Generating functions for the generalized Li's sums

TL;DR

This paper derives generating functions for the generalized Li's sums related to the Riemann zeta function's zeros, providing a new analytical tool to study the Riemann hypothesis through these sums.

Contribution

It introduces generating functions for the generalized Li's sums, linking their Taylor expansions to the Riemann hypothesis and extending previous criteria.

Findings

Derived explicit generating functions for generalized Li's sums.

Connected the non-negativity of sums to the Riemann hypothesis.

Provided analytical expressions to test zero distribution of the Riemann zeta function.

Abstract

Recently, in arXiv:1304.7895; Ukrainian Math. J. - 2014.- 66. - P. 371 - 383, we presented the generalized Li's criterion. This is the statement that the sums /lambda_(n, b, /sigma)=Sum_(rho)((1-((/rho+b)/(/rho-b-2*/sigma))^n)), taken over all Riemann xi-function zeroes taking into account their multiplicity (complex conjugate zeroes are to be paired when summing whenever necessary) for any n=1, 2, 3... and any real b>(-/sigma), are non-negative if and only if there are no Riemann function zeroes with Re(b)>/sigma. For any n=1, 2, 3... and any real b<(-/sigma), these sums are non-negative if and only if there are no Riemann function zeroes with Re(b)</sigma; correspondingly, for /sigma=1/2 and b not equal to 1/2 such non-negativity is equivalent to the Riemann hypothesis. In this Note we obtain generation functions for this generalized criterion demonstrating the Taylor expansion (b is not equal to (-/sigma)): Ln(/xi(b+2*/sigma+(2*b+2*/sigma)*z/(1-z))) =ln(/xi(b+2*/sigma))+Sum_(n=1)^(infinity)(/lambda_(n, b, /sigma)*z^n/n))