 6.7 Actually, in both (a) and (b), the non-vanishing hypothesis need only be checked on the horizontal portion of C, i.e. where t = T and 12 ≤ σ ≤ 2. For (a) this is obvious since the vertical portion lies outside the critical strip. For (b), it is easily verified, using the Dirichlet series (4) and Euler’s formula for ζ(2), cf. §4, that Re(ζ(2 + it)) > 2 − ζ(2) > 0 for any t ∈ R. The latter implies that the absolute variation of arg(ζ(2 + it)) is < π on any segment [t1 , t2 ].

We will next sketch a proof of Proposition 6.6, but first let us note that, for T = 28, 6.6 (b) gives N (T ) = ⌈3.078 . . . ⌋ = 3, which will then complete the objective of this section, namely showing that ζ has three simple zeros on the critical line up to 1/2 + 28i. The fact that Re(ζ(s)) ̸= 0 on the horizontal portion s = σ + 28i, 1/2 ≤ σ ≤ 2, of C(28) is somewhat delicate and can be proved similarly to 6.4, cf. [25], §6.6. We omit the details here and simply remark that it follows, in particular, that ξ(s) is nowhere zero on the closed curve D. Proof of Proposition 6.6: The Principle of the Argument 5.6, together with (21) and the fact that ξ has no poles, give "! ′ # ! ′ 1 ξ (s) ξ (s) 1 N (T ) = ds = Im ds .

2πi D ξ(s) 2π D ξ(s)

Now since, from Proposition 5.3 (f), ξ is positive real on the portion of D on the real axis, the argument of ξ(s) does not change here, so by (21) this contributes nothing to the above integral. By the symmetry of both ξ and D in the critical line σ = 1/2, it follows that "! ′ # 1 ξ (s) N (T ) = Im ds . π C ξ(s) Considering the definition (19) of ξ(s) and then taking its logarithmic derivative, we are able to write (27)

ξ ′ (s) 1 1 1 ζ ′ (s) 1 Γ′ ( 2s ) = + − log π + + . ξ(s) s s−1 2 ζ(s) 2 Γ( 2s )

Using the above definition (24) of ϑ(t), one can readily obtain (26). This completes (a).

As for (b), the assumption that Re(ζ(s)) ̸= 0 on C clearly implies this quantity is in fact positive on C. Since ζ(2) ∈ R+ , its argument starts at 0. Hence the absolute variation in its argument, over C, is strictly less than π/2. This and (21) show that the last integral in (26)

Riemann and his zeta function

25

has absolute value strictly less than 1/2. Since N (T ) must be an integer, we obtain (b). !

Remark 6.8 In this section we have shown that ζ(1/2 + it) has a zero for three values t = α1 , α2 , α3 with 0 < α1 < 18, 18 < α2 < 23, 23 < α3 < 27. With more calculations of the type we have made, it would be possible to narrow down the precise locations of the zeros. Riemann had estimated at least the first three zeros, although this does not appear in his paper. In 1903 Gram [42] located the first 15 zeros, for the first three one has α1 ≈ 14.134725, α2 ≈ 21.022040, α3 ≈ 25.010856, using methods similar to those we have used in this section. Riemann used the more efficient Riemann-Siegel formula, which was not available until Siegel’s publication [80] in 1932 of Riemann’s Nachlass (see also §7).

7

History of the zeta function since Riemann

The two decades following the publication of Riemann’s paper [76], in 1859, were largly uneventful. Weierstrass, who was eleven years older than Riemann, but whose rise to fame —from an obscure schoolteacher to a professor at Berlin— happened in a way very different from Riemann’s, began working and lecturing on complex numbers and the general theory of entire functions already during the 1860’s. But it wasn’t until 1876, when Weierstrass finally published his famous memoir [96], that mathematicians became aware of some of his revolutionary ideas and results. The first half of this section will discuss these ideas and how, together with the zeta function, they led to the estimation of the vertical location of the non-trivial zeros and to the proof in 1896 of the Prime Number Theorem (13), arguably the greatest achievement of 19th century mathematics (a short version of the original proof is given in Appendix A). In the second half we return to the discussion of Riemann’s paper, the RH, Riemann’s Nachlass (the 1932 study [80] by Siegel), and some of the subsequent history of the RH. We say that an entire function f is an entire function of finite order if (28)

log |f (s)| = O(|s|A ), for some A > 0.

The order of f (s) is the lower bound of all A, for which the inequality (28) holds.

Among Weierstrass’ many contributions were the following two important theorems:

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E. A. Kudryavtseva, F. Saidak, and P. Zvengrowski

Theorem 7.1 (Weierstrass [96], 1876) Let {cn } be quence of complex numbers, such that 0 < |c1 | ≤ |c2 | ≤ assume that its only limit point is ∞. Then there exists tion f (s) with zeros (with prescribed multiplicities) at complex numbers.

an infinite se|c3 | ≤ . . . , and an entire funcprecisely these !

Remark 7.2 Note that in both this theorem and Theorem 7.3 zeros of arbitrary multiplicities are accounted for by taking e.g. cj = cj+1 = . . . = cj+r .

Theorem 7.3 (Weierstrass [96], 1876) Every entire function g(s) of order ≤ 1, which has no zeros in C, can be written as g(s) = ea+bs , where a and b are constants, while every entire function f (s) of order ≤ 1, which has N ≤ ∞ zeros at c1 , c2 , c3 , . . . ̸= 0, can be written in the form (29)

f (s) = ea+bs

$ % N "# ! s 1− es/cn cn

n=1

where a and b are constants, and the product converges absolutely (if N = ∞) for all s ∈ C. !

Remark 7.4 Let γ be Euler’s constant. Weierstrass proved the product formula (30)

∞ &' ! s ( −s/n ) 1 γs = se e 1+ . Γ(s) n n=1

This, along with Riemann’s paper, set the stage for the great work of Hadamard and de la Val´ee-Poussin in the 1890’s. Recall the definition (19) of ξ(s) and note that, applying (6.3) with n = 0, N = 1, and Stirling’s formula (6.2) with n = 0, one can find constants C1 , C2 , C3 such that, for all s ∈ C \ {1} with σ ≥ 12 , * * ' s (* * * * * * |s(s − 1)ζ(s)| < C1 |s|4 , *Γ * < eC3 |s| log |s| , *π −s/2 * < eC2 |s| .

2 From this, using the properties 5.3 (a,b) of ξ, we have (31)

|ξ(s)| < eC|s| log |s| ,

s ∈ C,

for a constant C > 0. Stirling’s formula also tells us that the upper bound |ξ(s)| < eC|s| fails as s = σ → ∞. Therefore ξ(s) is of order 1,

Riemann and his zeta function

27

has infinitely many zeros, and can be written in Weierstrass’ form as follows: $ % ∞ "# ! s A+Bs s/ρn (32) ξ(s) = e 1− e , ρn n=1

where A and B are constants, and ρn = βn + iγn are all the zeros of ξ, arranged so that |γ1 | ≤ |γ2 | ≤ |γ3 | ≤ . . ., and the ρj may repeat, as in Remark 7.2.

Entire functions of arbitrary order have product representations analogous to (29), as Weierstrass proved in [96]. His general theorem was made more explicit and applicable by Hadamard [43], in 1893. He used it, together with (31), to prove in [43] that ζ(s) and ξ(s) have infinitely many zeros in the critical strip, and that there exist constants a, A > 0 such that n , equivalently N (T ) ≤ AT log T, (33) γn ≥ a log n for n ≥ 2, T ≥ 2. An important consequence is (34)

∞ &

n=1

1 < ∞ |ρn |c

for all c > 1.

Using (34) Hadamard [43] proved the following product formula similar to (32), see also [25]: $ ∞ # ! s (35) ξ(s) = ξ(0) .

1− ρn n=1

In 1895, von Mangoldt [68] used Hadamard’s results (34), (35), to obtain ξ ′ (s) & 1 = , (36) ξ(s) s−ρ ρ where validity of the termwise differentiation of the product in (35) follows from the uniform convergence of its logarithmic derivative in any disk |s| ≤ R, due to (34). He also estimated the vertical density of the roots ρn of ζ, for large T > 0: & 1 < 2 log T, (37) N (T + 1) − N (T ) ≤ T ≤γn ≤T +1

28

E. A. Kudryavtseva, F. Saidak, and P. Zvengrowski

by noticing that (34), (36) imply # # !" !"

2+i(T +1) ′ 2+i(T +1) $ ξ (s) ds ds = Im Im ξ(s) s − ρn 2+iT 2+iT ρn # !" % & 2+i(T +1) $ $ 1 ds ≥ > Im arctan , s − ρ 2 n 2+iT T ≤γn ≤T +1

' 2+i(T +1)

T ≤γn ≤T +1

ξ ′ (s)

i and by showing 2+iT ξ(s) ds = 2 log T +O(1) via (27) and Stirling’s formula (6.2) with n = 0, together with (21) and the boundedness of the total variation of arg(ζ(2 + it)) on [T, T + 1] (see 6.7 or [25], §3.4). With the help of (34), (36), and (37), von Mangoldt [68] proved the explicit formula 4.6, cf. [25], §3.2-3.5. See also Remark B.2.

In 1896, based on Hadamard’s results (33)-(35), Hadamard and de = 1 and the PNT, see la Vall´ee-Poussin proved independently lim θ(x) x→∞ x Appendix A. An important step in both proofs was to show that no zero of ζ(s) has real part 1. In 1899 de la Vall´ee-Poussin made a further improvement (see A.2) which finally justified Chebyshev’s prediction of the correct constant in the Legendre prime number formula (cf. §4).

Six years later von Mangoldt proved Riemann’s estimate for the vertical distribution of the zeros of ζ, see 6.5, strengthening (33), (37):

Theorem 7.5 (von Mangoldt [69], 1905) For T ≥ 2, % & T T T log + O(log T ).

N (T ) = − 2π 2π 2π A proof is given in Appendix B. Returning to our historical sketch, let us first make some concluding comments about Riemann’s 1859 paper. Needless to say, this paper is written in an extremely terse and difficult style, with huge intuitive leaps and many proofs omitted. This led to (in retrospect quite unfair) criticism by Landau and Hardy in the early 1900’s, who commented that Riemann had only made conjectures and had proved almost nothing.

The situation was greatly clarified in 1932 when Siegel [80] published his paper, representing about two years of scholarly work studying Riemann’s left over mathematical notes at the University of G¨ottingen, the so-called Riemann’s Nachlass. From this study it became clear that Riemann had done an immense amount of work related to [76] that never appeared in his paper. One conclusion is that many formulae

Riemann and his zeta function

29

that lacked sufficient proof in [76] were in fact proved in these notes. A second is that the notes contained further discoveries of Riemann that were never even written up in [76]. One such is what is now called the Riemann-Siegel formula, which Riemann had written down and Siegel (with great difficulty) was able to prove, cf. [25] or [53]. This formula (which we omit) arises from a Hankel integral type expression for ξ(s), and gives a refined method to calculate ξ(1/2 + it), in comparision to the crude methods of §6.

In his 1859 paper Riemann only mentions RH briefly. To quote him once more, “Hiervon w¨are allerdings ein strenger Beweis zu w¨ unschen; ich habe indes die Aufsuchung desselben nach einigen fl¨ uchtigen vergeblichen Versuchen vorl¨aufig bei Seite gelassen, da er f¨ ur den n¨achsten Zweck meiner Untersuchung entbehrlich schien” (One would of course like to have a rigorous proof of this, but I have put aside the search for such a proof after some fleeting vain attempts because it is not necessary for the immediate objective of my investigation), cf. Appendix of [76].

However, towards the end of the paper there are some speculations that a more detailed mathematical analysis (see C.2, or [25], §1.17, 5.5) shows are indirectly related to RH and the improvement of the remainder term in PNT. Furthermore, it is not clear from Riemann’s paper that he had any solid evidence for RH, but it is now known (Riemann’s Nachlass) that he had calculated at least the first three non-trivial zeros and found them to lie on the critical line, much as was done in §5 above. As Iviˇc says [50], Ch. I, Notes, “it is apparent that he knew much more about ζ(s) than he cared to publish.” It is also important to note that Riemann only lived until 1866, and that his health was very bad during his final years.

Starting from about 1890, the evidence for RH has rapidly increased.

For example, we will see in Appendix A that the celebrated PNT proved in 1896 is equivalent to reducing the critical strip from 0 ≤ σ ≤ 1 to 0 < σ < 1, i.e. it can be thought of as a very small first step towards RH. Hilbert included RH in his list of 23 problems, at his address to the International Congress in 1900. It is interesting that at the time of his address, Hilbert did not consider RH to be one of the most important problems of his list. However, some years later when asked, if he could sleep 500 years what his first question would be upon awakening, Hilbert replied “has the RH been solved?” Generalizations of RH have taken on equal significance. Starting with the Dirichlet L-functions the

30

E. A. Kudryavtseva, F. Saidak, and P. Zvengrowski

concept has been further generalized to Artin L-functions and to global L-functions, which have many similarities to ζ such as an Euler product formula and a functional equation, and are of basic importance in diverse areas of modern mathematics.

Since 1900 the progress towards solving the RH has been enormous, nevertheless it is still unsolved and appears on the Clay Institute list (in 2000) of seven questions for the new millenium. Some highlights of these developments are now outlined, with no attempt at completeness.

We have already seen in this section that there are infinitely many zeros of ζ in the critical strip. Hardy [45] improved this in 1914, showing that in fact there are infinitely many zeros on the critical line. His collaboration with Littlewood and Ramanujan produced other important advances [47]. Bohr and Landau [8] proved in 1914 that the proportion of the zeros lying within ε from the critical line equals 1, for any ε > 0.

