Generating functions for the generalized Li's sums

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))