A discrete mean value of the Riemann zeta function

Abstract

In this work, we estimate the sum \begin{align*} \sum_{0 < \Im(ρ) \leq T} ζ(ρ+α)X(ρ) Y(1\!-\! ρ) \end{align*} over the nontirival zeros $\rho$ of the Riemann zeta funtion where $\alpha$ is a complex number with $\alpha \ll 1/\log T$ and $X(\cdot )$ and $Y(\cdot )$ are some Dirichlet polynomials. Moreover, we estimate the discrete mean value above for higher derivatives where $\zeta (\rho +\alpha )$ is replaced by ${\zeta }^{(m)}(\rho )$ for all $m\in \mathbb{N}$. The formulae we obtain generalize a number of previous results in the literature. As an application, assuming the Riemann Hypothesis we obtain the lower bound \begin{align*} \sum_{0 < \Im(ρ) < T} | ζ^{(m)}(ρ)|^{2k} \gg T(\log T)^{k^2+2km+1} \quad \quad (k,m\in\mathbb{N}) \end{align*} which was previously known under the Generalized Riemann Hypothesis, in the case $m=1$.

42 pages, comments welcome