Dirichlet series with periodic coefficients, Riemann's functional equation and real zeros of Dirichlet $L$-functions

Abstract

In this paper, we give Dirichlet series with periodic coefficients that have Riemann's functional equation and real zeros of Dirichlet $L$-functions. The details are as follows. Let $L(s,\chi)$ be the Dirichlet $L$-function and $G(\chi)$ be the Gauss sum associate with a primitive Dirichlet character $\chi$ (${\rm{mod}} \,\, q$). Put $f (s,\chi) := q^s L(s,\chi) + i^{-\kappa (\chi)} G(\chi) L(s,\overline{\chi})$, where $\overline{\chi}$ is the complex conjugate of $\chi$ and $\kappa (\chi) :=(1-\chi (-1))/2$. Then we prove that $f (s,\chi)$ satisfies Riemann's functional equation appearing in Hamburger's theorem if $\chi$ is even. In addition, we show that $f (\sigma,\chi) \ne 0$ all $\sigma \ge 1$. Moreover, we prove that $f(\sigma,\chi) \ne 0$ for all $1/2 \le \sigma < 1$ if and only if $L(\sigma,\chi) \ne 0$ for all $1/2 \le \sigma < 1$. When $\chi$ is real, all zeros of $f(s,\chi)$ with $\Re (s) >0$ are on the line $\sigma =1/2$ if and only if GRH for $L(s,\chi)$ is true. However, $f (s,\chi)$ has infinitely many zeros off the critical line $\sigma =1/2$ if $\chi$ is non-real.