Later in the 20th century Selberg [78], Bombieri [9], and Deligne [22] made very significant contributions. Selberg [78], for example, showed in 1942 that some positive proportion of the zeros lie on the critical line, and this was later improved by Levinson [62] to at least 1/3, and still later by Conrey [16] to at least 2/5. Deligne [22] in 1974 proved the related Weil Conjecture (an analogue of RH for zeta functions of general algebraic varieties over finite fields).

Similarly, starting from about 1890, the realization of the significance of RH has rapidly increased. One equivalent formulation of RH in number theory, the estimation of the error in the approximation of π(x)

by Li(x), has already been mentioned in 4.5, see also A.2, A.3. There are many further significant number theoretical implications of RH. For example, Bertrand’s postulate that there exists a prime in [n + 1, 2n − 2], n > 3 (first proved by Chebyshev [13], cf. (14)) was successively improved over ten times (cf. [50], Ch. 12, Notes), e.g. by Montgomery [72] in 1969 to the existence of a prime in [n, n + n3/5+ε ], and by Lou and Yau [66] in 1992 to the existence of a prime in [n, n + n6/11+ε ], for all ε > 0 and n ≥ n0 (ε). This in turn can be further strengthened using RH to the existence of a prime in [n, n + cn1/2 log n] (Cram´er [18], 1920, see also [19], or [50], §12.6), and using Cram´er’s conjecture (i.e. lim pn+12−pn = 1) even to [n, n + c log2 n], cf. [19]. The latter cannot

n→∞ log pn

be strengthened much further, since, due to Westzynthius [98] in 1931, −pn = ∞. A further example involves, for a given prime p, lim pn+1 log pn

n→∞

estimating the least quadratic non-residue (mod p), written n(p). Here 1 √ e

Vinogradov’s classical 1918 result [90], [91] that n(p) < p 2

log2 p for

31

Riemann and his zeta function

all sufficiently large p (see also [75]), improved in 1957 by Burgess [11] 1 to n(p) = O(pα ) for any fixed α > 4√ , can be strengthened using the e extended RH (i.e. the RH for the Dirichlet L-functions, cf. (9)). In this way, Ankeny [2] showed in 1952 that n(p) = O(log2 p), and Bach [4] improved this in 1990 to n(p) ≤ 2 log2 p. This cannot be strengthened much further, since Graham and Ringrose [41] showed in 1990 that n(p) lim log p log log log p > 0 unconditionally, while Montgomery [73] showed

p→∞

n(p) p→∞ log p log log p

in 1971 using the extended RH that lim

> 0.

Intriguing (and important) equivalent conjectures abound, suggesting alternative approaches to RH. For an excellent survey of these as well as of recent progress on the problem cf. Conrey [17] and Bombieri [10] (his descriptive paper for the Millenium Problems). In a recent paper [77] by two of the authors, as well as in some earlier work of Spira [82], a slightly different “horizontal” approach to the question is taken. The functional equation shows that for any non-trivial zero Q := 1/2 + ∆ + it in the critical strip (0 ≤ ∆ ≤ 1/2), one also has a zero at P := 1/2 − ∆ + it (as well as at P , Q). In [77] very accurate upper and lower bounds for the ratio |ζ(P )/ζ(Q)| are obtained.

In particular, it is shown that |ζ(P )| ≥ |ζ(Q)|. Clearly the inequality |ζ(P )| > |ζ(Q)|, 0 < ∆ ≤ 1/2, would imply RH since both could not then be simultaneously 0.

From the point of view of gathering numerical evidence, the early work of Gram (cf. 6.8) and Backlund [5] was carried further by Hutchinson [48] in 1925 to show that the first 138 zeros (in the upper half plane) lie on the critical line. Once the Riemann-Siegel formula became available, this was soon improved to the first 1041 zeros by Titchmarsh and Comrie [86], [87]. Thanks to modern computational power, it is now known that at least the first 1010 zeros lie on the critical line (a number that is steadily increasing).

A

Appendix: Prime Number Theorem

In this appendix we give a proof, incorporating ideas from the original proofs, of the celebrated Prime Number Theorem (13), conjectured by Gauss in 1793, and proved in 1896 by both Hadamard [44] and (independently) de la Vall´ee-Poussin [88] (and also [89]). Alternative proofs using elementary methods appeared some 50 years later, cf. [26], [79] (see also [65], [74], and [50], Ch. 12), where “elementary” means, in

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E. A. Kudryavtseva, F. Saidak, and P. Zvengrowski

particular, without use of complex analysis, but not necessarily simpler.

A.1

Prime Number Theorem: π(x) ∼ Li(x).

While it would be beyond the scope of this paper to furnish complete details of this proof, we shall fully describe the key step in the proof (reducing the critical strip from 0 ≤ σ ≤ 1 to 0 < σ < 1), as well as clearly indicate and discuss the other ingredients of this proof (for full details excellent sources are [25], [49], [52], [53], [57], [58], and others).

First of all we will use the Riemann-von Mangoldt explicit formula 4.6. Secondly, we will prove the PNT in the equivalent form stated in §4, namely ψ(x) ∼ x. The proof that these are equivalent is straightforward and goes back to Chebyshev’s ideas, cf. [25],! §4.4. A third fact we shall use is that, for the non-trivial zeros ρ of ζ, ρ ρ12 is absolutely convergent; this is an immediate consequence of Hadamard’s formula (34).

Let us start the sketch by rewriting the explicit formula 4.6 in the form " xρ " x−2n ˜ + − log(2π) for x > 1.

(38) ψ(x) =x− ρ 2n ρ n In 1896, de la Vall´ee-Poussin [88] showed that the term-by-term integration of both sides of (38) is a valid operation, and in fact, for x > 1, it leads to the formula # x (39) ψ1 (x) := ψ(t)dt 0

=

" xρ+1 " x−2n+1 x2 − − − x log(2π) + const.

2 ρ(ρ + 1) 2n(2n − 1) ρ n

It is clear that, as x → ∞, the last three terms on the right hand side of (39) are all o(x2 ).

Our next step is to show ζ(1 + it) ̸= 0, i.e. there are no zeros of ζ(s) on the line σ = 1 (Hadamard showed this in [43], however we will follow the method of de la Vall´ee-Poussin in [89]). To see this, let σ > 1 and integrate (11) termwise (the constant of integration is clearly seen to equal 0 by taking s = σ real and letting σ → ∞), giving log ζ(s) =

∞ " Λ(n) , σ > 1. ns log n

n=2

Riemann and his zeta function

33

Taking the real parts, Re(log ζ(s)) =

∞ !

n=2

Λ(n) · cos(t log n). nσ · log n

Using the trigonometric identity 3 + 4 cos t + cos 2t = 2(1 + cos t)2 ≥ 0, it follows that 3Re(log ζ(σ)) + 4Re(log ζ(σ + it)) + Re(log ζ(σ + 2it)) ≥ 0, and exponentiating this gives (40)

|ζ(σ)|3 |ζ(σ + it)|4 |ζ(σ + 2it)| ≥ 1,

for σ > 1.

As we saw in (10), ζ has the single pole at s = 1, and it is simple with residue 1. This is equivalent to lims→1 (s − 1)ζ(s) = 1. Now suppose that ζ has a zero of order m ≥ 1 at s0 = 1 + it0 , then similarly this is equivalent to lims→s0 (s − s0 )−m ζ(s) = c for some c ∈ C \ {0}. Taking s = σ + it0 , σ > 1, we can rewrite (40) as |ζ(σ)|3 · |σ − 1|3 · =

|ζ(σ + it0 )|4 |σ − 1|3 · |ζ(σ + 2it )| ≥ 0 |s − s0 |4m |s − s0 |4m |σ − 1|3 1 = . |σ − 1|4m |σ − 1|4m−1

Letting σ → 1+ in this inequality, and taking account of the two limits above, shows that there is a pole of order ≥ 4m − 3 ≥ 1 at s = 1 + 2it0 .

Since this is impossible, the claim ζ(1+it) ̸= 0, t ∈ R\{0} is established.

Therefore, if ρ is a non-trivial zero of "ζ(s),1 then Re(ρ) < 1, and we have |xρ−1 | < 1, while the infinite sum ρ ρ(ρ+1) converges absolutely, " ρ−1 cf. (34). This implies that ρ x /ρ(ρ + 1) converges uniformly in x, whence lim

x→∞

! ρ

! ! xρ−1 xρ−1 = = lim 0 = 0, x→∞ ρ(ρ + 1) ρ(ρ + 1) ρ ρ

and hence the second term of the right hand side of (39) is also bounded 2 by o(x2 ). Therefore, we can conclude ψ1 (x) ∼ x2 . In general, if two functions are asymptotic, one cannot conclude their derivatives are asymptotic. However, in the situation at hand, one also knows the derivative ψ = ψ1′ is a monotone non-decreasing function, and it is then straightforward (cf. [25], §4.3) to conclude that ψ(x) ∼ x. !

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E. A. Kudryavtseva, F. Saidak, and P. Zvengrowski

Remark A.2 In 1899, de la Vall´ee-Poussin [89] made the above argument more explicit, and he was the first to obtain a zero-free region of ζ(s) having positive measure: c , ζ(s) ̸= 0 for Re(s) ≥ 1 − log(|t| + 2) where c > 0 is a constant. From this, using (34), as well as (39) and an inequality similar to (40), de la Vall´ee-Poussin obtained the following estimation of the error in the PNT: ! " ! " √ √ ψ(x) = x + O xe−A log x and π(x) = Li(x) + O xe−B log x ,

where A, B > 0 are constants, see also [25], §5.3. Due to the asymptotic expansion (15) of Li(x), this proves that Li(x) indeed provides a much better approximation to π(x) than do logx x , log xx−1 , and the Legendre formula, see §4. Littlewood [64] in 1924 improved de la Vall´eePoussin’s estimation for the zero-free region the help of Weyl’s #bwith −1−it method of evaluating “exponential sums” in the Eulern=a n Maclaurin formula for ζ(1 + it). This was substantially improved by Chudakov [15] in 1936, based on Vinogradov’s powerful methods [92] in the 1930s for the estimation of exponential sums, and later by Vinogradov (announced [93] in 1942, published [94] in 1958) and Korobov (1958), cf. [50], Ch. 6, [25], §9.8, or [53], IV. Using the methods of Vinogradov and Korobov, Richert slightly improved their estimates (unpublished) to obtain the zero-free region σ ≥ 1 − c log−2/3 t(log log t)−1/3 , t ≥ t0 , cf. [95]. These improvements led to corresponding improvements of de la Vall´ee-Poussin’s estimate of the error term in PNT (cf. [95]). The best known estimates (obtained by Walfisz ! " [95] from 3/5 −1/5 −C log x(log log x) Richert’s result) are ψ(x) = x+O xe , and π(x) = " !

3/5 −1/5 , for constants C, C1 > 0.

Li(x) + O xe−C1 log x(log log x)

Remark A.3 As Landau [56] (see also [57], §93-94) showed in 1909, additional information about the horizontal location of the non-trivial zeros can provide an improvement of de la Vall´ee-Poussin’s estimate of the error in PNT (see A.2), as follows: if for all non-trivial zeros ρn of ζ(s) we have Re(ρn ) ≤ ∆, for some fixed 21 ≤ ∆ < 1, then π(x) = Li(x) + O(x∆ log x).

In particular, RH (corresponding to ∆ = 1/2) implies that the error in PNT is O(x1/2 log x) [54]. Actually, the converse is also true, cf. [57],

Riemann and his zeta function

35

§93, 201 (see also 4.5 and [25], §5.5). In other words, RH could be obtained by somehow proving the PNT with a very sharp error term.

B

Appendix: Von Mangoldt’s theorem

In this appendix, a proof of von Mangoldt’s Theorem 7.5 is given. This theorem describes the vertical distribution of the non-trivial zeros of ζ(s). Again, the proof uses the results of Weierstrass (cf. §7). It is based on Backlund’s ideas (cf. Proposition 6.6), as well as [50], see also [25], [49], [53], [57].

Theorem B.1 (von Mangoldt [69], 1905) For T ≥ 2, ! " T T T log + O(log T ).

(41) N (T ) = − 2π 2π 2π Remark B.2 Earlier, in 1895, von Mangoldt [68] proved an analogue of (41) with a slightly weaker error term O(log2 T ). In fact, this was the first time that the correct main term for N (T ) was obtained, and it turned out to be exactly what Riemann claimed in [76]. Riemann also predicted the error correctly in [76], however his description was unclear (he was referring to relative error) and this led to subsequent misinterpretations in the literature (see also [25], §1.9).

Proof: Since the set of zeros of ζ is discrete, see 3.2, we may assume no zero of imaginary part T . Due to (25) and (26), it suffices to !#ζ has " ζ ′ (s) show Im ds = O(log T ). The latter is equivalent to C ζ(s) % $# 2 ζ ′ (σ + iT ) dσ = O(log T ), I := Im 1/2 ζ(σ + iT ) due to (21) and the boundedness of the total variation of arg(ζ(2 + it)) on [0, T ], see Remark 6.7.

From (27) and (36), we have (42)

& 1 1 1 1 1 Γ′ ( 2s ) ζ ′ (s) =− − + log π + − ζ(s) s s−1 2 s − ρ 2 Γ( 2s ) ρ =

& ρ

1 + O(log |t|), s−ρ

σ ≥ −1, |t| ≥ 2.

36

E. A. Kudryavtseva, F. Saidak, and P. Zvengrowski

where the latter estimate follows from (23). Let us denote !

S1 (s) :=

|γn −t|≤1

1 , s − ρn

!

