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Open AccessArticle

Mean Values of Products of L-Functions and Bernoulli Polynomials

1
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
2
Department of Mathematics, Institute of Pure and Applied Mathematics, Chonbuk National University, 567 Baekje-daero, Deokjin-gu, Jeonju-si 54896, Korea
*
Author to whom correspondence should be addressed.
Mathematics 2018, 6(12), 337; https://doi.org/10.3390/math6120337
Received: 17 November 2018 / Revised: 12 December 2018 / Accepted: 14 December 2018 / Published: 19 December 2018
(This article belongs to the Special Issue Special Functions and Applications)
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. View Full-Text
Keywords: Dirichlet character; Bernoulli polynomials; mean value of the L-function Dirichlet character; Bernoulli polynomials; mean value of the L-function
MDPI and ACS Style

Bayad, A.; Kim, D. Mean Values of Products of L-Functions and Bernoulli Polynomials. Mathematics 2018, 6, 337.

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