Mean Values of Products of L-Functions and Bernoulli Polynomials

Let m1, · · · , mr be nonnegative integers, and set: Mr = m1 + · · ·+ mr. In this paper, first we establish an explicit linear decomposition of:


Notations and Introduction
As usual, we define the Bernoulli polynomials B n (x) through the generating function: and the Bernoulli numbers are given by B n = B n (0).Moreover, from (1), we have: B 2k+1 = 0 for all k 1, and Let q be a positive integer, we denote by χ a Dirichlet character modulo q, and L(s, χ) a Dirichlet L-series is given by: for Re(s) > 0 if χ is non-principal and Re(s) > 1 if χ is the principal character modulo q.
For a Dirichlet character χ modulo q, the generalized Bernoulli numbers: B n,χ ∈ Q(χ(1), χ(2), . . ., χ(q − 1)) are given through the generating function: Therefore, we have: The main interest of the numbers B n,χ is that they give the value at non-positive integers of Dirichlet L-series.In fact, there is a well-known formula proven by Hecke in [1]: We are motivated by the arithmetic properties satisfied by the finite product of several generalized Bernoulli numbers B n,χ and also the product of Bernoulli polynomials.
Nielsen gave the following important result for the product of two Bernoulli polynomials.Theorem 1 ([2], p. 75).For m 1 , m 2 ≥ 1, we have the formula: At the same time, Nörlund [3] gave formulas for the integral of the product of two Bernoulli polynomials.
Theorem 2 ([3], p. 31).For m 1 , m 2 ≥ 1, we have the formula: Note that Theorem 2 can be obtained directly from Theorem 1.However, the proofs of Nörlund and Nielsen are different.
Carlitz [7] studied the integrals of the product of three and four Bernoulli polynomials.Furthermore, the results by Carlitz-Mikolas-Mordell-Nörlund were generalized by Hu, Kim, and Kim in [8] as follows.
Similar integral evaluations have also been studied by Espinosa and Moll [9][10][11].The purpose of this paper is to prove a generalization of Nielsen's Theorem 1 and to study the mean values of L-functions at negative integers and their connections to Bernoulli-Dedekind sums.
We consider: 2! , ..., As the first goal of this paper, we prove explicit formulas for the coefficients a k (m 1 , ..., m d ) with 0 ≤ k ≤ M d .In the second part of this paper, we establish relationships between these coefficients and mean values of Dirichlet L-series at negative integers.

Statement of the Main Results
Now, we state our main results.Theorem 4. Let m 1 , . . ., m d 1 be positive integers.We have the explicit formula: where: We have our second main result.
Theorem 5. Let q ≥ 2, m 1 , . . ., m d 0 be nonnegative integers and d ≥ 2. We have the mean values: with: where ϕ is the Jordan totient function, J k (q) = q k ∏ p prime|q 1 − p −k is the k th Jordan function, the summation is over all Dirichlet characters χ 1 , • • • , χ d modulo q, and the coefficients a j are given by Formula (9).
Theorem 5 can be viewed as a complement to the recent work of Bayad-Raouj [12] on the Mean values of L-functions at positive integers.
We can restate Theorem 5 in terms of the generalized Bernoulli numbers B n,χ as follows.Theorem 6.Let q ≥ 2, m 1 , . . ., m d 0 be non-negative integers and d ≥ 2. We have the mean values: Let us state some special cases of Theorem 5.

Proof of Theorem 4
We start this section with some useful lemmas.

Three Lemmas
Let m 1 , ..., m d be non-negative integers and: From Equation (1), we obtain the following results.
x k i .Then, we have: Lemma 3. Let d be a positive integer.Then, we have:

Proof of Theorem 4
By Lemmas 1, 2, and 3, we have: where M I = ∑ i∈I m i and I = {1, • • • , d}\I.Then, we get: On the other hand, we consider: Therefore, we obtain: . By Lemma 2, we have: and: Thus, we get: Using this identity, we get the following.
Proposition 1.Let d be a positive integer.Then, we obtain: Using these results, we have: We have also: with: We finish the proof of Theorem 4 by using Proposition 1 and Theorem 7.

Further Examples and Consequences of Theorem 4
We restate Theorem 4 explicitly in the cases d = 2, 3, 4, and we get some new recurrence formulas for Bernoulli numbers.
Example 1.For d = 2, we have: We thus recover Nielsen's theorem.We give its generalized formulation in the cases d = 3, 4.
Example 2. For d = 3, with M 3 = m 1 + m 2 + m 3 , we obtain: , we have: where: From Example 1 and Example 2, with m 1 = m 2 = m 3 = m, we have the recurrence formulas: Corollary 3.For m a positive integer greater than one, we have:
Therefore, we have the following.