The discrete mean square of Dirichlet \(L\)-function at integral arguments
Let \(q\geq 3\) be an integer, \(\mu\) the Möbius function, \[ J_k(q)=\sum_{d\mid q}\mu\left(\frac{q}{d}\right)d^k. \] The authors prove: Theorem 1. For a non-negative integer \(n\), we have \[ \begin{multlined} \sum_{\chi\bmod q}|L(-n,\chi)|^2=\varphi(q)q^{2n}\frac{B_{2n+2}}{(n+1)^2}J_{-2n-1}(q)\\ +\frac{2\varphi(q)q^{2n}}{n+1}\sum_{r=0}^{n-1}\binom{n+1}{r}\frac{B_{n-r+1}B_{n+r+1}}{n+r+1}J_{-n-r}(q)\\ +(-1)^n\varphi^2(q)q^{2n}\frac{1}{(n+1)^2}\frac{B_{2n+2}}{\binom{2n+2}{n+2}}, \end{multlined} \] where \(B_l\) denote the Bernoulli numbers and \(\varphi(q)\) Euler's function. Theorem 2. For integers \(N,k>1\) we have \[ \begin{multlined}\sum_{\chi\bmod q}|L(k,\chi)|^2=\frac{\varphi(q)}{q^{2k}}J_{2k}(q)\zeta(2k) +\frac{2\varphi(q)}{q^{2k}} \sum_{\substack{ r=0\\r\neq k-1}}^N\binom{k+r-1}{r-1}\zeta(k+r)\zeta(k-r)J_{k-r}(q)\\ +\frac{2(-1)^{k-1}}{q^{2k}}\binom{2k-2}{k-1}\zeta(2k-1)\varphi^2(q) \times\left(\log q+\sum_{p\mid q}\frac{\log p}{p-1} -\frac{\zeta'}{\zeta}(2k-1)+\gamma-\sum_{n=k}^{2k-2}\frac{1}{n}\right)\\ +\frac{\varphi(q)}{q^{2k}}R_N(q), \end{multlined} \] where \(\gamma\) is Euler's constant, \[ \begin{multlined} R_N(q)=\sum_{d\mid q}\mu\left(\frac{q}{d}\right) \sum_{r=0}^{k-1}\binom{k+r-1}{r}\binom{r-k}{N-k+1}\frac{1}{d^{N-k}}\\ \times\sum_{m=1}^\infty\frac{1}{m^{k+N}}\int_1^\infty\bar{B}_{N-k+1}(dmz)z^{r-N-1}\,dz, \end{multlined} \] where \(\bar{B}_k(x)=B_k(x-[x])\), \(B_k(x)\) the Bernoulli polynomials.
scientific article; zbMATH DE number 5871102