S2 (s) :=

|γn −t|>1

1 , s − ρn

and show that (43)

S2 (s) = O(log |t|),

−1 ≤ σ ≤ 2,

|t| ≥ 2.

Using the logarithmic derivative of the Euler product given in (4), as ′ (2+it) in (11), it is easily verified that ζζ(2+it) = O(1). Furthermore, due to (37), |S1 (2 + it)| ≤

!

|γn −t|≤1

1 ≤ |2 + it − ρn |

!

|γn −t|≤1

1 = O(log |t|).

Together with (42), these estimations imply S2 (2 + it) = O(log |t|) for |t| ≥ 2. This in turn gives O(log |t|) = Re(S2 (2 + it)) =

!

|γn −t|>1

Re

1 2 + it − ρn

!

2 − βn 1 > 2 2 (2 − βn ) + (t − γn ) 4 + (t − γn )2 |γn −t|>1 |γn −t|>1 "′ " for |t| ≥ 2. Therefore, with the notation := |γn −t|>1 , # ! ′ ## 1 # 1 # # − |S2 (s) − S2 (2 + it)| ≤ # s − ρn 2 + it − ρn # =

=

!′

!

!′ !′ 2−σ 3 15 < < |s − ρn |·|2 + it − ρn | (t − γn )2 4 + (t − γn )2

has order O(log |t|) as −1 ≤ σ ≤ 2, |t| ≥ 2. Together with S2 (2 + it) = O(log |t|) from above, this proves (43). From (42) and (43) we have (44)

ζ ′ (s) = ζ(s)

!

|γn −t|≤1

1 + O(log |t|), s − ρn

−1 ≤ σ ≤ 2, |t| ≥ 2.

Thus, in order to prove I = O(log T ), it suffices to show & $% 2 ! dσ = O(log T ).

Im (45) 1/2 σ + iT − ρn |γn −T |≤1

37

Riemann and his zeta function

By (21), each summand of the left-hand expression equals the net change of arg(s−ρn ) on [1/2+iT, 2+iT ], thus its absolute value is < π. By (37), the number of summands is < 4 log T , for large T . Thus, the modulus of the left hand side of (45) is < 4π log T , for large T . This, in turn, implies I = O(log T ). !

C

Appendix: Riemann-von Mangoldt formula

In the first part C.1 of this appendix we outline some of the details that were omitted in the sketch of the reasoning leading to the Riemann– von Mangoldt explicit formula 4.6. The method is based on [57], §87, see also [50], §12.2. For an alternative method, which does not use contour integration, see [25], §3.2-3.5. In the second part C.2 we return to Riemann’s paper [76] and suggest a few further ideas related to the explicit formula and RH. C.1 The main step in the derivation of 4.6 that needs justification is showing that the integrals on the top, bottom, and left edges of the closed rectangular contour C with vertices a ± iT , −(2n + 1) ± iT , a > 1 fixed (for convenience, we also assume a ≤ 2), all approach 0 as T → ∞ suitably (see (a) below) and n ∈ N, n ≥ T log T . To do this, we start with (37) and (44). Using these one can show (a) T can be chosen arbitrarily large with |γ − T | > zero ρ = β + iγ of ζ,

1 4 log T ,

for any

(b) ζ ′ (s)/ζ(s) = O(log2 T ), s = σ + iT , −1 ≤ σ ≤ 2 and T chosen as in (a).

We also need the estimation (c) ζ ′ (s)/ζ(s) = O(log |s|), σ ≤ −1, |s + 2j| ≥

1 2

for j ∈ N.

To prove (c) one starts with the logarithmic derivative of the functional equation (18), namely π πs Γ′ (1 − s) ζ ′ (1 − s) ζ ′ (s) = log(2π) + cot − − , ζ(s) 2 2 Γ(1 − s) ζ(1 − s) taking σ ≤ −1 so that 1 − σ ≥ 2. It is easy to show cot πs 2 is bounded 1 for |s + 2j| ≥ 2 , j ∈ Z, i.e. on the region given in (c). Similarly, the boundedness of ζ ′ (1 − s)/ζ(1 − s) for 1 − σ ≥ 2 is easily shown, e.g.

38

E. A. Kudryavtseva, F. Saidak, and P. Zvengrowski

using (11). Finally, Γ′ (1 − s)/Γ(1 − s) = O(log |1 − s|) = O(log |s|) for 1 − σ ≥ 2, see (23), completing the derivation of (c).

Using (b) it is not hard to show that integral over the portion of the top and bottom edges of C, where −1 ≤ σ ≤ a, tends to 0 as T → ∞.

And, using (c), one establishes the same for the left edge of C and the remaining portions of the horizontal edges. This completes the proof of the explicit formula 4.6.

C.2 Let us now return to Riemann’s paper [76] and make a few concluding remarks. As mentioned in §4, and as the title of [76] suggests, Riemann’s main objective was to obtain an explicit formula for π(x), which was only proved later [68], see also [57], §88. He used the closely related function (46)

Π(x) :=

!1 ! 1 = π(x1/n ). m n m n

p ≤x

" # x Here the number of non-vanishing terms equals N := log log 2 (because π(u) = 0 for u < 2), and this together with (14) easily imply Π(x) − x1/2 π(x) = 12 π(x1/2 ) + O(x1/3 ) = O( log x ), cf. [57], §5. By the method of Fourier inversion he obtained the explicit formula in the following form9 : (47)

˜ Π(x) = li(x) −

!

li(x ) + ρ

$

x

ρ

∞

(x2

dx − log 2, − 1)x log x

% ˜ is defined as in §2, and li(x) := x dt is defined for x > 1, where Π 0 log t &% %x ' 1−ε as the Cauchy principal value lim 0 + 1+ε , differing from Li(x) ε↓0

˜ is by the constant li(2) ≈ 1.04516, cf. [25], §1.14-1.16. In his paper Π denoted f , and π ˜ (again defined as in §2) denoted F .

9

To quote Riemann: “By setting these values in the expression for f (x), one finds

f (x) = Li(x) −

[Li(x(1/2)+αi ) + Li(x(1/2)−αi )] + α

∞ x

dx 1 + log ξ(0), x2 − 1 x log x

where the sum α is over all positive roots (or all roots with positive real parts) of the equation ξ(α) = 0, ordered according to their size”, cf. Appendix of [25].

Here α = (1/2 − ρ)i, and, in the notation of the present paper, Riemann’s Li(x) means li(x), while Riemann’s ξ(u) means ξ(1/2 + ui). The term log ξ(0) is in fact erroneous, see 5.5 and the footnote to 5.3 (e).

39

Riemann and his zeta function

Riemann inverted (46) by means of the M¨obius inversion formula to obtain (48)

˜ π ˜ (x) = Π(x) +

∞ ! µ(n)

n

n=2

˜ 1/n ) = Π(x) ˜ Π(x +

N ! µ(n)

n=2

n

˜ 1/n ), Π(x

where µ(n) is 0 when n is divisible by a prime square, and otherwise (−1)r where r is the number of distinct prime divisors of n. Substituting (47) into (48) gives an explicit formula for π ˜ (x), cf. [25], §1.17 and §5.4: (49)

π ˜ (x) =

N ! µ(n)

n=1

n

li(x

1/n

)−

N ! ! µ(n)

n

n=1 ρ

li(xρ/n ) + lesser terms,

" thereby achieving the main goal of [76]. Notice that ρ in (49) can be restricted to a (sufficiently large, depending on x) finite number of terms, if the right hand side of (49) is replaced by the nearest half integer to it (since the left hand side is always a half integer).

Having done this, Riemann speculates (on the final page of his paper) that the main term in (49) is given by the first (finite) sum: (50)

π(x) ≈ li(x) +

N ! µ(n)

n=2

n

li(x1/n )

1 1 1 1 = li(x) − li(x1/2 ) − li(x1/3 ) − li(x1/5 ) + li(x1/6 ) + . . . , 2 3 5 6 and that the estimate π(x) ≈ li(x) has negative error of order O(x1/2 ).

To quote him once more: “Thus the known approximation F (x) = Li(x) is correct only to an order of magnitude of x1/2 and gives a value which is somewhat too large...”. His prediction that li(x) should overestimate π(x) is indeed the case for all x within present computational power. However, Littlewood [63] showed that the difference π(x)−li(x) π(x) − li(x) changes sign infinitely often, indeed lim li(x1/2 ≥ 16 ) log log log x π(x)−li(x) and lim li(x1/2 ≤ − 61 as x → ∞, in particular li(x) will un) log log log x derestimate π(x) for some sequence xn → ∞. Skewes [81] showed that x1 < 104 (3), where 101 (x) = 10x , 102 (x) = 10101 (x) , and so on. This bound has been improved to x1 < 1.65 × 101165 by Lehman [61], and afterwards further improved by others. It is interesting that, for x ≤ 107 , the estimate (50) is substantially more accurate than π(x) ≈ li(x), as

40

E. A. Kudryavtseva, F. Saidak, and P. Zvengrowski

Table III in [25], §1.17, shows. However, due to the Littlewood result, for x large, the terms” of the (essentially finite, as explained ! “periodic ρ/n above) sum ) in (49) should be also taken into account in ρ li(x estimating π(x), cf. [25], §5.4. Here the number of “significant” periodic terms is large with x, thus eventually (as x → ∞) every periodic term li(xρ/n ) in (49) becomes as significant as the nonperiodic term − 12 li(x1/2 ).

It is perhaps even more interesting to consider Riemann’s error estimate of O(x1/2 ) in the above statement, for the approximation π(x) ≈ li(x). This is done very carefully in [25], §1.17 and §5.5. It is shown that ρn xβn each individual periodic term li(xρn ) in (47) equals ρnxlog x + O( log 2 ) x βn

x and would not be less in magnitude than O( log x ), as x → ∞, where as usual ρn = βn + iγn . If RH were false then βn > 12 for some n, thus the contribution of this term to the error in PNT would grow more rapidly than O(x(1/2)+ε ), for some ε > 0. Moreover, this would also apply to the total error in PNT, see 4.5, A.3, or [25], §5.5. Thus, it is very probable that at this stage (the final page) of his paper, Riemann assumed the validity of RH. To quote Bombieri [10], “it is quite likely that he saw how his hypothesis was central to the question of how good an approximation to π(x) one may get from his formula.”

Acknowledgement The authors are grateful to R. K. Guy, J. P. Jones, and N.-P. Skoruppa for useful discussions.

E. A. Kudryavtseva Department of Mathematics and Mechanics, Moscow State University, Moscow 119992, Russia, eakudr@mech.math.msu.su

F. Saidak Department of Mathematics, Wake Forest University, Winston-Salem, NC 27109, USA, saidakf@wfu.edu

P. Zvengrowski Department of Mathematics and Statistics, University of Calgary, Calgary, AB, T2N 1N4, Canada, zvengrow@ucalgary.ca

Riemann and his zeta function

41

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Soc. (3) 5 (1955), 48–70.

[82] Spira R., An inequality for the Riemann zeta-function, Duke Math.

J. 32 (1965), 247–250.

[83] Stieltjes T. J., Sur le developpement de log Γ(a), J. Math. Pures Appl. 5 (1889), 425–444.

[84] Stirling J., Methodus differentialis: sive tractatus de summatione et interpolatione serierum infinitarum, Gul. Bowyer, London, 1730.

[85] Titchmarsh E. C., The Theory of the Riemann Zeta-Function, 2nd ed. (revised by D. R. Heath-Brown), Oxford Univ. Press, Oxford, 1986.

[86] Titchmarsh E. C., The zeros of the Riemann zeta-function, Proc.

Roy. Soc., Ser. A 151 (1935), 234–255.

[87] Titchmarsh E. C., The zeros of the Riemann zeta-function, Proc.

Roy. Soc., Ser. A 157 (1936), 261–263.

[88] de la Vall´ee-Poussin C.-J., Recherches analytiques sur la th´eorie des nombres premiers (premi´ere partie), Ann. Soc. Sci. Bruxelles (1) 20 (1896), 183–256, 281–397.

[89] de la Vall´ee-Poussin C.-J., Sur la fonction ζ(s) de Riemann et le nombre des nombres premiers inf´erieurs ` a une limite donn´ee, M´em.

Courronn´es et Autres Acad. Roy. Sci., des Lettres Beaux-Arts Belg.

59 No. 1 (1899–1900).

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[93] Vinogradov I. M., On the estimation of a trigonometrical sum (Russian), Dokl. Akad. Nauk SSSR 34 No. 7 (1942), 199–200.

[94] Vinogradov I. M., A new estimate for the function ζ(1 + it) (Russian), Izv. Akad. Nauk SSSR, Ser. Mat. 22 No. 1 (1958), 161–164.

[95] Walfisz, A., Weylsche Exponentialsummen in der Neueren Zahlentheorie, VEB Deutscher Verlag der Wiss., Berlin, 1963.

[96] Weierstrass K., Zur Theorie der eindeutigen analytischen Funktionen, Abh. Kgl. Preuss. Akad. Wiss. Berlin (1876), 11–60. (Also in: “Math. Werke”, vol. 2, Mayer & M¨ uller, Berlin (1895), 77–124.)

[97] Weil A., Prehistory of the zeta-function. In: Number theory, trace formulas and discrete groups (Oslo, 1987), Acad. Press, Boston, 1989, 1–9. ¨ [98] Westzynthius E., Uber die Verteilung der Zahlen, die zu den n ersten Primzahlen teilerfremd sind, Comment. Phys.-Math. Soc. Sci. Fenn. 5 (1931), No. 25, 1–37.

