ON A FUNCTION IN THE NUMBERS THEORY

Page 1

ON A FUNCTION IN THE NUlVIBERS THEORY by Ion Cojocaru and Sorin Cojocaru

Abstract. In the present paper we study some series concerning the following function of the Numbers Theory [1]:

"s : N->N

such that Sen) is the smallest k with property that k! is

divisible by n". 1. Introduction. The following functions in Numbers Theory are well - known : the

function J..l.(n) of Mobius, the function A(n) ofMangoldt

(A(n) = {

~(s) of Riemann (s(s) = f ~,s=cr+it e

log~f'ifn=pr

) .

0,1 n:;t:

etc.

n=1

C),

the function

n

The purpose of this paper is to study some series concerning the following function of the Numbers Theory "[1] S : N->N such that Sen) is the smallest integer k with the propriety that k! is divisible by n". We first prove the divergence of some series involving the S function, using an unitary method, and then we prove that the series

f

1

0=2

S(2)S(3) ... S(n)

is convergent to a number S e

(711100, 1011100) and we study some applications of this series in the Numbers Theory .

Then we prove that series

f

n=2

- 1S, is convergent to a real numbers s e(0.717, 1.253) (n).

and that the sum of the remarkable series

L

Sen) is a irrational number.

~n!

2. The main results

Proposition 1. If (X n)r21 is strict increasing sequence of natural numbers, then the senes:

L co

11=1

Xn+l - Xn S(Xn) ,

(1)

is divergent.

Proof. We consider the function f[xn, xn+d -+R, defined by f(x) = In In x is meets the conditions of the Lagrange's theorem of finite increases. Therefore there is Cn e (xn, Xn+l) such that: InInXiM-l -Inlnx n =

/

Cn nCn

(X n -

X n+l).

Because Xn < Cn < Xn+l, we have: Xn+l - Xn Xn+l - Xn In <lnInxn+l-lnInx n < I ,(V)neN, Xn+l Xn+l Xn n Xn

(2) (3)

ifxn:;t:. 1. We know that for each n eN*\{l}, 0< Sen) < _1_ nInn - Inn'

S~n) ~ 1, i.e. (4)


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ON A FUNCTION IN THE NUMBERS THEORY by Don Hass - Issuu