On some mean value results for the zeta-function in short intervals

TL;DR

This paper investigates mean value results for the Riemann zeta-function in short intervals, linking them to error terms in divisor problems and deriving bounds and formulas for moments of |e(1/2+it)|.

Contribution

It introduces new connections between zeta-function moments and divisor problem error terms, providing bounds and asymptotic formulas for short interval integrals.

Findings

Derived bounds for moments of |e(1/2+it)| in short intervals.

Established asymptotic formulas for integrals involving zeta in short intervals.

Connected zeta moments to divisor problem error terms.

Abstract

Let $\Delta(x)$ denote the error term in the Dirichlet divisor problem, and let $E(T)$ denote the error term in the asymptotic formula for the mean square of $|\zeta(1/2+it)|$. If $E^*(t) := E(t) - 2\pi\Delta^*(t/(2\pi))$ with $\Delta^*(x) := -\Delta(x) + 2\Delta(2x) - \frac{1}{2}\Delta(4x)$ and $\int_0^T E^*(t)\,dt = \frac{3}{4}\pi T + R(T)$, then we obtain a number of results involving the moments of $|\zeta(1/2+it)|$ in short intervals, by connecting them to the moments of $E^*(T)$ and $R(T)$ in short intervals. Upper bounds and asymptotic formulas for integrals of the form $$ \int_T^{2T}\left(\int_{t-H}^{t+H}|\zeta(1/2+iu)|^2\,du\right)^k\,dt \qquad(k\in N, 1 \ll H \le T) $$ are also treated.