Dirichlet Series with Periodic Coefficients, Riemann’s Functional Equation, and Real Zeros of Dirichlet L-Functions

Abstract

ABSTRACT In this paper, we provide 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, χ) be the Dirichlet L-function and G(χ) be the Gauss sum associated with a primitive Dirichlet character χ (mod q). We define f ( s , χ ) : = q s L ( s , χ ) + i − κ ( χ ) G ( χ ) L ( s , χ ¯ ) , where χ ¯ is the complex conjugate of χ and κ(χ) := (1 – χ(−1))/2. Then, we prove that f (s, χ) satisfies Riemann’s functional equation in Hamburger’s theorem if χ is even. In addition, we show that f (σ, χ) ≠ 0 for all σ ≥ 1. Moreover, we prove that f(σ, χ) ≠ 0 for all 1/2 ≤ σ < 1 if and only if L(σ, χ) ≠ 0 for all 1/2 ≤ σ < 1. When χ is real, all zeros of f (s, χ) with ℜ ( s ) > 0 are on the line σ = 1/2 if and only if the generalized Riemann hypothesis for L(s, χ) is true. However, f (s, χ) has infinitely many zeros off the critical line σ = 1/2 if χ is non-real.