Study of coefficients of finite Dirichlet series vanishing at some zeros of Riemann's zeta function

Abstract

In 2013, Yu. V. Matiyasevich numerically studied finite Dirichlet series vanishing at multiple non-trivialzeros of the Riemann zeta function. He fixed the first coefficient of these series to 1 and observedthat the initial coefficients of the considered Dirichlet series closely resemble those of the alternating zeta function.This study was extended by fixing multiple coefficients of such finite Dirichlet series, and patterns were identified in the remaining initial coefficients. Specifically, these coefficients approximate the coefficients of the product ζ(s) · ^f+1_k=1 Σ b_k k^-s , where f denotes the number of fixed coefficients, and the b_k values are determined from the series coefficients.The results of this analysis provide new insights into the relationship between the Riemann zeta function and number theory.