Sum of the triple divisor function of mixed powers

2022, Volume 7, Issue 7: 12885-12896. doi: 10.3934/math.2022713

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Research article

1.
Department of Mathematics, Nanchang University, Nanchang 330031, Jiangxi, China
2.
Institute of Mathematics and Interdisciplinary Sciences, Nanchang University, Nanchang 330031, Jiangxi, China

Received: 05 January 2022 Revised: 17 April 2022 Accepted: 27 April 2022 Published: 05 May 2022
MSC : 11P32, 11P05, 11P55

Let $ d_3(n) $ denote the 3-th divisor function. In this paper, we study the asymptotic formula of the sum

$ \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) $

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.

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