We found a match
Your institution may have access to this item. Find your institution then sign in to continue.
- Title
Dirichlet Series with Periodic Coefficients, Riemann's Functional Equation, and Real Zeros of Dirichlet L-Functions.
- Authors
Nakamura, Takashi
- 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.
- Subjects
DIRICHLET series; RIEMANN hypothesis; GAUSSIAN sums; L-functions
- Publication
Mathematica Slovaca, 2023, Vol 73, Issue 5, p1145
- ISSN
0139-9918
- Publication type
Article
- DOI
10.1515/ms-2023-0084