Mean values of products of L-functions and Bernoulli polynomials

Abdelmejid Bayad; Daeyeoul Kim

doi:10.3390/math6120337

Mean values of products of L-functions and Bernoulli polynomials

Abdelmejid Bayad
, Daeyeoul Kim^*

^*Corresponding author for this work

Department of Mathematics

University of Evry

Research output: Contribution to journal › Journal article › peer-review

2 Scopus citations

Abstract

Let m₁,..., m_r be nonnegative integers, and set: M_r=m₁+...+m_r In this paper, first we establish an explicit linear decomposition of: 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: where the summation is over Dirichlet characters X_i modulo q. Incidentally, a part of our work recovers Nielsen's theorem, Nörlund's formula, and its generalization by Hu, Kim, and Kim.

Original language	English
Article number	337
Journal	Mathematics
Volume	6
Issue number	12
DOIs	https://doi.org/10.3390/math6120337
State	Published - 2018.12.19

Keywords

Bernoulli polynomials
Dirichlet character
Mean value of the L-function

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title = "Mean values of products of L-functions and Bernoulli polynomials",

abstract = "Let m1,..., mr be nonnegative integers, and set: Mr=m1+...+mr In this paper, first we establish an explicit linear decomposition of: in terms of Bernoulli polynomials Bk(x) with 0 ≤ k ≤ Mr. Second, for any integer q ≥ 2, we study the mean values of the Dirichlet L-functions at negative integers: where the summation is over Dirichlet characters Xi modulo q. Incidentally, a part of our work recovers Nielsen's theorem, N{\"o}rlund's formula, and its generalization by Hu, Kim, and Kim.",

keywords = "Bernoulli polynomials, Dirichlet character, Mean value of the L-function",

author = "Abdelmejid Bayad and Daeyeoul Kim",

note = "Publisher Copyright: {\textcopyright} 2018 by the authors.",

year = "2018",

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day = "19",

doi = "10.3390/math6120337",

language = "English",

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T1 - Mean values of products of L-functions and Bernoulli polynomials

AU - Bayad, Abdelmejid

AU - Kim, Daeyeoul

PY - 2018/12/19

Y1 - 2018/12/19

N2 - Let m1,..., mr be nonnegative integers, and set: Mr=m1+...+mr In this paper, first we establish an explicit linear decomposition of: in terms of Bernoulli polynomials Bk(x) with 0 ≤ k ≤ Mr. Second, for any integer q ≥ 2, we study the mean values of the Dirichlet L-functions at negative integers: where the summation is over Dirichlet characters Xi modulo q. Incidentally, a part of our work recovers Nielsen's theorem, Nörlund's formula, and its generalization by Hu, Kim, and Kim.

AB - Let m1,..., mr be nonnegative integers, and set: Mr=m1+...+mr In this paper, first we establish an explicit linear decomposition of: in terms of Bernoulli polynomials Bk(x) with 0 ≤ k ≤ Mr. Second, for any integer q ≥ 2, we study the mean values of the Dirichlet L-functions at negative integers: where the summation is over Dirichlet characters Xi modulo q. Incidentally, a part of our work recovers Nielsen's theorem, Nörlund's formula, and its generalization by Hu, Kim, and Kim.

KW - Bernoulli polynomials

KW - Dirichlet character

KW - Mean value of the L-function

UR - https://www.scopus.com/pages/publications/85059472461

U2 - 10.3390/math6120337

DO - 10.3390/math6120337

M3 - Journal article

AN - SCOPUS:85059472461

SN - 2227-7390

VL - 6

JO - Mathematics

JF - Mathematics

IS - 12

M1 - 337

ER -

Mean values of products of L-functions and Bernoulli polynomials

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Keywords

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