Non-zero values of a family of approximations of a class of $L$-functions

TL;DR

This paper investigates the distribution of non-zero $a$-values of a family of approximations to the Riemann zeta function, showing that none of these $a$-values actually lie on the critical line, contrasting with previous results on zeros.

Contribution

It proves that 0 ext{ }% of non-zero $a$-values of the approximation lie on the critical line, extending the analysis to broader $L$-function approximations.

Findings

0 ext{ }% of non-zero $a$-values lie on the critical line

$a$-values cluster near the critical line but do not lie on it

Results extend to wider class of $L$-function approximations

Abstract

Consider the approximation $\tilde{Z}_N(s) = \sum_{n=1}^N n^{-s} + \chi(s) \sum_{n=1}^N n^{1-s}$ of the Riemann zeta function $\zeta(s)$, where $\chi(s)$ is the ratio of the gamma functions. This arise from the approximate functional equation of $\zeta(s)$. Gonek and Montgomery have shown that $\tilde{Z}_N(s)$ has 100\% of its zeros lie on the critical line. Recently, $a$-values of $\tilde{Z}_N(s)$ for non-zero complex number $a$ are studied and it has been shown that the $a$-values of $\tilde{Z}_N(s)$ are cluster arbitrarily close to the critical line. In this paper, we show that, despite the above, 0\% of non-zero $a$-values of $\tilde{Z}_N(s)$ actually lie on the critical line itself. For $\zeta(s)$ at most $50\%$ non-zero $a$-values lie on the critical line is known due to Lester. We also extend our results to approximations of a wider class of $L$-functions.