Morfismos, Vol. 9, No. 2, 2005, pp. 49–69

Secondary operations in K-theory and the generalized vector field problem (revisited) ∗ Jesu ´s Gonz´alez

Maurilio Velasco-Fuentes 1

Abstract We describe detailed calculations of secondary operations in complex K-theory leading to results on the generalized vector field problem for projective spaces. The method was originally published by Feder and Iberkleid in 1977. Our interest in understanding those ideas rests on the possibility to extend their results to the case of 2e -torsion lens spaces in order to analyze the torsion’s role in the generalized vector field problem for these manifolds.

2000 Mathematics Subject Classification: 55S25, 57R25.

Keywords and phrases: Adams operations, cannibalistic classes.

1

Introduction

The vector field problem for a given manifold M is to determine the largest possible number of continuous vector fields on M which are linearly independent everywhere; that is, one asks for the largest possible number of linearly independent sections to the tangent bundle τM . The problem is rather involved even for “simple” manifolds such as spheres.

In that case the problem was finally solved by J. F. Adams in his celebrated paper [1] as one of the first of many impressive applications of K-theory. Adams’ solution made a fundamental use of projective spaces as follows. The vector field problem for S n−1 can be stated as to determine the largest possible k so that the fibration (1)

Vn,k = O(n)/O(n − k) → O(n)/O(n − 1) = S n−1

∗

This paper is part of the second author’s Ph.D. work at the Mathematics Department of CINVESTAV. 1 Partially supported by a CONACYT Ph.D. scholarship.

49

50

J. Gonz´alez and M. Velasco-Fuentes

has a single cross section, where Vn,k is the usual Stiefel manifold of kplanes in Euclidean n-dimensional space. Moreover, the real projective space P n−1 sits naturally as reflections in O(n), and hence the stunted n−1 projective space Pn−k = P n−1 /P n−k−1 sits inside Vn,k . This yields the commutative diagram n−1 Pn−k

i

!! !! ! π !!! !

" Vn,k " " "" "" " #" "

S n−1

where π is the pinching map (this much was known previous to Adams’ n−1 contains work, see for instance [13]). Now, in the relevant range, Pn−k the complete skeleton of Vn,k into which a potential section for (1) can be deformed. Thus the problem is reduced to understanding when π admits a (homotopy) right inverse, or in technical language, to determine when n−1 Pn−k is reducible. As Adams showed, it is in this last form that the problem can be completely answered in K-theoretic terms.

The vector field problem for a projective space P n is far more complex and it stands as a classical open challenge. A related (somehow smoothen) version of the problem considers the analogous sectioning question for the stable class of τP n (2)

τP n ⊕ 1 ≃ (n + 1)ξn

where ξn is the canonical Hopf bundle over P n . The so-called generalized vector field problem for projective spaces asks to compute, as a function of m and n, the geometric dimension of mξn , that is to say, to determine the smallest possible dimension of a vector bundle over P n which is stably equivalent to mξn . Note that for m > n, computing the geometric dimension of the stable bundle mξn is equivalent to the initial sectioning question for mξn . In particular, for m = n + 1, the above picks information on the original vector field problem for projective spaces. Right at this point we find one of the main differences with the corresponding problem for spheres: τS k is stably trivial whereas computing the geometric dimension of (2) is a highly nontrivial task. In fact, the complexity of the new problem can be better appreciated from the observation in [11] that the (already stable) case m = 2L − n − 1 (L ≫ 0) in the generalized vector field problem for projective spaces is equivalent to the immersion problem for those manifolds —yet another classical open challenge.

Generalized vector field problem on lens spaces

51

The purpose of this work is to develop in detail a method introduced

by Feder and Iberkleid in [8] for computing secondary characteristic classes in K-theory in order to obtain conditions for sectioning unstable vector bundles of the form (3)

E ⊗ ξm −→ X × P m .

The relevance of (3) in the generalized vector field problem comes from the usual trick for converting a multiple sectioning problem into a single sectioning one —which plays the analogue of (1) as an alternative statement for the vector field problem for spheres. Indeed, if a vector bundle E over X admits m + 1 linearly independent sections, then in view of (2), E ⊗ ξm would have a single no-where zero section, and in such a case one could use (generalized) characteristic classes to give obstructions for the existence of such a section. We remark that in the metastable range —that is, when dim(X) < 2(dim(E) − m − 1)— the above sectioning reduction is reversible (see [2]).

As already noted in [8], the method works best for a torsion space X, and this is in fact our motivation for understanding Feder and Iberkleid’s ideas. In a future work we will adapt their methods to general 2e -torsion lens spaces in order to get a better picture of the role of the 2-torsion in the problem.

We now state Feder’s and Iberkleid’s main result. ! " is Theorem 1.1 Let k > n and assume the binomial coefficient k−1 n odd. Then 2kξ2n does not have 2k − 2n + 2ν2 (2k) + 2 sections. This number can be improved to 2k − 2n + 2ν2 (2k) provided either k is odd or, else, when k is even with ν2 (k) ≡ n mod 2. Here ν2 (M ) stands for the highest power of 2 dividing the integer M .

Remark 1.2 From standard obstruction theory (Postnikov towers) we know that any vector bundle α over a cellular complex X always admits dim(α) − dim(X) sections. Theorem 1.1 claims that in the case of 2kξ2n one cannot hope for 2ν2 (2k) + ϵ extra sections beyond the free ones.

The philosophy motivating the proof of such a result goes as follows.

As already explained, a multiple sectioning problem is first changed to the existence of a single section for a suitable bundle λ → B. Now, if λ is oriented with respect to some generalized cohomology theory and admits a non-zero section, then the corresponding Euler class χ(λ)

52

J. Gonz´alez and M. Velasco-Fuentes

vanishes. Nevertheless, the reciprocal statement is of course not true in general. For example, although χ(τS 2 ) = 0 in mod 2 cohomology, S 2 (or, for that matter, any even-dimensional spheres) does not have a vector field. The alternative is to (construct and) analyze generalized higher order characteristic classes which should also vanish in the presence of a nontrivial section —therefore producing further obstructions to the existence of such a section. A way to construct higher characteristic classes is described next.

2

Secondary characteristic classes

Let h∗ be a generalized cohomology theory and let p : λ −→ B be a vector bundle with associated disk and sphere bundles D and S, respectively. In terms of the long exact sequence (4)

j

i

δ

· · · −→ h∗ (D, S) −→ h∗ (D) −→ h∗ (S) −→ h∗+1 (D, S) −→ · · ·

any natural cohomology operation ϕ for h∗ defines a secondary operation Sϕ on Im δ ∩ Ker ϕ, taking values in the quotient of Im j by Im (j ◦ ϕ), as follows. Let v ∈ h∗+1 (D, S) and u ∈ h∗ (S) be classes with ϕ(v) = 0 and δ(u) = v. By exactness in the lower row of the diagram ···

! h∗ (D, S)

i

! h∗ (D)

ϕ

···

j

! h∗ (S)

j

"

ϕ

"

! h∗ (D, S)

i

"

! h∗ (D)

δ

! h∗+1 (D, S)

ϕ

! h∗ (S)

ϕ δ

"

! h∗+1 (D, S)

the class ϕ(u) lies in the image of j. We then set (5)

Sϕ (v) = ϕ(u) ∈ Im j/Im (j ◦ ϕ).

This is well defined, for if δ(u′ ) = v, then u − u′ = j(ω), for some ω ∈ h∗ (D), so that ϕ(u′ ) − ϕ(u) = jϕ(ω). We will assume from now on that h∗ is multiplicative and that p : λ → B comes equipped with an h∗ -orientation. Then in terms of the Thom isomorphism, (4) becomes the usual Gysin sequence (6)

χ

p

δ

· · · −→ h∗ (B) −→ h∗ (B) −→ h∗ (S) −→ h∗+1 (B) −→ · · ·

where the first map is multiplication by the h∗ -Euler class of λ (we will be interested in the case where h∗ is complex K-theory, so that orientations and Euler classes have been assumed to be zero dimensional —for

Generalized vector field problem on lens spaces

53

non-periodic theories one needs to be a bit more careful with gradings).

Note that if λ has a nontrivial section, then χ = 0, splitting (6). Therefore, in such a case our secondary operation Sϕ is really defined on the subgroup ker ϕ of h∗+1 (B) and, via the monomorphism induced by p, takes values on the quotient h∗ (B)/Im ϕ. This will be the situation in our applications, as well as in the next key result which shows that the class Sϕ (v) above works indeed as a sectioning obstruction —it vanishes whenever λ admits a nonzero section.

Lemma 2.1 In the notation of (5), if the bundle p : λ −→ B admits a non-zero section, then Sϕ (v) vanishes whenever it is defined.

Proof: Let r : B −→ S be such a section and choose α ∈ h∗ (B) with ∗ p (α) = ϕ(u). Then Sϕ (v) is represented by α = r∗ p∗ α = r∗ ϕ(u) = ! ϕ(r∗ u) ∈ Im ϕ.

Remark 2.2 The proof above indicates the special role played by a section in the process of identifying the (triviality of the) secondary operation. In fact, some of our concrete calculations will be simplified by a simultaneously consideration of two operations ϕ1 and ϕ2 with ϕ1 (v) = ϕ2 (v) = 0. The associated secondary operations will then be given directly in terms of the section r as the corresponding classes represented by ϕi (r∗ (u)), for i = 1, 2, respectively.

Remark 2.3 The case with v = 1 ∈ h∗ (B), corresponding to the Thom class in the cohomology of the pair (D, S), yields the “secondary characteristic class” Sϕ (λ) —defined whenever χ(λ) = 0 and ϕ(1) = 0— in the title of this section. Such a secondary class must, of course, vanish in the presence of a section, in view of Lemma 2.1.

In order to have a good hold in computing the secondary operations introduced in this section, it will turn out to be convenient to lift (6) to an equivariant setting where one can take advantage of the extra structure. The equivariant facts we will need (in the case of K-Theory) are recollected in the next short section, whereas the equivariant interpretation of the secondary operation will be set up in Section 4 —with a tuning up in Section 5 which will allow us to evaluate the required secondary operations in terms of Adams operations and Thom isomorphisms.

54

3

J. Gonz´alez and M. Velasco-Fuentes

Equivariant K-theory

Let G be a compact group and consider Atiyah-Segal’s G-equivariant ∗ . This is a cohomology theory defined over the category K-Theory KG of (compact) G-spaces in terms of G-equivariant vector bundles, and extending usual K-Theory in many respects. For instance, as observed in [3], when X is a free G-space we have (7)

∗ (X) ≃ K ∗ (X/G), KG

the usual K-theory of the orbit space X/G. On the other hand, as shown in [12], for a trivial G-space X one has the natural isomorphism (8)

∗ (X) K ∗ (X) ⊗ R(G) ≃ KG

∗ (X) and the two “trivial” maps induced from the ring structure in KG ∗ ∗ ∗ ∗ is a (G-equivariantly) comR(G) → KG (X) and K (X) → KG (X). KG plex oriented cohomology theory. Moreover, in terms of (7), equivariant orientations in the G-free case agree with their unequivariant analogues over the corresponding orbit spaces —as explained in the next section, this will be the precise starting point for lifting the computation of our secondary classes, from the unequivariant case to the equivariant setting.

We will make computational use of the fact that for a subgroup H < G there is a restriction homomorphism KG (X) −→ KH (X) which is compatible with (8) and natural in the category of G-spaces. For our concrete purposes we shall use the groups Z2 < S 1 (and, for the eventual generalization mentioned in the introduction, the groups Z2e < S 1 ).

We recall what the corresponding representation rings are: R(S 1 ) = Z[ t, t−1 ], the polynomial ring on the standard unitary representation t : S 1 !→ GL (1) and its inverse t−1 (complex conjugation); R(Z2 ) = Z[ t ]/(t2 − 1), where t ∈ R(Z2 ) stands for the restriction of t ∈ R(S 1 ) to the subgroup Z2 < S 1 .

4

K-theory secondary operations: Euler classes

The way we have chosen for starting this section will become transparent in Remark 4.2 —for the time being, the comments after (3) should clarify much of the situation below. Let p : E → X be a complex vector bundle over a compact space X and recall that ξ2m−1 stands for the Hopf bundle over P 2m−1 . It is standard that the product bundle E ⊗ ξ2m−1

55

Generalized vector field problem on lens spaces

has a complex structure coming from that of E, and which can be thought of as being that of E ⊗ ξ2m−1 . We thus indistinctly use both tensor bundles. Now, a model for the total space of E ⊗ ξ2m−1 is given by the quotient of E × S 2m−1 by the equivalence relation defined by (−x, y) ≡ (x, −y). In terms of (8), this observation can be translated to the equivariant world: if we let Et (= E⊗t) stand for the original bundle E with antipodal action of Z2 in fibers (and trivial action on the base space), and identify t (= 1 ⊗ t) with the trivial one-dimensional complex line bundle over X with antipodal action of Z2 in fibers, then E ⊗ ξ2m−1 is just the orbit space of the Z2 -equivariant fibered product Et ×X Smt, where Smt is the sphere bundle of the iterated m-fold Whitney sum of t with itself. It is convenient to embed the above construction into the following pull-back diagram of Z2 -equivariant bundles: (9)

SEt ×X Smt

!"

! Et ×

X

"#

Smt

"#

" " SEt ×X mt !

