On mean square of the error term of a multivariable divisor function

Zhen Guo; Zhen Guo

doi:10.3934/math.20241415

AIMS Mathematics

2024, Volume 9, Issue 10: 29197-29219. doi: 10.3934/math.20241415

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On mean square of the error term of a multivariable divisor function

Zhen Guo ^,

Department of Mathematics, China University of Mining and Technology, Beijing 100083, China

Received: 29 August 2024 Revised: 09 October 2024 Accepted: 10 October 2024 Published: 15 October 2024
MSC : 11N37

Let $ \tau(n) $ be the Dirichlet divisor function and $ k\geqslant2 $ be a fixed integer. We give an asymptotic formula of the mean square of
$ \begin{equation*} \Delta_k(x) = \sum\limits_{n_1, \cdots, n_k\leqslant x}\tau(n_1 \cdots n_k)-x^kP_k(\log x). \end{equation*} $
- divisor function,
- mean square,
- Dirichlet series
Citation: Zhen Guo. On mean square of the error term of a multivariable divisor function[J]. AIMS Mathematics, 2024, 9(10): 29197-29219. doi: 10.3934/math.20241415

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Abstract

Let $ \tau(n) $ be the Dirichlet divisor function and $ k\geqslant2 $ be a fixed integer. We give an asymptotic formula of the mean square of

$ \begin{equation*} \Delta_k(x) = \sum\limits_{n_1, \cdots, n_k\leqslant x}\tau(n_1 \cdots n_k)-x^kP_k(\log x). \end{equation*} $

References

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