Sum Rules for Functions of the Riemann Zeta Type

TL;DR

This paper develops generalized sum rules for functions of the Riemann zeta type using infinite product representations and exponential convergence acceleration, linking zeros to Taylor coefficients, and discusses implications for the Riemann hypothesis.

Contribution

It introduces a more general approach to sum rules for zeta-type functions, extending previous work by Lehmer and Keiper, and explores new conditions related to the Riemann hypothesis.

Findings

Derived sum rules connecting zeros and coefficients of zeta-type functions.

Proposed a new sufficient condition for the Riemann hypothesis.

Enhanced convergence techniques for analyzing zeta functions.

Abstract

We consider analytic functions of the Riemann zeta type, for which, if $s$ is a zero, so is $1-s$. We use infinite product representations of these functions, assuming their zeros to be of first order. We use exponential factors to accelerate convergence, and by comparing the exponential factors with the Taylor series coefficients about $s=0$ of the original function, connect these coefficients with sums of powers of reciprocals of the zeros, in the form of sum rules. Such sum rules have been previously considered by Lehmer and Keiper, but the approach and applications taken here are more general. In related work, a new sufficient condition is found for the Riemann hypothesis, and the basis for this condition is discussed.