## Functiones et Approximatio Commentarii Mathematici

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

#### 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

#### 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

#### References

• X. Cao, Y. Tanigawa and W. Zhai, On a conjecture of Chowla and Walum, Science China Mathematics 53 (2010), 2755–2771.
• X. Cao, Y. Tanigawa and W. Zhai, Tong-type identity and the mean square of the error term for an extended Selberg class, to appear in Science China Mathematics, see arXiv:1501.04269.
• K. Corrádi and I. Kátai, A comment of K. S. Ganggadharan's paper entitled “Two classical lattice point problems”, Magyar Tud. Akad. Mat. Fiz.Oszt. Közl. 17 (1967), 89–97.
• H. Cramér, Über zwei Sätze von Hern G. H. Hardy, Math. Z. 15 (1922), 201–210.
• J.L. Hafner, New omega theorems for two classical lattice point problems, Invent. Math. 63 (1981), 181-186.
• M.N. Huxley, Exponential sums and lattice points III, Proc. London Math. Soc. 87 (2003), 591–609.
• A. Ivić, The Riemann Zeta-Function, John Wiley and Sons, 1985.
• A. Ivić, The general divisor problem, J. Number Theory 27 (1987), 73–91.
• A. Ivić, E. Krätzel, M. Kühleitner and W. G. Nowak, Lattice points in large regions and related arithmetic functions: recent developments in a very classic topic,(English summary) Elementare und analytische Zahlentheorie, 89–128, Schr.Wiss. Ges. Johann Wolfgang Goethe Univ.Frankfurt am Main, 20, Franz Steiner Verlag Stuttgart, Stuttgart, 2006.
• E. Krätzel, Lattice Points, Kluwer Academic Publishers, Dordrecht 1988.
• Y.-K. Lau and K.-M. Tsang, On the mean square formula of the error term in the Dirichlet divisor problem, Math. Proc. Camb. Phil. Soc. 146 (2009), 277–287.
• K.-C. Tong, On divisor problem III, Acta Math. Sinica 6 (1956), 515–541.
• W. Zhai and X. Cao, On the mean square of the error term for the asymmeric two-dimensional divisor problem (I), Monatsh. Math. 159 (2010), 185–209.