Mean Values of Products of L-Functions and Bernoulli Polynomials

Bayad, Abdelmejid; Kim, Daeyeoul

doi:10.3390/math6120337

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Mean Values of Products of L-Functions and Bernoulli Polynomials

Abdelmejid Bayad

¹ and

Daeyeoul Kim

^2,*

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

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)

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Abstract

Let

m_{1}, \dots, m_{r}

be nonnegative integers, and set:

M_{r} = m_{1} + \dots + m_{r} .

In this paper, first we establish an explicit linear decomposition of:

\prod_{i = 1}^{r} \frac{B_{m_{i}} (x)}{m_{i}!}

in terms of Bernoulli polynomials

B_{k} (x)

with

0 \leq k \leq M_{r}

. Second, for any integer

q \geq 2

, we study the mean values of the Dirichlet L-functions at negative integers:

\sum_{χ_{1}, \dots, χ_{r} (\mod q); χ_{1} \dots χ_{r} = 1} \prod_{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

This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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MDPI and ACS Style

Bayad, A.; Kim, D. Mean Values of Products of L-Functions and Bernoulli Polynomials. Mathematics 2018, 6, 337. https://doi.org/10.3390/math6120337

AMA Style

Bayad A, Kim D. Mean Values of Products of L-Functions and Bernoulli Polynomials. Mathematics. 2018; 6(12):337. https://doi.org/10.3390/math6120337

Chicago/Turabian Style

Bayad, Abdelmejid, and Daeyeoul Kim. 2018. "Mean Values of Products of L-Functions and Bernoulli Polynomials" Mathematics 6, no. 12: 337. https://doi.org/10.3390/math6120337

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