Asymptotic of the generalized Li's sums which non-negativity is equivalent to the Riemann Hypothesis

TL;DR

This paper analyzes the asymptotic behavior of generalized Li's sums, which are linked to the Riemann Hypothesis, providing explicit formulas under the assumption that RH is true.

Contribution

It derives the asymptotic form of generalized Li's sums for large n assuming the Riemann Hypothesis, extending previous criteria.

Findings

Asymptotic formula for large n under RH

Explicit dependence on parameter b in sums

Confirmation of non-negativity equivalence to RH

Abstract

Recently, we have established the generalized Li's criterion equivalent to the Riemann hypothesis, viz. demonstrated that the sums over all non-trivial Riemann function zeroes k_n,b=Sum_rho(1-(1-((rho+b)/(rho-b-1))**n) for any real b not equal to -1/2 are non-negative if and only if the Riemann hypothesis holds true; arXiv:1304.7895 (2013); Ukrainian Math. J., 66, 371 - 383, 2014. (Famous Li's criterion corresponds to the case b=0 (or b=1) here). This makes timely the detailed studies of these sums, and in particular also the study of their asymptotic for large n. This question, assuming the truth of RH, is answered in the present Note. We show that on RH, for large enough n, for any real b not equal to -1/2, one has: k_n,b=Sum_rho(1-(1-((rho+b)/(rho-b-1))**n)=0.5*abs(2b+1)*n*ln(n)+0.5*abs(2b+1)*(gamma-1-ln(2*pi/abs(2b+1))*n+o(n), where gamma is Euler-Mascheroni constant.