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

Takashi Nakamura

doi:10.1515/ms-2023-0084

Artikel

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

Takashi Nakamura

Veröffentlicht/Copyright: 7. Oktober 2023

Veröffentlicht von

Veröffentlichen auch Sie bei De Gruyter Brill

Informationen für Autor*innen Erkunden Sie dieses Fachgebiet

Aus der Zeitschrift Mathematica Slovaca Band 73 Heft 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.

REFERENCES

[1] Apostol, T. M.: Introduction to Analytic Number Theory. Undergraduate Texts in Math., Springer, New York, 1976.10.1007/978-1-4757-5579-4Suche in Google Scholar

[2] Bombéri, E.—Gosh, A: On the Davenport-Heilbronn function, Russian Math. Surveys 66(2) (2011), 221–270.10.1070/RM2011v066n02ABEH004740Suche in Google Scholar

[3] Cohen, H., Number Theory Vol. II: Analytic and Modern Tools. Graduate Texts in Math., Vol. 240, Springer, New York, 2007.Suche in Google Scholar

[4] Hamburger, H.: Uber die Riemannsche Funktionalgleichung der ζ-Funktion, Math. Z. 10(3–4) (1921), 240–254 (in German).10.1007/BF01211612Suche in Google Scholar

[5] Kaczorowski, J.—Perelli, A.: On the structure of the Selberg class. I. 0 ≤ d ≤ 1, Acta Math. 182(2) (1999), 207–241.10.1007/BF02392574Suche in Google Scholar

[6] Knopp, M.: On Dirichlet series satisfying Riemann’s functional equation, Invent. Math. 117(3) (1994), 361–372.10.1007/BF01232248Suche in Google Scholar

[7] Nakamura, T.: Functional equation and zeros on the critical line of the quadrilateral zeta function, J. Number Theory 233 (2022), 432–455.10.1016/j.jnt.2021.06.017Suche in Google Scholar

[8] Vaughan, R. C.: Zeros of Dirichlet series, Indag. Math. (N. S.) 26(5) (2015), 897–909.10.1016/j.indag.2015.09.007Suche in Google Scholar

[9] Saias, E.—Weingartner, A.: Zeros of Dirichlet series with periodic coefficients, Acta Arith. 140 (2009), 335–344.10.4064/aa140-4-4Suche in Google Scholar

[10] Titchmarsh, E. C.: The Theory of the Riemann Zeta-function, 2nd ed., edited and with a preface by D. R. Heath-Brown, The Clarendon Press, Oxford University Press, New York, 1986.Suche in Google Scholar

Received: 2022-07-13

Accepted: 2022-12-06

Published Online: 2023-10-07

Sie haben derzeit keinen Zugang zu diesem Inhalt.

Artikel in diesem Heft

https://doi.org/10.1515/ms-2023-0084

Schlagwörter für diesen Artikel

Dirichlet series with periodic coefficients; real zeros of Dirichlet L-functions; Riemann’s functional equation (Hamburger’s theorem)