On the sum of divisors of mixed powers in short intervals

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 $\mathfrak {L}_1(x,y)\asymp y^{3}x^{-2+1/k}\log x,\,\mathfrak {L}_2(x,y)\asymp y^3x^{-2+1/k}$.