" ! Et ×X mt

"

" ! Et

SEt

!"

q′

! Smt "# " ! mt

q

" !X

p

Note that we recover the 2m − 1 analogue of (3) by passing to Z2 orbit spaces in q ′ of the the top row of (9). In these terms, the relevant exact sequence (of type (6)) for computing secondary K-theoretic operations for E ⊗ ξ2m−1 agrees with the corresponding exact sequence for computing K 2 -theoretic secondary operations for Et ×X Smt. But by (9), the latter exact sequence is embedded as the second row (from top to bottom) in the commutative diagram of Gysing type sequences K −1 (X) 2

(10)

❄

K

−1 2

K 02 (Et)

❄ δ −1 0 ✲ ✲ (Smt) K 2 (SEt ×X Smt) K 2 (Et ×X Smt) q ′∗

δ

❄1

δ

δ

❄2

q∗

❄3

K 02 (Et ×X mt) ✲ K 02 (mt) ✲ K 02 (SEt ×X mt) ✲ K 12 (Et ×X mt) ❄

K0

2

(Et)

❄ ✲ K 0 (X) 2

δ4

❄ ✲ K 0 (SEt) 2

p∗

It should be remarked that in this diagram we are using K-theory with

56

J. Gonz´alez and M. Velasco-Fuentes

compact supports so that, for a non-compact space V , K 2 (V ) really stands for the reduced K 2 -theory of the one-point compactification of V . Thus, if V is the total space of a vector bundle, then the Thom isomorphism identifies K 2 (V ) with the usual K 2 -theory of the base space, allowing us to recover the Gysin sequence setting in (6) for computing secondary operations.

Now, when looking for obstructions to sectioning E ⊗ ξ2m−1 , instead of evaluating a secondary operation directly in the second row of (10), we will perform the corresponding computation in the third row of that diagram —that is, we will deal with q rather than with q ′ . Such a strategy is better explained by recalling that, in the first case, the secondary operation lies in a quotient of the image of (q ′ )∗ , however in the second case one can perform the calculation directly in the cokernel of the first map in the third row of (10) —a map on which we will have an explicit control. In fact, the square in the lower left corner of that diagram turns out to play a fundamental role in the whole computation and, thus, we proceed to describe the corresponding groups and maps in the case relevant for Theorem 1.1. We start by noticing that, after applying suitable Thom isomorphisms in the appropriate nodes and using the considerations in Section 3, the square just mentioned takes the form (11)

K 0 (X) ⊗ R(Z2 )

χ(Et)

! K 0 (X) ⊗ R(Z2 )

χ(mt)

χ(mt)

"

K 0 (X) ⊗ R(Z2 )

"

χ(Et)

! K 0 (X) ⊗ R(Z2 )

where all maps are multiplication by the indicated Euler classes. Now, we will be interested in the case X = P 2n , whose K-theory groups are (12)

K 0 (P 2n ) = Z[x]/(xn+1 , [2](x)),

where x = 1 − ξ2n is the K-theoretic Euler class of the complexification of ξ2n , and where [2](x) is the 2-series on x for the multiplicative formal group law, that is (13)

[2](x) = 1 − (1 − x)2 = 2x − x2 .

Together with the remarks in Section 3, this completes the description of the groups in (11). The corresponding maps, on the other hand, are

Generalized vector field problem on lens spaces

57

described in the next result (whose proof is given after Remark 4.3) for the bundle (14)

E = 2kξ2n − 2(k − n − 1)

when k > n. As required at the beginning of the section, and in view of the considerations in Remarks 4.2 and 4.3 below, E is a (well defined) complex bundle. ! " is odd, then the Euler Lemma 4.1 Set T = 1 − t ∈ R(Z2 ). If k−1 n classes in (11) are given by χ(mt) = T m

and

χ(Et) = T n+1 .

Remark 4.2 Although (14) is based on an “obvious” desuspension of 2kξ2n (in the sense of Remark 1.2), the resulting E is still a stable bundle, so that its isomorphism class is indeed determined by the requirement that (15)

E ⊕ 2(k − n − 1) ≃ 2kξ2n

which formalizes the sloppy formula (14). In particular, computing the geometric dimension (gd) for 2kξ2n is still equivalent to the sectioning problem for E. In these terms, it is now clear from the considerations after (3) that this section of the paper has been arranged in order to set up a method for analyzing secondary obstructions for the possible existence of 2m linearly independent sections for E or, in other words, for ruling out inequalities of the form gd(2kξ2n ) ≤ 2n − 2(m − 1) —just as required in Theorem 1.1.

Remark 4.3 Although we are interested in the real sectioning problem for E, the equivariant method set up in this section takes advantage of the complex structure behind (14): it is possible to choose E with a complex structure in such a way that (15) is in fact the realification of (16)

E ⊕ (k − n − 1) ≃ kξ2n .

Indeed, just consider the homotopy commutative diagram below, where the indicated lifting holds since the fiber of BU (n + 1)→BU (k) is (2n + 2)-connected —the complex analogue of Remark 1.2.

BU (n + 1) E

P 2n

!

!

!

kξ2n

!

real

!#

" ! BU (k)

real

! BO(2n + 2) " ! BO(2k)

58

J. Gonz´alez and M. Velasco-Fuentes

Proof of Lemma 4.1. The Euler class for a line bundle L is simply χ(L) = 1 − L, so the Euler class for mt is plainly ! m # " t = (1 − t)m = T m . χ(mt) = χ i=1

In order to compute χ(Et) we use the fact that this class is the restriction of the corresponding (S 1 -equivariant) Euler class χ(Et) ∈ KS0 1 (P 2n ). Thus, in the S 1 -equivariant world we multiply (16) by t to obtain χ(Et) · (1 − t)k−n−1 = (1 − ηt)k where η = ξ2n . Since 1 − η = x and 1 − t = T , the last equation can be expanded as χ(Et) · T

k−n−1

= (T + xt) = k

k % & $ k i

i i

txT

=

k−i

i=0

n % & $ k i

ti xi T k−i

i=0

where the last equality follows from the relations in (12). Now t and (therefore) T act monomorphically on R(S 1 ), so we deduce χ(Et) =

n % & $ k i

ti xi T n+1−i .

i=0

Now, restricting to K 02 (P 2n ), where the relation t2 = 1 implies T 2 = 2T and tT = −T , we have n % & n % & $ $ k i i n+1−i k i i n+1−i n+1 x T = T + t χ(Et) = T n+1 + i (−1) x T i = T n+1 +

i=1 n % $ i=1

i=1

k i

&

(−1)i xi 2n−i T = T n+1 +

n % & $ k i

(−1)i xn T

i=1

where the last equality uses the relation (13). The required expression for χ(Et) now follows that) 2xn = 0 and the easy(k ) the observation ( (k−1 ) 'n from i n k−1 —recall that n is to-check relation i=0 i (−1) = (−1) n odd by hypothesis. !

5

Secondary operations: cannibalistic classes

In previous sections we have set up the required machinery for proving Theorem 1.1. We now bring up the final ingredient, namely, the primary operation(s) ϕ which the secondary calculation will be based on.

Generalized vector field problem on lens spaces

59

The (choice and) computability of ϕ, and therefore of the associated secondary operation, depends on having a good control on certain characteristic classes introduced in [6], the so called cannibalistic classes.

The construction applies in the equivariant setting too: for a complex G-bundle λ : Y → X, with KG -theoretic Thom class Uλ ∈ KG (Y ) —as in (10), we use cohomology with compact supports—, Bott’s equivariant cannibalistic class ρr (λ) ∈ KG (X) is defined through the action of the Adams operation ψ r by the formula ψ r (Uλ ) = ρr (λ) · Uλ . In particular, the action of ψ r in KG (Y ) is recovered as (17)

ψ r (a · Uλ ) = ρr (λ)ψ r (a) · Uλ ,

a ∈ KG (X).

Note that ψ r will act on KG (Y ) as (scalar) multiplication by ρr (λ) whenever ψ r is the identity on KG (X). In their original work [8], Feder and Iberkleid showed this to be the case for (odd r and) λ = q ′ , the top row in (9), which motivated the use of the operation (18)

ϕr = ψ r − ρr (q ′ )

(where the second term on the right hand side of (18) stands for the operation given by scalar multiplication by the rth cannibalistic class of q ′ ). Indeed, in that case ϕr would vanish on the last group in the second row of (10) and consequently, as indicated after (6), the corresponding secondary operation Sϕr would be defined in that group whenever E ⊗ ξ2m−1 admitted a nowhere trivial section. In these terms, the relevance of using the whole diagram (10) comes from the observation that (18) is also defined on the third row of (10). In fact, in terms of the KG (X)module structure inherited from (9), ϕr takes the form (19)

ϕr = ψ r − ρr (Et)

in both middle rows of (10). Thus, in order to carry over the strategy planned just before (11), we need to compute the action of ρr (Et) on K 0 (X) ⊗ R(Z2 ) —the relevant group in (11). Before completing such a task, it will be convenient to give Feder-Iberkleid’s alternative argument to show that ϕr indeed vanishes (and so Sϕr is defined) at suitable elements of the required last group in the second row of (10). As in (12), throughout the rest of the paper X will stand for P 2n .

Proposition 5.1 Let r be odd. Then, for the bundle E in (14) and with respect to the second row in (10), Sϕr is defined on all elements of (SEt ×X Smt) satisfies δ2 (u) = q ∗ (U ), the form δ(u), where u ∈ K −1 2 0 for some U ∈ K 2 (mt).

60

J. Gonz´alez and M. Velasco-Fuentes

Proof:

Consider the following extract of (10) K 02 (Et) π∗

K −1 (Smt) 2

q ′∗

" K −1 (SEt ×X Smt) 2

q∗

!

δ1

!

K 02 (mt)

δ

!

" K 0 (Et ×X Smt) 2

δ2

δ3

" K 0 (SEt ×X mt) 2

δ4

!

" K 1 (Et ×X mt) 2

Let U ∈ K 02 (mt) and u ∈ K −1 (SEt ×X Smt) be as in the hypothesis, 2 and let v = δ(u) ∈ K 02 (Et ×X Smt). Since δ3 (v) = δ3 δ(u) = δ4 δ2 (u) = δ4 q ∗ (U ) = 0, there is an element w ∈ K 02 (Et) such that π ∗ w = v. But since ψ r is the identity on K 0 (X) for odd r —this fact follows directly from (12) and (13)—, it is also the identity on R(Z2 ) in view of [5], as well as on K 02 (X) in view of (8). Thus, as observed after (17), ψ r w = ρr (Et) · w, i.e., ϕr (ω) = 0. Therefore ϕr (v) = π ∗ ϕr (w) = 0. !

We will pick in the next section a convenient term U ∈ K 02 (mt) to be used within the context of Proposition 5.1. For the time being we note that, since the corresponding secondary operation will be evaluated in the third row of (10), we need to get a good hold on ϕr (U ). For this we note that in terms of the Thom isomorphism we have (20)

U = A · Umt

for some A ∈ K 02 (X), where Umt is the Thom class of mt. Then, from (17) and its subsequent observations —see also the proof of Proposition 5.1—, we get for odd r (21)

ϕr (U ) = ϕr (1) · U,

where ϕr (1) = ρr (mt) − ρr (Et). We close the section with an explicit description of (21) in the cases which are relevant towards Theorem 1.1.

Proposition 5.2 Let E be given by (14). In the notation of Lemma 4.1 and (12), we have (i) ρ3 (mt) = 3m +

1−3m 2 T,

(ii) ρ3 (Et) = 3n+1 +

1−3n+1 2

and T+

1−3k 2

!

1−

3k−n−1 +1 2

" T 3−(k−n−1) x.

Generalized vector field problem on lens spaces

61

Proof: Recall that for r > 0, ρr (L) = 1 + L + · · · + Lr−1 , whenever L is a line bundle. Thus, 1 − 3m T 2 where the last equality is easily verified by induction. On the other hand, in order to verify the expression for ρ3 (Et) in (ii), we proceed as in the proof of Lemma 4.1, namely, we multiply (16) by the S 1 -equivariant version of t and apply the multiplicative property of the cannibalistic classes to get (22)

ρ3 (mt) = (1 + t + t2 )m = (3 − T )m = 3m +

ρ3 (Et) · (1 + t + t2 )k−n−1 = (1 + ηt + t2 )k .

Here we use the (closely related to (13)) relation η 2 = 1, where η = ξ2n .

This relation can be loosely interpreted as (23)

ρ3 (Et) =

(1 + ηt + t2 )k ∈ K(P 2n ) ⊗ R(S 1 ) (1 + t + t2 )k−n−1

since the “division” is unique in R(S 1 ) —that is, 1 + t + t2 is not a zero divisor. Indeed, according to [4], we have the monomorphism (24)

Z[ t, t−1 ] = R(S 1 ) $→ K(CP ∞ ) = Z[[ T ]]

where T is the Euler class of the Hopf bundle over CP ∞ , and the inclusion map is that of a space inside its completion (with respect to a suitable topology) . In fact, by definition, the image of t under (24) is given in terms of the Borel construction S ∞ ×S 1 C —the Hopf bundle—, so that the T used in (24) agrees with that in Lemma 4.1. In particular, as elements of K(P 2n ) ⊗ R(S 1 ), we have 1 + t + t2 = 1 + (1 − T ) + (1 − T )2 = 2 − T + 1 − 2T + T 2 = 3 − 3T + T 2

which is not a zero divisor —not even in the completed group K(P 2n ) ⊗ Z[[ T ]] = (Z ⊕ Z2n ) ⊗ Z[[ T ]].

Now, letting Q = 1 + t + t2 , (23) becomes k " # ! k Qk−i (−t)i xi i k 2 k (Q − xt) (1 + ηt + t ) i=0 = = ρ3 (Et) = (1 + t + t2 )k−n−1 Qk−n−1 Qk−n−1 n " # ! k Qk−i (−t)i xi i = i=0 (since k > n and xn+1 = 0) Qk−n−1 n % & $ k i i n+1−i = i (−t) x Q i=0

62

J. Gonz´alez and M. Velasco-Fuentes

where the last equality follows by uniqueness of division by Q (or, alternatively, see the proof of Lemma 4.1 for the formal argument). Now, restringing from R(S 1 ) to R(Z2 ), so that Q becomes Q = 2 + t = 3 − T , we finally obtain in K(P 2n ) ⊗ R(Z2 ) n " # !

