Analytical recurrence formulas for non-trivial zeros of the Riemann zeta function

TL;DR

This paper introduces new analytical recurrence formulas for the non-trivial zeros of the Riemann zeta function, assuming the Riemann Hypothesis, and explores their connections with prime numbers and other L-functions.

Contribution

It develops four types of recurrence formulas for zeros assuming RH, and establishes direct relations between zeros and primes, extending to Dirichlet-L functions.

Findings

Formulas enable high-precision computation of zeros

Zeros can be generated from primes and vice versa

Results extend to Dirichlet beta function

Abstract

In this article, we develop four types of analytical recurrence formulas for non-trivial zeros of the Riemann zeta function on critical line assuming (RH). Thus, all non-trivial zeros up to the $n$th order must be known in order to generate the $n$th+1 non-trivial zero. All the presented formulas are based on certain closed-form representations of the secondary zeta function family, which are already available in the literature. We also present a formula to generate the non-trivial zeros directly from primes. Thus all primes can be converted into an individual non-trivial zero, and we also give a set of formulas to convert all non-trivial zeros into an individual prime. We also extend the presented results to other Dirichlet-L functions, and in particular, we develop an analytical recurrence formula for non-trivial zeros of the Dirichlet beta function. Throughout this article, we also numerically compute these formulas to high precision for various test cases and review the computed results.