Discrete mean value theorems for the Riemann zeta-function over its shifted nontrivial zeros

Let ζ(s) be the Riemann zeta function and ρ=β+iγ its nontrivial zero. In 1984 S.M. Gonek started to consider the mean value ∑0<γ<Tζ(ρ+δ)ζ(1-ρ+øverlineδ) for imaginary δ. Here we obtain asymptotic formulas for this mean value for general complex δ.