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

Aleksandar Ivić¹

¹ Katedra Matematike RGF-a Universiteta u Beogradu Dušina 7 11000 Beograd, Serbia

Acta Arithmetica, Tome 162 (2014) no. 2, pp. 141-158

Résumé

Let $\varDelta(x)$ 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 $E^*(t) := E(t) - 2\pi\varDelta^*(t/(2\pi))$ with $\varDelta^*(x) = -\varDelta(x) + 2\varDelta (2x) - \frac12\varDelta (4x)$ and $\int_0^T E^*(t)\,d t = \frac{3}{4}\pi T + R(T)$, 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 formulae for integrals of the form $$ \int_T^{2T}\Big(\int_{t-H}^{t+H}|\zeta(1/2+iu|^2\,d u\Big)^k\,d t\quad\ (k\in\mathbb{N},\, 1 \ll H \le T) $$ are also treated.

Détail

DOI : 10.4064/aa162-2-2

Keywords: vardelta denote error term dirichlet divisor problem denote error term asymptotic formula mean square zeta * vardelta * vardelta * vardelta vardelta frac vardelta int * frac obtain number results involving moments zeta short intervals connecting moments * short intervals upper bounds asymptotic formulae integrals form int int t h zeta quad mathbb treated