Mean values of the Riemann zeta function at shifted zeros under the Riemann Hypothesis
Assuming the Riemann hypothesis, we obtain asymptotic formulas for $\sum_{0<γ<T}ζ(ρ+δ)ζ(1-ρ+\overlineδ)$ in the region $-\frac{a}{\log T} \leq \Re δ\leq \frac{1}{2}+\frac{a}{\log T}$, $|\Im δ|\ll 1$. Unconditionally, this asymptotic formula was recently obtained by Garunkštis and Novikas in essentially the same region, with a slight incompleteness. Assuming RH, we obtain a sharper error term, and we also correct an inaccuracy in the unconditional error term there.
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