Functiones et Approximatio Commentarii Mathematici

Mean square of the error term in the asymmetric multidimensional divisor problem

Xiaodon Cao, Yoshio Tanigawa, and Wenguang Zhai

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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.

Article information

Source
Funct. Approx. Comment. Math., Volume 54, Number 2 (2016), 173-193.

Dates
First available in Project Euclid: 20 June 2016

Permanent link to this document
https://projecteuclid.org/euclid.facm/1466450666

Digital Object Identifier
doi:10.7169/facm/2016.54.2.4

Mathematical Reviews number (MathSciNet)
MR3513577

Zentralblatt MATH identifier
06862342

Subjects
Primary: 11N37: Asymptotic results on arithmetic functions

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

Citation

Cao, Xiaodon; Tanigawa, Yoshio; Zhai, Wenguang. Mean square of the error term in the asymmetric multidimensional divisor problem. Funct. Approx. Comment. Math. 54 (2016), no. 2, 173--193. doi:10.7169/facm/2016.54.2.4. https://projecteuclid.org/euclid.facm/1466450666


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References

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