Uniform distribution of the zeros of the Riemann zeta function and the mean value theorems of Dirichlet L-functions
The author assumes the Riemann hypothesis and states without proof several results concerning the distribution of imaginary parts of zeta- zeros and zeros of L-functions, including some mean value theorems. For instance, if \(\Delta =2\pi \alpha /\log (T/2\pi)\) (\(\neq 0)\) is bounded, \(\alpha\) is real and \(C_ 0\) denotes Euler's constant, then \[ \sum_{0<\gamma \leq T}| \zeta (+i\gamma +i\Delta)|^ 2=(1- (\frac{\sin \pi \alpha}{\pi \alpha})^ 2)\frac{T}{2\pi}\log^ 2(T/2\pi)+ \] \[ 2\{-1+C_ 0+(1-2C_ 0)\frac{\sin 2\pi \alpha}{2\pi \alpha}+Re(\frac{\zeta '}{\zeta}(1+i\Delta)\}\frac{T}{2\pi}\log \frac{T}{2\pi}+G(T,\alpha)+O(T^{9/10} \log^ 2 T), \] which improves an earlier result of \textit{S. M. Gonek} [Invent. Math. 75, 123-141 (1984; Zbl 0531.10040)]. G(T,\(\alpha)\) \((=O(T))\) has a complicated expression, given precisely in the text.
scientific article; zbMATH DE number 4035929