Mean values of the Riemann zeta function at shifted zeros under the Riemann Hypothesis
Ramūnas Garunkštis, Julija Paliulionytė
TL;DR
The paper investigates the mean values of the Riemann zeta function at shifted zeros under the Riemann Hypothesis, focusing on the sum S(δ)=∑_{0<γ<T} ζ(ρ+δ)ζ(1−ρ+δ̄). It sharpens the error term and extends the region of validity for the asymptotic formulas, deriving precise RH-based expansions in a broad δ-region and providing a corrected unconditional error bound. The results include a small-shift regime where S(δ) is governed by (|δ|^2/12)(T/2π) log^4 T, and a full-region expansion featuring terms with ζ(1+2δ1), ζ(1−2δ1), and ζ(1+δ), with an error of O(T^{1/2+ε}). A corollary ties the mean value to a main term (T/2π) Re g(δ) under suitable size restrictions on δ, highlighting the leading behavior and its dependence on the shift. Overall, the work refines prior unconditional results, clarifies error terms, and provides robust RH-based asymptotics for shifted zeta zeros.
Abstract
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.