Approximation of Riemann's zeta function by finite Dirichlet series: multiprecision numerical approach

TL;DR

This paper introduces a numerical method using finite Dirichlet series to approximate Riemann's zeta function, revealing remarkable properties of the coefficients with potential implications for understanding the zeta function.

Contribution

The paper presents a novel multiprecision numerical approach to approximate the zeta function using finite Dirichlet series that vanish at initial zeros, uncovering unexpected number-theoretical properties.

Findings

Finite Dirichlet series provide highly accurate approximations within the critical strip.

Coefficients exhibit remarkable number-theoretical properties.

No current theoretical explanation for observed phenomena.

Abstract

The finite Dirichlet series from the title are defined by the condition that they vanish at as many initial zeroes of the zeta function as possible. It turned out that such series can produce extremely good approximations to the values of Riemann's zeta function inside the critical strip. In addition, the coefficients of these series have remarkable number-theoretical properties discovered in large scale high accuracy numerical experiments. So far no theoretical explanation to the observed phenomena was found.