2020/08/06 by Takashi Nakamura, Nakamura, Takashi
Mathematics · #11M20 #11M26 #Analytic Number Theory Research #FOS: Mathematics #Mathematical Dynamics and Fractals #Meromorphic and Entire Functions #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.2008.02570
openalex publication_date 2020/08/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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,χ) be the Dirichlet L-function and G(χ) be the Gauss sum associate with a primitive Dirichlet character χ (\rmmod q). Put f (s,χ) := qs 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 appearing in Hamburger's theorem if χ is even. In addition, we show that f (σ,χ) ≠ 0 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 \Re (s) >0 are on the line σ=1/2 if and only if GRH 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.