Explicit formulae for the mean value of products of values of Dirichlet \(L\)-functions at positive integers (Q6621277)

Let \(m\geq2\) and \(f>2\). The mean value of \(|L(m, \chi)|^{2}\), where \(\chi\) ranges over the \(\phi(f)/2\) Dirichlet characters modulo \(f\) of the same parity as \(m\) (i.e., \(\chi(-1) = (-1)^{m}\)) is defined by \N\[\NM(m,f)=\frac{2}{\phi(f)}\sum_{\chi(-1) = (-1)^{m}}|L(m, \chi)|^{2}.\N\]\NIn this paper under review, the author proves in Theorem 1 that for \(m=1\) \N\[\NM(m,f)=\frac{\pi^{2}}{6}\left(\phi_{2}(f)-3\frac{\phi_{1}(f)}{f}\right)\N\]\Nand for \(m\geq2\) \N\[\NM(m,f)=\zeta(2m)\left(\phi_{2m}(f)+(-1)^{m-1}\binom{2m}{m}\sum_{k=1}^{[m/2]}\frac{m}{m-k}\binom{m}{2k}\frac{B_{2k}B_{2(m-k)}}{B_{2m}}\frac{\phi_{2k}(f)}{f^{2m-2k}}\right),\N\]\Nwhere \(\phi_{l}(f)=\prod_{p\mid f}\left(1-\frac{1}{p^{l}}\right)\), \(B_{k}\) denotes the Bernoulli rational numbers and \(\zeta\) denotes the Riemann zeta function. This implies that \(M(m,f)\) is asymptotic to \(\zeta(2m)\phi_{2m}(f)\) as \(f\) goes to infinity. Some expression of \(M(m,f)\) for \(m=2,3,4,5\) are given. Furthermore, he proves by the same argument used to prove Theorem 1 a general explicit formula of the means values for products of Dirichlet \(L\)-functions.