Mean values of the Riemann zeta function at shifted zeros under the Riemann Hypothesis

Assuming the Riemann hypothesis, we obtain asymptotic formulas for ${\sum }_{0<\gamma <T}\zeta (\rho +\delta )\zeta (1-\rho +\bar{\delta })$ in the region $-\frac{a}{\log T}\le \mathrm{\Re }\delta \le \frac{1}{2}+\frac{a}{\log T}$, $\mathrm{\mid }\mathrm{\Im }\delta \mathrm{\mid }\ll 1$. 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.