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

  • Takashi Nakamura
Published/Copyright: October 7, 2023
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Mathematica Slovaca
From the journal Mathematica Slovaca Volume 73 Issue 5

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):=qsL(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.

2020 Mathematics Subject Classification: Primary 11M20; 11M26
Keywords: Dirichlet series with periodic coefficients; real zeros of Dirichlet L-functions; Riemann’s functional equation (Hamburger’s theorem)

(Communicated by Marco Cantarini)

Funding statement: This work was partially supported by JSPS grant 16K05077.

Acknowledgement.

The author would like to thank the referee for a careful reading of the manuscript and for their valuable comments, which helped improve the quality of the paper.

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Received: 2022-07-13
Accepted: 2022-12-06
Published Online: 2023-10-07

© 2023 Mathematical Institute Slovak Academy of Sciences

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https://doi.org/10.1515/ms-2023-0084
Keywords for this article
Dirichlet series with periodic coefficients; real zeros of Dirichlet L-functions; Riemann’s functional equation (Hamburger’s theorem)

Articles in the same Issue

  1. Remembering Professor Štefan Znám, 9.2.1936–17.7.1993
  2. Chordal and Perfect Zero-Divisor Graphs of Posets and Applications to Graphs Associated with Algebraic Structures
  3. Enlargements of Quantales
  4. Multiplicative Dependent Pairs in the Sequence of Padovan Numbers
  5. Dirichlet Series with Periodic Coefficients, Riemann’s Functional Equation, and Real Zeros of Dirichlet L-Functions
  6. On the Rational Parametric Solution of Diagonal Quartic Varieties
  7. Theory of Certain Non-Univalent Analytic Functions
  8. Initial Coefficients and Fekete-Szegő Inequalities for Functions Related to van der Pol Numbers (VPN)
  9. A Conjecture on H3(1) for Certain Starlike Functions
  10. Coefficient Problems of Quasi-Convex Mappings of Type B on the Unit Ball in Complex Banach Spaces
  11. Complete Monotonicity and Inequalities Involving the k-Gamma and k-Polygamma Functions
  12. Study of Oscillation Criteria of Odd-Order Differential Equations with Mixed Neutral Terms
  13. On a System of Difference Equations Defined by the Product of Separable Homogeneous Functions
  14. On the Existence of Bi-Lipschitz Equivalent Metrics in Semimetric Spaces
  15. The Lehmann Type II Teissier Distribution
  16. Asymptotic Predictive Inference of Negative Lower Tail Index Distributions
  17. On Numerical Problems in Computing Life Annuities Based on the Makeham–Beard Law
  18. The Rational Zero-Divisor Cup-Length of Oriented Partial Flag Manifolds
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