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

Artur Kawalec

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arXiv · 2012.06581

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

Artur Kawalec

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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.

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Artur Kawalec. 2020-12-13. Analytical recurrence formulas for non-trivial zeros of the Riemann zeta function. https://arxiv.org/abs/2012.06581

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