On the mean square of the divisor function in short intervals

Abstract

We provide upper bounds for the mean square integral $$ \int_X^{2X}(\Delta_k(x+h) - \Delta_k(x))^2 dx \qquad(h = h(X)\gg1, h = o(x) {\roman{as}} X\to\infty) $$ where $h$ lies in a suitable range. For $k\ge2$ a fixed integer, $\Delta_k(x)$ is the error term in the asymptotic formula for the summatory function of the divisor function $d_k(n)$, generated by $\zeta^k(s)$.