On a certain sum of the derivative of Dirichlet \(L\)-functions (Q2080661)

The paper produces an asymptotic sum of the derivatives of Dirichlet \(L\)-functions over their zeros. Theorem. Let \(c_1\) be a positive constant and \(\chi\) a primitive character modulo \(q\). Then, uniformly for \(q\leq \exp(c_1\sqrt{\log T})\), as \(T\to\infty\), we have \[ \sum_{0<\gamma_\chi\leq T} L'(\rho_\chi, \chi)=\frac{1}{4\pi}T\left(\log \frac{qT}{2\pi}\right)^2+a_1\frac{T}{2\pi}\log \frac{qT}{2\pi}+a_2\frac{T}{2\pi}+a_3 \] \[ +O(T \exp (-c\sqrt{\log T})), \] where the implicit constant is absolute, \(c\) is a positive absolute constant depending on \(c_1\) and \[ a_1 = \sum_{p|q}\frac{\log p}{p-1} + \gamma_0-1, \] \[ a_2 = \frac{1}{2}\left(\sum_{p|q}\frac{\log p}{p-1}\right)^2+( \gamma_0-1)\sum_{p|q}\frac{\log p}{p-1}-\frac{3}{2}\sum_{p|q}p\left(\frac{\log p}{p-1}\right)^2+1-\gamma_0-\gamma_0^2+3\gamma_1 \] with the Stieltjes constants \(\gamma_0\), \(\gamma_1\) and \[ a_3 = \frac{\omega\chi(-1)\tau(\overline{\chi})\tau(\overline{\omega}\chi)}{q\phi(q)}\frac{L'(\beta, \omega)}{\beta}\left(\frac{qT}{2\pi}\right)^\beta \] when \(L(s, \omega)\) with a quadratic character \(\omega\ (\mathrm{mod}\ q)\) has an exceptional zero \(\beta\), otherwise \(a_3=0\). Assuming the GRH, for \(\varepsilon > 0\), we can replace the error term by \((qT)^{1/2+\varepsilon}\), uniformly for \(q\ll T^{1-\varepsilon}\).