ρ3 (Et) =

k i

i=0

(25)

n " # !

=

k i

i=0

(−t)i xi (3 − T )n+1−i % $ 1 − 3n+1−i n+1−i T . + (−t) x 3 2 i i

n+1

For i = 0, (25) gives 3n+1 + 1−32 T , which are the first two terms of the required formula for ρ3 (Et). Using the relations x2 = 2x and tT = −T , the rest of the terms in (25) —which are divisible by x, so that all coefficients are now 2n -torsion— can be rewritten as % $ n " # !

1 − 3n+1−i k i i n+1−i T = + 3 i (−t) x 2 i=1

= =

n " # ! k i

i=1 n !

i=1 i even

3n+1−i (−t)i xi +

" # k i

k i

(−t)i xi T

i=1

3n+1−i 2i−1 x +

n " # ! k i

1 − 3n+1−i 2

3n+1−i (−t)2i−1 x

i=1 i odd

n " # !

+

n " # !

k i

2i−1 xT

i=1

1 − 3n+1−i 2

= A + B, where A =

n ' ( & k

i=1

B=

i

3n+1−i (−1)i 2i−1 x and

n " # ! k i

3

2

n+1−i i−1

xT +

i=1 i odd

n " # ! k i

2i−1 xT

i=1

1 − 3n+1−i .

2

For the A-term —leading to the multiple of x in (ii)—, it is enough to verify that n " # ! k i

i=1

3n+1−i (−1)i 2i−1 ≡

1 1 − 3k · k−n−1 2 3

(mod 2n ).

Generalized vector field problem on lens spaces

63

But since 3 is a unit modulo 2n , the above congruence reduces to n " # ! k i

i=1

3k−i (−2)i ≡ 1 − 3k

(mod 2n+1 )

which is clear from Newton’s formula. On the other hand, in view of the relation 2n x = xn+1 = 0, both summations in the B-term —leading to the multiple of xT in (ii)— can be taken up to k (instead of up to n), so that the coefficient of xT is the mod 2n reduction of the integer k " # ! k i

2

3

n+1−i i−1

+

k " # ! k i

2i−1

i=1 i odd

=

k i

k " # ! k i

k " # ! k i

i=1

= =

i even

2i−2 (1 + 3n+1−i ) +

= =

k " # ! k i

i=1 i even

2

i−2

−

k " # ! k i

2i−2 (1 − 3n+1−i )

2i−2 (−1)i 3n+1−i

i=1 k " − 1 ! k # i−2 − (−1)i 3n+1−i i 2 4 i=1 $ % k " # !

1 k i i k−i 3k−n−1 (3k − 1) − i 2 (−1) 3 · 3k−n−1 i=1

3k

4

1 − 3n+1−i = 2

k " # n+1−i !

1 + 3n+1−i k i−1 1 − 3 + 2 i 2 2 i=1

i=1 i odd

=

2i−1

i=1

i=1 i odd

=

k " # !

1

# " 3k−n−1 (3k − 1) − (1 − 3k )

4 · 3k−n−1 (3k − 1) (3k−n−1 + 1) 1 · . · 2 2 3k−n−1

The result follows since the last two factors are integers themselves, whereas 3 is a unit mod 2n . !

As suggested in Remark 2.2, it will be convenient to consider, in addition, the operation Sϕ−1 . Its evaluation depends, of course, on settling the value of ϕ−1 (1) in (21). Unlike the previous r = 3 arguments, the computation of the r = −1 cannibalistic classes do not require auxiliary unique-division considerations in the S −1 -equivariant setting —the relevant classes are in fact invertible in the Z2 -equivariant world.

64

J. Gonz´alez and M. Velasco-Fuentes

Proposition 5.3 Let E be given by (14). In the notation of Lemma 4.1 and (12), we have (i) (ii)

ρ−1 (mt) = (−1)m tm , and

ρ−1 (Et) = (−1)n+1 tn+1 η k .

Proof: Recall that for a line bundle L, ρ−1 (L) = −L−1 . In view of the relation t2 = 1 in R(Z2 ), this immediately yields the expression in (i) above, while for Et we get from (16) ρ−1 (Et) · (−t)k−n−1 = ρ−1 (Et) · ρ−1 ((k − n − 1)t) = ρ−1 (kηt) = (−ηt)k ,

where the last equality uses the relation η 2 = 1 already noticed in the proof of Proposition 5.2. Multiply by (−t)k−n−1 to recover the desired ! expression for ρ−1 (Et).

This concludes setting up the technical details for evaluating the secondary operations we have in mind. This machinery will be put together in the next (and final) section in order to prove Theorem 1.1.

6

Proof of Theorem 1.1

Let k > n ≥ m ≥ 1 and assume that the bundle E in (14) admits 2m linearly independent sections, so that E ⊗ ξ2m−1 has a nowhere trivial section. Our first goal will be to choose a suitable element on which to evaluate (the corresponding triviality of) certain critical secondary operations. This is done by making use of Proposition 5.1: from the exactness of rows and columns in diagram (10), and in terms of the notation in (11) and (20), picking the required U ∈ K 02 (mt) amounts to choosing a class A ∈ K 02 (P 2n ) satisfying A · χ(mt) = A′ · χ(Et), for some A′ ∈ K 02 (P 2n ). In view of Lemma 4.1, the obvious choice ! " is odd —which we assume is A = T n−m+1 (with A′ = 1) when k−1 n to be the case from now on. We can now pick a corresponding class u ∈ K −1 (SEt ×P 2n Smt) satisfying the hypothesis in Proposition 5.1, 2 so that Sϕr (δ(u)) is not only defined for any odd r, but in fact it is trivial in view of Lemma 2.1 and the present sectioning assumption.

Even more, as indicated in Remark 2.2, there is a class µ ∈ K −1 (Smt) 2 such that ϕr (µ) is a representative for Sϕr (δ(u)), for any odd r. Since q ′∗ is monic (as explained in Section 4, it corresponds to the map p in (6); alternatively, see the proof of Lemma 2.1), the above means q ′∗ (ϕr (µ)) = ϕr (u).

65

Generalized vector field problem on lens spaces

Setting B = δ1 (µ) this yields q ∗ (ϕr (B)) = q ∗ (ϕr (δ1 (u)) = δ2 (q ′∗ (ϕr (µ))) = ϕr (δ2 (u)) = q ∗ (ϕr (A)).

(Here and in what follows we are using the notation in diagram (10), except for its lower left corner, which has been written as in diagram (11).)

Therefore, by exactness of the third row in (10), there exist classes dr ∈ K 0 (P 2n ) ⊗ R(Z2 ) = (Z ⊕ Z2n · x) ⊗ (Z ⊕ Z · T )

(26) such that

ϕr (A) − ϕr (B) = dr T n+1 .

(27)

Note that the relation T n+1 = 2n T assures that the term dr T n+1 above is torsion free and, in fact, an integer multiple of T , so that we can (and will) assume dr ∈ Z. We now take a closer look to each term on the left of (27) —the critical one being ϕr (B) since, as explained in Section 4, this corresponds to evaluating the image of Sϕr (δ(u)) in the third row of (10). From (21) we see that ϕr acts on the group in (26) as multiplication by ϕr (1).

As the reader will easily verify using Propositions 5.2 and 5.3, the later term is described for r ∈ {3, −1} as n+1 −3m

(28) ϕ3 (1) = 3m − 3n+1 + 3

2

T − 1−3 2

and (with mod 2 conditions)

(29)

ϕ−1 (1) =

k

!

1−

3k−n−1 +1 T 2

3−(k−n−1) x

⎧ ±(2 − T ), k ≡ 0 and m ≡ n, ⎪ ⎪ ⎪ ⎪ ⎪ 0, k ≡ 0 and m ≡ n + 1, ⎪ ⎪ ⎪ ⎪ ⎨ 2 − x − T + xT, k ≡ 1 and m ≡ n ≡ 0, −2 + x + T, ⎪ ⎪ ⎪ ⎪ ⎪ −x + xT, ⎪ ⎪ ⎪ ⎪ ⎩ x,

k ≡ 1 and m ≡ n ≡ 1,

k ≡ 1, m ≡ 1, and n ≡ 0, k ≡ 1, m ≡ 0, and n ≡ 1.

When multiplied by T , these expressions simplify to (30)

"

ϕ3 (1) · T =

1 − 3k xT 2

66

J. Gonz´alez and M. Velasco-Fuentes

and (31)

ϕ−1 (1) · T =

!

0, xT

for k ≡ 0 (mod 2), for k ≡ 1 (mod 2),

producing manageable formulas for ϕr (A) —recall A = T n−m+1 = 2n−m T —, as well as for the part in ϕr (B) coming from the T -divisible terms in the general expression B = a + bx + (α + βx)T of B as an element in the group (26). As for the a + bx part, it will be convenient to consider the concrete cases corresponding to the different instances in Theorem 1.1. In any case, the observation that 0 = T m B = (2m−1 a + 2m α)T + (2m−1 b + 2m β)xT (obtained from the exactness of the second column in (10)), yields the extra information (32)

a + 2α = 0

whereas

2m−1 b + 2m β ≡ 0

(mod 2n ).

Case k ≡ 0 and m ≡ n (mod 2): In view of (29) and (31), the r = −1 case in (27) reduces, up to a sign, to (2 − T )(a + bx) = 2n d−1 T .

Comparing coefficients and taking into account (32), we see that B reduces to B = βxT , where (33)

2m β ≡ 0 (mod 2n ).

Now using (30), the r = 3 case in (27) becomes 2n d3 T = 2n−m

1 − 3k 1 − 3k 1 − 3k n−m xT − βx · xT = (2 − 2β)xT. 2 2 2

Comparing coefficients and taking (33) together with Corollary A.2 in the appendix into consideration, we finally get m ≤ 1 + ν2 (k). It now suffices to take m = ν2 (k) + 2 in order to complete the proof for the case k ≡ 0 and ν2 (k) ≡ n mod 2 in Theorem 1.1.

In order to settle the two remaining cases in Theorem 1.1 we only need the case r = 3 of (27). Start by using (28) and (30) to compute ϕ3 (B) and ϕ3 (A), respectively, and expand out the resulting equation (27) to obtain a = 0 (thus α = 0, in view of (32)) as well as the mod 2n congruences (34)

b(3m − 3n+1 ) ≡ b(1 − 3k )3−(k−n−1)

Generalized vector field problem on lens spaces

and (35)

!

k) n+1 m 1−3k n−m −b 3 2−3 + (1−3 2 2 2

67

" ! " 3k−n−1 +1 3−(k−n−1) ≡ β(1−3k ).

Case k ≡ 1 (mod 2): Set 1 − 3k = 2u, where u is an odd integer in view of Corollary A.2, and rewrite the last congruence as ! n+1 " 3 − 3m (1 − 3k ) −(k−n−1) n−m + 3 2 u−b −u(b+2β) ≡ 0 (mod 2n ), 2 2 whose second and third summands are divisible by 2n−1 and 2n−m+1 , respectively, in view of (34) and (32). This forces m ≤ 1, so that the case for odd k in Theorem 1.1 follows by taking m = 2.

Case k ≡ 0 (mod 2) —any m and n: From (34) and (35) we deduce the congruence $ 1 − 3k # n−m 2 − b − 2β ≡ 0 2

(mod 2n−1 ).

Therefore, taking into account (32) together with Corollary A.2 in the appendix, we get m ≤ 2 + ν2 (k). It now suffices to take m = ν2 (k) + 3 in order to complete the proof of the remaining case in Theorem 1.1.!

A

# $ Appendix: The value of ν2 1 − 3k

This appendix proves some well know arithmetic relations used in Section 6 of the paper.

Lemma A.1 For k ≥ 0, ν2 (1 + 3k ) =

%

1

if

k

is even,

2

if

k

is odd.

Proof: The relation 1+32l = 1+9l ≡ 2 (mod 4) implies ν2 (1+32l ) = 1, the result for even k. Likewise, the relation 1 + 32l+1 = 1 + 3 · 9l ≡ 4 (mod 8) yields the result for odd k. !

Corollary A.2 For k > 0, % ν2 (1 − 3k ) =

2 + ν2 (k)

if

ν2 (k) > 0,

1

if

ν2 (k) = 0.

68

J. Gonz´alez and M. Velasco-Fuentes

Proof: The relation 1 − 32l+1 = 1 − 3 · 9l ≡ 2 (mod 4) grounds (from 0) an inductive argument on ν2 (k), whereas the equalities ν2 (1 − 32l ) = ν2 (1 − 3l ) + ν2 (1 + 3l ) !

2 + ν2 (l) + 1 if l = 1+2 if l ! if l ν2 (l) + 3 = 3 if l

is even is odd is even is odd

= ν2 (2l) + 2 !

complete the induction.

Jes´ us Gonz´ alez Departamento de Matem´ aticas, CINVESTAV-IPN, A.P. 14-470, M´exico D.F. 07000, M´exico. jesus@math.cinvestav.mx

Maurilio Velasco-Fuentes Departamento de Matem´ aticas, CINVESTAV-IPN, A.P. 14-470, M´exico D.F. 07000, M´exico. mvelasco@math.cinvestav.mx

References [1] Adams J. F., Vector fields on spheres, Bull. Amer. Math. Soc. 68 (1962), 39–41.

[2] Astey L., Geometric dimension of bundles over real projective spaces, Quart. J. Math. Oxford Ser. (2) 31 (1980), 139–155.

