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On the sum of divisors of mixed powers in short intervals

  • Min Zhang
  • Jinjiang Li
Article

Abstract

Let d(n) denote the Dirichlet divisor function. Define
$$\begin{aligned} \mathcal {S}_k(x,y):= \sum _{\begin{array}{c} x-y<m_3^k\leqslant x+y \\ x-y<m_i^2\leqslant x+y\\ i=1,2 \end{array}}d(m_1^2+m_2^2+m_3^k),\qquad 3\leqslant k\in \mathbb {N}. \end{aligned}$$
Let \(y=x^{1-\delta _k+4\varepsilon }\) with \(\delta _3=\frac{2}{15},\,\delta _k=\frac{1}{k(2^{k-2}+1)}\) for \(4\leqslant k\leqslant 7\), and \(\delta _k=\frac{1}{k(k^2-k+1)}\) for \(k\geqslant 8\). In this paper, we establish an asymptotic formula of \(\mathcal {S}_k(x,y)\) and prove that
$$\begin{aligned} \mathcal {S}_k(x,y)= \mathcal {K}_1\mathfrak {L}_1(x,y)+2(\gamma \mathcal {K}_1-\mathcal {K}_2)\mathfrak {L}_2(x,y)+O\big (y^3x^{-2+\frac{1}{k}-\varepsilon }\big ), \end{aligned}$$
where \(\mathcal {K}_j\,(j=1,2)\) are two constants and \(\mathfrak {L}_j(x,y)\,(j=1,2)\) satisfying \(\mathfrak {L}_1(x,y)\asymp y^{3}x^{-2+1/k}\log x,\,\mathfrak {L}_2(x,y)\asymp y^3x^{-2+1/k}\).

Keywords

Divisor problem Circle method Mixed powers Short intervals 

Mathematics Subject Classification

11P05 11P55 11E20 11L03 

Notes

Acknowledgements

The authors would like to express the most sincere gratitude to Professor Wenguang Zhai for his valuable advice and constant encouragement.

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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsChina University of Mining and TechnologyBeijingPeople’s Republic of China

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