An Analytic Form for Riemann Zeta Function at Integer Values

Abstract

An original definition of the generalized Euler-Mascheroni constants allowed us to prove that their infinite sum converges to the number (1-Ln2) . By considering this number is the Lebesgue measure of a set defined as the difference between the area of the square unit and the area under the curve y=1/x 1≤x≤2 ; we introduced a partition of this set such that each Lebesgue measure of the subsets can be related to values of Riemann zeta function at integers. From this relationship, we proved that the Lambert W function can produce all ζ(s)  values whatever is the parity of s . Finally, by considering that ζ(s)  values allow calculation of the probability, for s  integers chosen in an interval [1,n] n∈N , to be coprime; we proved that Lambert W function can describe prime numbers distribution.