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

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.