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)