Explicit formulas for sums related to Dirichlet L-functions

Brahim Mittou
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 28, 2022, Number 4, Pages 744–748
DOI: 10.7546/nntdm.2022.28.4.744-748
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Authors and affiliations

Brahim Mittou  
Department of Mathematics, University Kasdi Merbah, Ouargla
EDPNL & HM Laboratory, ENS of Kouba, Algiers, Algeria

Abstract

Let p\geq3 be a prime number and let m, n and l be integers with \gcd(l,p)=1. Let \chi be a Dirichlet character modulo p and L(s,\chi) be the Dirichlet L-function corresponding to \chi. Explicit formulas for:

    \[\dfrac{2}{p-1} \sum \limits\sb{\underset{\chi(-1)=+1}{\chi\hspace{-0.2cm} \mod p}} \chi(l) L(m,\chi)L(n,\overline{\chi}) \text{ and }\dfrac{2}{p-1} \sum \limits\sb{\underset{\chi(-1)=-1}{\chi\hspace{-0.2cm} \mod p}} \chi(l) L(m,\chi)L(n,\overline{\chi})\]

are given in this paper by using the properties of character sums and Bernoulli polynomials.

Keywords

  • Character sum
  • Dirichlet L-function
  • Bernoulli number
  • Generalized Bernoulli number

2020 Mathematics Subject Classification

  • 11M06
  • 11B68

References

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  2. Arakawa, T., Ibukiyama, T., & Kaneko, M. (2014). Bernoulli Numbers and Zeta Functions. Springer, Japan.
  3. Liu, H. (2015). On the mean values of Dirichlet L-functions. Journal of Number Theory, 147, 172–183.
  4. Louboutin, S. (1993). Quelques formules exactes pour des moyennes de fonctions L de Dirichlet. Bulletin canadien de mathématiques, 36(2), 190–196.
  5. Louboutin, S. (2015). Twisted quadratic moments for Dirichlet L-functions. Bulletin of the Korean Mathematical Society, 52(6), 2095–2105.
  6. Louboutin, S. (2019). Twisted quadratic moments for Dirichlet L-functions at s = 2. Publicationes Mathematicae Debrecen, 95(3–4), 393–400.
  7. Walum, H. (1982). An exact formula for an average of L-series. Illinois Journal of
    Mathematics, 26(1), 1–3.

Manuscript history

  • Received: 11 May 2022
  • Revised: 20 August 2022
  • Accepted: 10 November 2022
  • Online First: 11 November 2022

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Cite this paper

Mittou, B. (2022). Explicit formulas for sums related to Dirichlet L-functions. Notes on Number Theory and Discrete Mathematics, 28(4), 744-748, DOI: 10.7546/nntdm.2022.28.4.744-748.

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