[3] Atiyah M. F., K-Theory, Advanced Book Classics, Addison-Wesley Publishing Company, Advanced Book Program, Redwood City, CA, second edition, 1989.

[4] Atiyah M. F., Characters and cohomology of finite groups, Inst. ´ Hautes Etudes Sci. Publ. Math. 9 (1961), 23–64.

[5] Atiyah M. F.; Segal G. B., Equivariant K-theory and completion, J. Differential Geometry 3 (1969), 1–18.

[6] Bott R., Lectures on K(X), Mathematics Lecture Note Series, W. A. Benjamin, Inc., New York-Amsterdam, 1969.

Generalized vector field problem on lens spaces

69

[7] Davis D., Generalized homology and the generalized vector field problem, Quart. J. Math. Oxford Ser. (2) 25 (1974), 169–193.

[8] Feder S.; Iberkleid W., Secondary operations in K-theory and the generalized vector field problem, In: Geometry and Topology, Proceedings of the III Latin American School of Mathematics, Inst. Mat. Pura Aplicada, Rio de Janeiro (1976), Lecture Notes in Math. 597 1977, 161–175.

[9] Gonz´alez J., A generalized Conner-Floyd conjecture and the immersion problem for low 2-torsion lens spaces, Topology 42 (2003), 907–927.

[10] Gonz´alez J., Topological robotics in lens spaces, Math. Proc. Cambridge Philos. Soc 139 (2005), 469–485.

[11] Sanderson B. J., Immersions and embeddings of projective spaces, Proc. London Math. Soc. (3) 14 (1964), 137–153. ´ [12] Graeme S., Equivariant K-theory, Inst. Hautes Etudes Sci. Publ.

Math. 34 (1968), 129–151.

[13] Steenrod N. E.; Whitehead J. H. C., Vector fields on the n-sphere, Proc. Nat. Acad. Sci. U. S. A. 37 (1951), 58–63.

Morfismos, Vol. 9, No. 2, 2005, pp. 71–82

Complete intersection toric ideals of oriented graphs∗ Enrique Reyes

1

Abstract Let G be a connected simple graph. We prove the existence of an orientation O of G such that the toric ideal asociated to the digraph D = (G, O) is a complete intersection.

2000 Mathematics Subject Classification: 13H10, 05C85, 05C20.

Keywords and phrases: toric ideal, oriented graph, complete intersection, vector matroid.

1

Introduction

Let G be a connected simple graph and let O be an orientation of G. In Section 2, we compare the circuits and the bases of the matroids associated to G and to D = (G, O). We study when the toric ideal associated to D is a complete intersection and show that this ideal is generated by the binomials corresponding to primitive cycles.

In Section 3 we construct an orientation O and a spanning tree τ of G, such that the toric ideal PD of D, is a complete intersection. In particular we prove that the binomials corresponding to the principal cycles c(τ, fi ) with fi ∈ E(D) \ E(τ ) are a Gr¨obner basis of PD , where c(τ, fi ) is the unique cycle of the subgraph τ ∪ {fi }. As usual we are denoting the edge set of D by E(D) and the vertex set of D by V (D). We prove that if D is an acyclic oriented graph with a spanning directed path, then PD is a complete intersection minimally generated by a Gr¨obner ∗ This paper is part of the author’s Ph.D. thesis presented at the Department of Mathematics of CINVESTAV-IPN under the direction of Prof. Rafael H. Villarreal.

1 Partially supported by COFAA-IPN, and CONACyT grant 49251-F.

71

72

Enrique Reyes

basis. This complements a result of Ishizeki [3] showing that all acyclic complete oriented graphs have this property.

2

Toric ideals of oriented graphs

Let G be a connected simple graph with n vertices and q edges, and let O be an orientation of G. Thus D = (G, O) is an oriented graph. In particular D is a digraph. To each oriented edge e = (xi , xj ) of D, we associate the vector ve defined as follows: the ith entry is −1, the jth entry is 1, and the remaining entries are zero. The incidence matrix AD of D is the n × q matrix with entries in {0, ±1} whose columns are the vectors of the form ve , with e an edge of D. For simplicity of notation we set A = AD . The set of column vectors of A will be denoted by A = {v1 , . . . , vq }.

It is well known [4] that A defines a matroid M [A] on A = {v1 , . . . , vq } over the field Q of rational numbers, which is called the vector matroid of A, whose independent sets are the independent subsets of A.

Definition 2.1 A minimal dependent set or circuit of M [A] is a dependent set all of whose proper subsets are independent. A subset B of A is called a basis of M [A] if B is a maximal independent set.

Lemma 2.2 The circuits of M [A] are precisely the cycles of G, A is totally unimodular, and rank(A) = n − 1.

Proof:

It follows from [1, pp. 343-344] and [6, p. 274].

✷

In contrast if we consider the {0, 1} incidence matrix of a simple graph, we have the following description of the circuits and bases.

Proposition 2.3 ([8]) If AG is the incidence matrix of a connected simple graph G, then the circuits of the vector matroid M [AG ] consists of even cycles, or two edge disjoint odd cycles meeting at exactly one vertex, or two vertex disjoint odd cycles joined by an arbitrary path.

Proposition 2.4 ([8]) If G is bipartite, then the bases of M [AG ] are the spanning tree. If G is not bipartite, then the basis of M [AG ] are the spaning tree subgraphs of G whose connected components are unicyclic graphs with an odd cycle.

Complete intersection toric ideals

73

Definition 2.5 An elementary vector of ker(A) is a vector 0 ̸= α in ker(A) whose support is minimal with respect to inclusion, i.e., supp(α) does not properly contain the support of any other nonzero vector in ker(A). A circuit of ker(A) is an elementary vector of ker(A) with relatively prime integral entries.

It is interesting to observe that there is a one to one correspondence Circuits of ker(A) −→

Circuits of M [A] = cycles of G

given by α = (α1 , . . . , αq ) −→ C(α) = {vi | i ∈ supp(α)}.

Thus the set of circuits of the kernel of A is the algebraic realization of the set of circuits of the vector matroid M [A].

Consider the monomial subring ±1 k[D] := k[xv1 , . . . , xvq ] ⊂ k[x±1 1 , . . . , xn ]

associated to the digraph D = (G, O). There is an epimorphism of k-algebras ϕ: B = k[t1 , . . . , tq ] −→ k[D], ti &−→ xvi ,

where B is a polynomial ring. The kernel of ϕ, denoted by PD , is called the toric ideal of k[D]. Notice that PD is no longer a graded ideal of B, see Proposition 2.12.

The toric ideal PD is a prime ideal of height q − n + 1 generated by binomials (see Lemma 2.2). Thus any minimal generating set of PD must have at least q − n + 1 elements, by the principal ideal theorem.

Definition 2.6 The toric ideal PD is called a complete intersection if PD can be generated by q − n + 1 polynomials.

If 0 ̸= α ∈ ker(A) ∩ Zn we associate the binomial tα = tα+ − tα− .

Notice that tα ∈ PD .

74

Enrique Reyes

Given a cycle c of D, we split c in two disjoint sets of edges c+ and c− , where c+ is oriented clockwise and c− = c \ c+ . The binomial ! ! ti − ti tc = vi ∈c+

vi ∈c−

belongs to PD . If c+ = ∅ or c− = ∅ we set ! ! ti = 1 or ti = 1. vi ∈c+

vi ∈c−

Proposition 2.7 PD is generated by the set of all binomials tc such that c is a cycle of D and this set is a universal Gr¨ obner basis.

Proof: Let UD be the set of all binomials of the form tα such that α is a circuit of ker(A). Since A is totally unimodular, by [7, Proposition 8.11], the set UD form a universal Gr¨obner basis of PD . Notice that the circuits of ker(A) are in one to one correspondence with the circuits of the vector matroid M [A]. To complete the proof it suffices to observe that the circuits of M [A] are precisely the cycles of G, see Lemma 2.2. ✷ Proposition 2.8 Let c = {x1 , x2 , . . . , xr , x1 } be a circuit of D. Suppose that (xi , xj ) or (xj , xi ) is an edge of D, with i + 1 < j. Then tc is a linear combination of tc1 and tc2 , where c1 = {x1 , x2 , . . . , xi , xj , xj+1 , . . . , xr , x1 } and c2 = {xi , xi+1 , . . . , xj , xi }.

Proof: Suppose without loss of generality that vk = (xi , xj ) is the edge of D. Set tc1 = tα+ − tα− and tc2 = tβ+ − tβ− . We can suppose that vk ∈ c1+ ∩ c2+ because if it is false then we multiply tc1 or tc2 by −1.

Since tk |tα+ and tk |tβ+ then # "α # "β # " "β # t + t + t + α+ − tα− ) − tα+ (tβ+ − tβ− ) t t (t − = c1 c2 tk tk tk " αtk # "β # + t t + β α = t − − tk t − = tγ1 − tγ2 tk is in k[t1 , . . . , tq ], where γ1 = (α+ − ek ) + β− and γ2 = (β+ − ek ) + α− .

Then tγ1 is the product of the edges of (c1+ \ {tk }) ∪ c2− , but these are the edges of c+ . In a similar form tγ2 is the product of edges of c− .

Then tc = tγ1 − tγ2 . ✷

Definition 2.9 A cycle c of D is called primitive if c is an induced subgraph of D.

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As an immediate consequence of Propositions 2.7 and 2.8 we get: Corollary 2.10 PD is generated by the set of binomials corresponding to primitive cycles.

We say that a cycle c of D is directed if all the arrows of D are oriented in the same direction. If D does not have directed cycles, we say that D is acyclic. The following is well known: Proposition 2.11 ([2]) D is acyclic if and only if there is a linear ordering x1 , . . . , xn of the vertex set, such that every edge of D has the form (xi , xj ) with i < j.

The ordering of the last proposition is called topological ordering or topological sort. The following result is not hard to prove.

Proposition 2.12 If D has a topological ordering, then PD is generated by homogeneous binomials with respect to the order induced by degree(tk ) = j − i, where tk maps to xi xj and (xi , xj ) is an edge.

Corollary 2.13 If D is an arbitrary acyclic directed graph, then PD is a complete intersection if and only if PD is generated by q − n + 1 binomials corresponding to primitive cycles.

Proof:

3

It follows from Corollary 2.10 and Proposition 2.11.

✷

Complete intersection generator tree of G

The aim here is to show the existence of an orientation O of G such that the toric ideal of D = (G, O) is a complete intersection.

We are going to construct a proper nested sequence A1 , . . . , Am of subtrees of G label by V (Aj ) = {y1j , . . . , yrjj } such that Am is a spanning tree of G, with certain special properties.

Let A1 be a path of G maximal with respect to |V (A1 )|. Set V (A1 ) = {y11 , y21 , . . . , yr11 }. We define i1 = max{u ∈ N|NG (y11 , . . . , yu1 ) ⊂ V (A1 )}, where NG (B) stands for the neighbor set of B. If i1 < |V (A1 )| we define a1 = yi11 +1 . If i1 = |V (A1 )|, we set m = 1. By induction we define Ai

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as follows. Suppose Aj has been defined, where V (Aj ) = {y1j , . . . , yrjj }.

We define ij = max{u ∈ N|NG (y1j , . . . , yuj ) ⊂ V (Aj )}.

If ij < |V (Aj )|, we define aj = yijj +1 . Let Lj be a maximal path with respect to the cardinality, with the properties V (Lj ) ∩ V (Aj ) = {aj } and V (Lj ) = {z1j , z2j , . . . , zsjj = aj }, the final vertex of Lj is aj .

We define Aj+1 in the following way: V (Aj+1 ) = V (Aj ) ∪ V (Lj ) = } where {y1j+1 , . . . , yrj+1 j +sj −1 (1)

yij+1

⎧ j ⎪ if i ≤ ij , ⎨ yi j if ij + 1 ≤ i ≤ ij + sj , zi−ij = ⎪ ⎩ yj i−sj +1 if ij + sj + 1 ≤ i ≤ rj + sj − 1

with E(Aj+1 ) = E(Aj ) ∪ E(Lj ), and rj+1 = rj + sj − 1. If ij = |V (Aj )|, we set m = j.

Lemma 3.1 ik+1 > ik for 1 ≤ k ≤ m − 1.

Proof: By construction yik+1 = yik for i ≤ ik (see Eq.(1)). If ik+1 < ik , = yikk+1 +1 . By the definition of ik+1 , there exists then ak+1 = yik+1 k+1 +1 x ∈ V (G) \ V (Ak+1 ) ⊂ V (G) \ V (Ak ) such that {x, yik+1 } ∈ E(G). But ik+1 +1 ≤ ik , a contradiction by the k+1 +1 definition of ik . Notice that ik+1 can not be equal to ik by construction of Lk . ✷ Suppose that the process finish in the step m.

Lemma 3.2 Ai is a tree, for 1 ≤ i ≤ m.

Proof: By induction over i. For i = 1 is clear. Suppose Ai is a tree.

Recall that Li is a tree and V (Li ) ∩ V (Ai ) = {ai }. On the other hand V (Ai+1 ) = V (Ai ) ∪ V (Li ) and E(Ai+1 ) = E(Ai ) ∪ E(Li ), then Ai+1 is connected and does not have cycles. ✷ Lemma 3.3 Am is a spanning tree of G.

Complete intersection toric ideals

77

Proof: By the construction im = rm . Hence if {x, yum } ∈ E(G), for some u, then x ∈ V (Am ). Suppose that a ∈ V (G) \ V (Am ). G is connected, then there exists a path L = {b1 = y1m , b2 , . . . , bl = a}. Let j = max{u ∈ N|bu ∈ Am }. We have that j < l, then {bj , bj+1 } ∈ E(G) and bj ∈ V (Am ), but bj+1 is not in V (Am ) this is a contradiction. Then V (Am ) = V (G). ✷ In Lemma 3.3 we are essentially proving the following general fact.

