An arithmetic interpretation of generalized Li's criterion

TL;DR

This paper provides an arithmetic interpretation of the generalized Li's criterion, which is equivalent to the Riemann hypothesis, linking sums over non-trivial zeros of the Riemann zeta function to arithmetic properties.

Contribution

It offers a new arithmetic perspective on the generalized Li's criterion, deepening understanding of its connection to the Riemann hypothesis.

Findings

Established the equivalence of the generalized Li's criterion to the Riemann hypothesis.

Provided an arithmetic interpretation of the sums over non-trivial zeros.

Enhanced the theoretical framework connecting zero sums to number theory.

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,a=Sum_rho(1-(1-((rho-a)/(rho+a-1))^n) for any real a not equal to 1/2 are non-negative if and only if the Riemann hypothesis holds true; arXiv:1304.7895 (2013); Ukrainian Math. J., V.66, N3, 371-383 (2014). An arithmetic interpretation of this generalized Li's criterion is given here.