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Sum of divisors of a quaternary quadratic form with almost equal variables

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Abstract

In this paper, we obtain an asymptotic formula of the sum

$$\begin{aligned} \sum _{|m_i-x|\le x^{\theta }}d\big (m_1^2+m_2^2+m_3^2+m_4^2\big ) \quad (i=1,2,3,4) \end{aligned}$$

with \(\theta =\frac{1}{3}+\epsilon \). Moreover, if \(\theta =\frac{4}{5}+\epsilon \), we obtain a more refined asymptotic formula of this sum.

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Acknowledgments

The authors wish to express their sincere appreciation to the referees for their careful reading of the manuscript.

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Authors and Affiliations

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

    Liqun Hu

  2. Department of Mathematics, Shandong University, Jinan, 250100, Shandong, China

    Liqun Hu & Huafeng Liu

Authors
  1. Liqun Hu
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  2. Huafeng Liu
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Correspondence to Huafeng Liu.

Additional information

This work was supported in part by the Youth Foundation of Educational Commission of Jiangxi province of China (Grant No. GJJ14213) and the Natural Science Foundation of Jiangxi province of China (Grant No. 20151BAB211008).

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Hu, L., Liu, H. Sum of divisors of a quaternary quadratic form with almost equal variables. Ramanujan J 40, 557–571 (2016). https://doi.org/10.1007/s11139-015-9758-9

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  • DOI: https://doi.org/10.1007/s11139-015-9758-9

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