# 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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