Sum of the triple divisor function of mixed powers-Reference-Cited by-同舟云学术

Sum of the triple divisor function of mixed powers

Published:2022 Issue:7 Volume:7 Page:12885-12896
ISSN:2473-6988
Container-title:AIMS Mathematics
language:
Short-container-title:MATH

Author:

Zhou Li¹,Hu Liqun¹²

Affiliation:

1. Department of Mathematics, Nanchang University, Nanchang 330031, Jiangxi, China

2. Institute of Mathematics and Interdisciplinary Sciences, Nanchang University, Nanchang 330031, Jiangxi, China

Abstract

<abstract>Let $ d_3(n) $ denote the 3-th divisor function. In this paper, we study the asymptotic formula of the sum <disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \sum\limits_{\substack{1 \leqslant n_1,n_2 \leqslant X^{\frac{1}{2}} \\ 1 \leqslant n_3 \leqslant X^{\frac{1}{k}}}} d_3(n_1^2+n_2^2+n_3^k) $\end{document} </tex-math> </disp-formula> with $ n_1, n_2, n_3\in \mathbb{Z}^+ $ and $ k \geqslant 3 $ be an integer. Previously only the case of $ k=2 $ is studied.</abstract>

Publisher

American Institute of Mathematical Sciences (AIMS)

Subject

General Mathematics

Reference10 articles.

1. C. E. Chace, The divisor problem for arithmetic progressions with small modulus, Acta Arith., 61 (1992), 35–50.

2. C. E. Chace, Writing integers as sums of products, Trans. Am. Math. Soc., 345 (1994), 367–379. https://doi.org/10.1090/S0002-9947-1994-1257641-3

3. R. T. Guo, W. G. Zhai, Some problems about the ternary quadratic form $m_1^2 +m_2^2 +m_3^2$, Acta Arith., bf 156 (2012), 101–121.

4. L. Q. Hu, An asymptotic formula related to the divisors of the quaternary quadratic form, Acta Arith., 166 (2014), 129–140. https://doi.org/10.4064/aa166-2-2

5. L. Q. Hu and L. Yang, Sums of the triple divisor function over values of a quaternary quadratic form, Acta Arith., 183 (2018), 63–85. https://doi.org/10.4064/aa170120-20-10