Mean square of the error term in the asymmetric many dimensional divisor problem

Abstract

Let $\ba=(a_1,a_2,\ldots,a_k)$, where $a_j(j=1,\dots ,k)$ are positive integers such that $a_1\le a_2\le \cdots \le a_k$. Let $d(\ba;n)=\sum_{n_1^{a_1}\cdots n_k^{a_k}=n}1$ and $Δ(\ba;x)$ be the error term of the summatory function of $d(\ba;n)$. In this paper we show an asymptotic formula of the mean square of $Δ(\ba;x)$ under a certain condition. Furthermore, in the cases $k=2$ and 3, we give unconditional asymptotic formulas for these mean squares.

References in corpus (1)

• Lattice points in large regions and related arithmetic functions: Recent developments in a very classic topic