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June 2016 Mean square of the error term in the asymmetric multidimensional divisor problem
Xiaodon Cao, Yoshio Tanigawa, Wenguang Zhai
Funct. Approx. Comment. Math. 54(2): 173-193 (June 2016). DOI: 10.7169/facm/2016.54.2.4

Abstract

Let $\mathbf{a}=(a_1,\cdots,a_k)$ denote a $k$-tuple of positive integers such that $a_1 \leq a_2 \leq\break \cdots \leq a_k$. We put $d(\mathbf{a};n)=\sum_{n_1^{a_1}\cdots n_k^{a_k}=n}1$ and let $\Delta(\mathbf{a};x)$ be the error term of the corresponding asymptotic formula for the summatory function of $d(\mathbf{a};n)$. In this paper we show an asymptotic formula of the mean square of $\Delta(\mathbf{a};x)$ under a certain condition. Moreover, when $k$ equals $2$ or $3$, we give unconditional asymptotic formulas for these mean squares.

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Xiaodon Cao. Yoshio Tanigawa. Wenguang Zhai. "Mean square of the error term in the asymmetric multidimensional divisor problem." Funct. Approx. Comment. Math. 54 (2) 173 - 193, June 2016. https://doi.org/10.7169/facm/2016.54.2.4

Information

Published: June 2016
First available in Project Euclid: 20 June 2016

zbMATH: 06862342
MathSciNet: MR3513577
Digital Object Identifier: 10.7169/facm/2016.54.2.4

Subjects:
Primary: 11N37

Keywords: asymmetric multidimensional divisor problem , Dirichlet series , functional equation , mean square of the error term , the Tong-type representation

Rights: Copyright © 2016 Adam Mickiewicz University

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Vol.54 • No. 2 • June 2016
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