Mean Values of Products of <i>L</i>-Functions and Bernoulli Polynomials

Abstract

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.

Keywords

• Dirichlet character
• Bernoulli polynomials
• mean value of the L-function