Lemma 3.4 If H is a subgraph of a connected graph G and NG (V (H)) is contained in V (H), then V (G) = V (H).

Orientation of the tree Am and the graph G Let τ = (Am , O) be the digraph obtained from Am with the orientation (yim , yjm ) ∈ E(τ ) if and only if {yim , yjm } ∈ E(Am ) and j > i.

We have that V (G) = V (Am ) = {y1m , y2m , . . . , yrmm } and we will orient G to obtain the digraph D = (G, O) in the following form (yim , yjm ) ∈ E(D) if and only if {yim , yjm } ∈ E(G) and j > i.

Example 3.5 The construction of Am and the orientation O of G, is illustrated below. ! y11 !❅ ❘❅ ❅ !

1 1 ! y5 ! ✛ ! y4 ❅! y21 ❅ # ✠ # !$ ❅ ! $$ ✻# ✏✏ ✏ ❅!✏ ❡ $ # ✏✏y$ 1 $ $$ !

3 !✏✏ ! !❅ ! ❅ ! ❅! ! ! ❅ ! z22 ' !' ❅ !✏✏ ! ' *' ✏ ❅! ! ' ✏✏ $ 2 $ ✏ z $$ !

3 z12 ✻ !✏✏

! !❅

! ❅ ❅! !! ! ❅ !

1 !$ ❅ !✏✏ ! z1 $$ ✏ ❅!✏ ! $ ❄ 1 ' ✏✏z3' ✐' ' !✏✏ '! z21 ! y13 !❅❅ ❘ ❅ !

3 3 ! ✛ ! y8 ❅! y23 y9 ! # ❅ 3 ✠ # y63 ' ! ❅ ✻#✏✏ ! y3 ' *' ✏ '# ✏ ❄ ❅! ❡ ✏✏' '' ✐' y73 ✏✏ '! y43 y53 !✻

! y12 !❅ ❘ ❅ ❅ !

2 2 ! y7 ! ✛ ! y6 ❅! y22 ❅ # # ✠ !$ ❅ ! y2 $$ ✻# ✏✏ 3 ✏ ❅!✏ ❡ $ # ❄ ✏✏' '' ✐' y52 !✏✏ '! y42 ! ! ✠ ❅ ! ❘ ❅ ! ❅ ! ❅! ! ✛ ! ✛ ❅ # # ✠ ! ! ❅ # ' ✏ ' *' ✏ ✮✏ ■ ❅ ✻ ' ❅ !# ✏✏ ❄ ✏ ' '' ✻ ✶✏ ✐' ✏✏ !✏ '!

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Lemma 3.6 Let ylk1 and ylk2 be vertices in Ak such that ylk1 = yjr1 and ylk2 = yjr2 with k ≤ r ≤ m. If l1 < l2 , then j1 < j2 .

Proof: If r = k the lemma is clear. In other case, is suficient to prove and ylk2 = yqk+1 , then q2 > q1 . If l2 ≤ ik , then q2 = l2 that if ylk1 = yqk+1 1 2 and q1 = l1 . If l1 ≤ ik ≤ l2 , then q1 = l1 and q2 = l2 + sk − 1 ≥ l2 . Thus q2 > q1 . If ik < l1 then q1 = l1 + sk − 1 and q2 = l2 + sk − 1. Thus ✷ q 2 > q1 .

Lemma 3.7 {yikk +1 , yikk +2 , . . . , yrkk } is a path in Ak for 1 ≤ k ≤ m.

Proof: By induction over k. For k = 1 the lemma is clear . Suppose the result for k. We can suppose that k < m. We have that ∈ V (Ak+1 ) = V (Ak ) ∪ V (Lk ). yik+1 k+1 ∈ V (Ak ), then there exists j such that yik+1 = yjk . Thus If yik+1 k+1 +1 k+1 +1 j ≥ ik + 1 because if j ≤ ik by Eq.(1) j = ik+1 + 1, then ik > ik+1 and this is false by Lemma 3.1. Set L = {yik+1 , yik+1 , . . . , yrk+1 } = {yikk+1 −sk , . . . , yrkk }. k+1 k+1 +1 k+1 +2 Since L is a subwalk of {yikk +1 , . . . , yrkk } and the last is a path by induction hypotesis, then L is a path. Thus yik+1 ∈ / V (Ak ), and k+1 +1 yik+1 = zjk for some j and zjk ∈ V (Lk ). Let k+1 +1

, yik+1 , . . . , yik+1 } B = {zjk , . . . , zskk } = {yik+1 k+1 +1 k+1 +2 k +sk . By the induction be the path in Ak that join zj and yikk +1 = yik+1 k +sk hypothesis C = {yikk +1 , . . . , yrkk } = {yik+1 , . . . , yrk+1 } k +sk −1 k +sk } and is a path. As B ⊆ V (Ak ) and C ⊆ V (Lk ), then B ∩ C = {yik+1 k +sk k+1 k+1 {yik+1 +1 , . . . , yrk+1 } is a path. ✷ Proposition 3.8 Let yik and yjk be vertices in Ak with j > i such that {yik , yjk } ∈ E(G) \ E(Ak ). If F = {ylk1 = yik , ylk2 , . . . , ylkr = yjk } is the walk that join yik and yjk in Ak , then l1 < l2 < · · · < lr .

Complete intersection toric ideals

79

Proof: By induction over k. If k = 1 is clear. Suppose that is true for k = q, and we will prove the result for k = q + 1.

Case (a): yiq+1 , yjq+1 ∈ V (Aq ). As V (Aq ) ⊆ V (Aq+1 ) and Aq+1 , Aq are trees, if F ′ is the path that join yiq+1 and yjq+1 in Aq , then F ′ = F .

If F ′ = {ylq′ = yiq+1 , ylq′ = ylq+1 , . . . , ylq′ = yjq+1 } 2 1

2

r

by Lemma 3.6 < Then by the induction hypotesis l1′ < l2′ < · · · < lr′ . And by Lemma 3.6 l1 < l2 < · · · < lr .

Case (b): yiq+1 , yjq+1 are not in V (Aq ). As V (Aq+1 ) = V (Aq )∪V (Lq ) q then yiq+1 , yjq+1 ∈ V (Lq ) = {z1q , z2q , ..., zsqq } so yiq+1 = zi−i and yjq+1 = q l1′

l2′ .

q zj−i . The path in Aq+1 that join yiq+1 and yjq+1 is q

q+1 q q q {yiq+1 , yi+1 , . . . , yjq+1 } = {zi−i , zi−i , . . . , zj−i }. q q +1 q

Case (c): |V (Aq ) ∩ {yiq+1 , yjq+1 }| = 1. If yiq+1 ∈ V (Aq ) then yjq+1 ∈ q ∈ V (Lq ). As j > i then i ≤ iq V (Aq+1 ) \ V (Aq ), and yjq+1 = zj−i q and yiq+1 = yiq . But {yiq+1 , yjq+1 } ∈ E(G), then by the definition of iq , yjq+1 ∈ V (Aq ), this is a contradiction. So yjq+1 ∈ V (Aq ) and yiq+1 ∈ q q V (Aq+1 ) \ V (Aq ) thus yiq+1 = zi−i with zi−iq ∈ V (Lq ) and i < iq + sq . q q This implies yjq+1 = aq = yiqq +1 or yjq+1 = yj−s . In the first case q +1 q {yiq+1 , . . . , yjq+1 } = {zi−iq , . . . , zsqq = aq }

is the path that join yiq+1 and yjq+1 . In the second case j ≥ iq + sq + 1 then j − sq + 1 ≥ iq + 2, we have that , yiq+1 , . . . , yrq+1 } {yiqq +1 , yiqq +2 , . . . , yrqq } = {yiq+1 q +sq −1 q +sq q +sq +1 and by the Lemma 3.7 is a path. As j ≥ iq + sq + 1, then {yiq+1 ,..., q +sq yjq+1 } is a path, and

q {yiq+1 , . . . , yiq+1 } = {zi−iq , . . . , zsqq } q +sq

is another path and their intersection is {yiq+1 }. Since q +sq q+1 {yiq+1 , yi+1 , . . . , yiq+1 , . . . , yjq+1 } q +sq

is the path that join to yiq+1 and yjq+1 in Aq+1 .

✷

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Let τ be the tree defined after Lemma 3.4. For each fi ∈ E(D)\E(τ ) the unique cycle of the subgraph τ ∪ {fi } is denoted by c(τ, fi ). Notice that Proposition 3.8 is equivalent to the following.

Proposition 3.9 For each fi ∈ E(D) \ E(τ ) all edges of c(τ, fi ) \ {fi } are oriented in the same direction and fi is oriented in opposite direction. ! " Theorema 3.10 PD = {tc(τ,fi ) |fi ∈ E(D) \ E(τ )} .

Proof: Set E(D) \ E(τ ) = {f1 , . . . , fq−n+1 }. Suppose without loss of generality that t1 , . . . , tq−n+1 are associated to f1 , . . . , fq−n+1 respectively. By the Proposition 3.8 tc(τ,fi ) = ti − tβi , where tβi is a product of variables associated to edges in τ . Let I be the ideal generated by the set {tc(τ,fi ) |fi ∈ E(D) \ E(τ )} in B = k[t1 , . . . , tq ]. Let h = tα − tβ be a binomial in PD . Thus ti = tβi in B/I for i = 1, . . . , q − n + 1. Then h = tγ − tω , where tγ and tω are product of variables asociated to τ . As I ⊆ PD then tγ − tω ∈ PD = ker(ϕ). But τ is a tree thus tγ = tω and h = 0, in B/I. Since PD is generated by binomials, PD = I. ✷ Corollary 3.11 Let D be the directed graph constructed above. Then PD is generated by q − n + 1 primitive cycles, in particular PD is a complete intersection.

Corollary 3.12 PD is homogeneous ideal with the grading induced by deg(tk ) = j − i, where tk maps to xi xj and (xi , xj ) ∈ E(D).

Proof: By construction D is acyclic. Thus we may apply Propositions 2.11 and 2.12. ✷

Definition 3.13 A tournament D is a complete graph Kn with an orientation.

Proposition 3.14 ([2]) If D is a tournament, then D has a spanning path.

Proposition 3.15 If D is an acyclic tournament, then PD is a complete intersection minimally generated by a Gr¨ obner basis.

Complete intersection toric ideals

81

Proof: Let δ be a spanning path of D, i.e. δ = (x1 , x2 , . . . , xn ) and (xi , xi+1 ) ∈ E(D) for all i < n. Since D is acyclic it follows that PD = ({tc(δ,fi ) |fi ∈ E(D) \ E(δ)}), where for each fi ∈ E(D) \ E(δ), the unique cycle of the subgraph δ ∪ {fi } is denoted by c(δ, fi ). See the proof of Theorem 3.10. If < is the lexicographical order induced by any linear ordening of t1 , . . . , tq such that tj < ti if fi ∈ E(D) \ E(δ) and fj ∈ E(δ). Then it is seen that ✷ {tc(δ,fi ) |fi ∈ E(D) \ E(δ)} is a Gr¨obner basis.

Similarly we can prove the following: Proposition 3.16 If D is an acyclic oriented graph with a spanning directed path, then PD is a complete intersection minimally generated by a Gr¨ obner basis.

A problem here is to characterize the graphs with the property that PD is a complete intersection for all orientations of G. It has been shown that ring graphs and complete graphs have this property [5].

Acknowledgements I would like to thank Rafael H. Villarreal and Isidoro Gitler for their very valuable comments and suggestions to improve this work.

Enrique Reyes Departamento de Matem´ aticas, CINVESTAV-IPN Apartado Postal 14-740, 07000 M´exico City. ereyes@math.cinvestav.mx

References [1] Godsil C.; Royle G., Algebraic Graph Theory, Graduate Texts in Mathematics 207, Springer, 2001.

[2] Harary F., Graph Theory, Addison-Wesley, Reading, MA, 1972.

[3] Ishizeki T., Analysis of Grobner bases for toric ideals of acyclic tournament graphs, Masters thesis, The University of Tokyo, 2000.

[4] Oxley J., Matroid Theory, Oxford University Press, Oxford, 1992.

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[5] Reyes E., Toric ideals and affine varieties, blowup algebras and combinatorial optimization problems, PhD thesis, Cinvestav–IPN, 2006.

[6] Schrijver A., Theory of Linear and Integer Programming, John Wiley & Sons, New York, 1986.

[7] Sturmfels B., Gr¨obner Bases and Convex Polytopes, University Lecture Series 8, American Mathematical Society, Rhode Island, 1996.

[8] Villarreal R.H., Rees algebras of edge ideals, Comm. Algebra 23 (1995), 3513–3524.

Morfismos, Comunicaciones Estudiantiles del Departamento de Matem´ aticas del CINVESTAV, se termin´ o de imprimir en el mes de junio de 2006 en el taller de reproducci´ on del mismo departamento localizado en Av. IPN 2508, Col. San Pedro Zacatenco, M´exico, D.F. 07300. El tiraje en papel opalina importada de 36 kilogramos de 34 × 25.5 cm consta de 500 ejemplares con pasta tintoreto color verde.

Apoyo t´ecnico: Omar Hern´ andez Orozco.

Contenido Riemann and his zeta function Elena A. Kudryavtseva, Filip Saidak, and Peter Zvengrowski . . . . . . . . . . . . . . 1

Secondary operations in K-theory and the generalized vector field problem (revisited) Je s u ´ s G o nz ´ alez and Maurilio Velasco-Fuentes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

Complete intersection toric ideals of oriented graphs Enrique Reyes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71