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

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.