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arXiv · 2512.03297

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

Ramūnas Garunkštis·Julija Paliulionytė
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Abstract

Assuming the Riemann hypothesis, we obtain asymptotic formulas for $\sum_{0<\gamma<T}\zeta(\rho+\delta)\zeta(1-\rho+\overline{\delta})$ in the region $-\frac{a}{\log T} \leq \Re \delta \leq \frac{1}{2}+\frac{a}{\log T}$, $|\Im \delta|\ll 1$. Unconditionally, this asymptotic formula was recently obtained by Garunk\v{s}tis 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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Ramūnas Garunkštis, Julija Paliulionytė. 2025-12-02. Mean values of the Riemann zeta function at shifted zeros under the Riemann Hypothesis. https://arxiv.org/abs/2512.03297

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