Moments of Hardy's function over short intervals

TL;DR

This paper investigates the behavior of Hardy's function over short intervals, providing bounds and asymptotic formulas related to the zeros of the Riemann zeta function, assuming the Riemann hypothesis.

Contribution

It establishes new bounds for integrals involving Hardy's function and its derivative, and generalizes previous results on the distribution of zeta zeros over short intervals.

Findings

Bounds for integrals involving Z(t) and Z'(t) under the Riemann hypothesis

Asymptotic estimates for sums over zeros of zeta function

Generalization of Milinovich's results on zeta zeros

Abstract

Let as usual $Z(t) = \zeta(1/2+it)\chi^{-1/2}(1/2+it)$ denote Hardy's function, where $\zeta(s) = \chi(s)\zeta(1-s)$. Assuming the Riemann hypothesis upper and lower bounds for some integrals involving $Z(t)$ and $Z'(t)$ are proved. It is also proved that $$ H(\log T)^{k^2} \ll_{k,\alpha} \sum_{T<\gamma\le T+H}\max_{\gamma\le \tau_\gamma\le \gamma^+} |\zeta(1/2 + i\tau_\gamma)|^{2k} \ll_{k,\alpha} H(\log T)^{k^2}. $$ Here $k>1$ is a fixed integer, $\gamma, \gamma^+$ denote ordinates of consecutive complex zeros of $\zeta(s)$ and $T^\alpha \le H \le T$, where $\alpha$ is a fixed constant such that $0<\alpha \le 1$. This sharpens and generalizes a result of M.B. Milinovich.