The mean square of the product of a Dirichlet \(L\)-function and a Dirichlet polynomial (Q2285753)

For a prime \(q\), let \(\chi\) mod \(q\) be an even primitive character and \(L(s, \chi) = \sum_{n=1}^{\infty} \chi(n)n^{-s}\) the associated Dirichlet \(L\)-function. The completed \(L\)-function is defined as: \[\Lambda\left(\frac{1}{2} + s, \chi\right) = \left(\frac{q}{\pi}\right)^{\frac{s}{2}}\Gamma \left(\frac{1}{4} + \frac{s}{2}\right) L\left(\frac{1}{2} + s, \chi\right) \]. Also, let \(W\) denote a fixed \(C^{\infty}\) function, with compact support on \([1, 2]\) and let \(\alpha\) and \(\beta\) be small real numbers, called \textit{shifts}. Define \[\Delta_{\alpha, \beta}(h, k; Q):=\sum_q W\left(\frac{q}{Q}\right)\sum_{\chi\ \bmod q}^\flat \Lambda\left(\frac{1}{2} + \alpha, \chi\right) \Lambda\left(\frac{1}{2} + \beta, \overline{\chi}\right) \chi(h)\overline{\chi}(k),\] the symbol \(\flat\) indicating summation over even primitive characters. Main result. Theorem. Let \(Q\) be large and the shifts \(\alpha\) and \(\beta\) are \(\ll 1 / \log Q\). Then \[\Delta_{\alpha, \beta}(h, k; Q)=\sum_{(q, hk)=1} W\left(\frac{q}{Q}\right)\left(\sum_{\chi\bmod q}^\flat 1\right) \] \[\times \Big(\left(\frac{q}{\pi}\right)^{\frac{\alpha + \beta}{2}}\Gamma\left(\frac{1}{4} + \frac{\alpha}{2}\right) \Gamma\left(\frac{1}{4} + \frac{\beta}{2}\right) \frac{(h,k)^{1+\alpha + \beta}}{h^{\frac{1}{2}+\beta} k^{\frac{1}{2}+\alpha}}\zeta_q(1+\alpha+\beta)\] \[+ \left(\frac{q}{\pi}\right)^{\frac{-\alpha - \beta}{2}}\Gamma\left(\frac{1}{4} - \frac{\alpha}{2}\right) \Gamma\left(\frac{1}{4} - \frac{\beta}{2}\right) \frac{(h,k)^{1-\alpha - \beta}}{h^{\frac{1}{2}-\alpha} k^{\frac{1}{2}-\beta}}\zeta_q(1-\alpha-\beta) \Big) + \xi_{h,k},\] where \[\zeta_q(s) = \zeta(s)\prod_{p|q}\left(1 - \frac{1}{p^s}\right)\] and the remainder terms \(\xi_{h,k}\) satisfy \[\sum_{h,k\leq Q^{\upsilon}}\frac{\lambda_h\overline{\lambda_k}}{\sqrt{hk}}\xi_{h,k} = O(Q^{2-(1-\upsilon)/2+\varepsilon}),\] uniformly for arbitrary complex numbers \(\lambda_h\) with \(\lambda_h \ll h^\varepsilon\), and \(\upsilon < 1\).