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Mean Values of Products of L -Functions and Bernoulli Polynomials

Author

Listed:
  • Abdelmejid Bayad

    (Laboratoire de Mathematiques et Modélisation d’Évry (UMR 8071), Université d’Évry Val d’Essonne, Université Paris-Saclay, I.B.G.B.I., 23 Bd. de France, 91037 Évry CEDEX, France)

  • Daeyeoul Kim

    (Department of Mathematics, Institute of Pure and Applied Mathematics, Chonbuk National University, 567 Baekje-daero, Deokjin-gu, Jeonju-si 54896, Korea)

Abstract

Let m 1 , ⋯ , m r be nonnegative integers, and set: M r = m 1 + ⋯ + m r . In this paper, first we establish an explicit linear decomposition of: ∏ i = 1 r B m i ( x ) m i ! in terms of Bernoulli polynomials B k ( x ) with 0 ≤ k ≤ M r . Second, for any integer q ≥ 2 , we study the mean values of the Dirichlet L -functions at negative integers: ∑ χ 1 , ⋯ , χ r ( mod q ) ; χ 1 ⋯ χ r = 1 ∏ i = 1 r L ( − m i , χ i ) where the summation is over Dirichlet characters χ i modulo q . Incidentally, a part of our work recovers Nielsen’s theorem, Nörlund’s formula, and its generalization by Hu, Kim, and Kim.

Suggested Citation

  • Abdelmejid Bayad & Daeyeoul Kim, 2018. "Mean Values of Products of L -Functions and Bernoulli Polynomials," Mathematics, MDPI, vol. 6(12), pages 1-11, December.
    • Handle: RePEc:gam:jmathe:v:6:y:2018:i:12:p:337-:d:191784
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