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# Mean Values of Products of L-Functions and Bernoulli Polynomials

by Abdelmejid Bayad 1 and Daeyeoul Kim 2,*
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
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