Mean Values of Products of L-Functions and Bernoulli Polynomials

Let $m_1,\cdots ,m_r$ be nonnegative integers, and set: $$M_r=m_1+\cdots +m_r.$$ In this paper, first we establish an explicit linear decomposition of: $${\displaystyle \prod _{i=1}^r\frac{B_{m_i}\left(x\right)}{m_i!}}$$ in terms of Bernoulli polynomials $B_k\left(x\right)$ with $0\le k\le M_r$ . Second, for any integer $q\ge 2$ , we study the mean values of the Dirichlet L-functions at negative integers: $${\displaystyle \sum _{{\chi }_1,\cdots ,{\chi }_r\left(\mathrm{mod}q\right);{\chi }_1\cdots {\chi }_r=1}\prod _{i=1}^{r}L(-m_i,{\chi }_i)}$$ where the summation is over Dirichlet characters ${\chi }_i$ 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