On the Sum of Divisors of Mixed Powers

TL;DR

This paper derives an asymptotic formula for the sum of the divisor function over mixed powers involving sums of squares and k-th powers, revealing detailed growth behavior depending on the parameter k.

Contribution

The paper establishes a new asymptotic expression for the sum of divisors over a specific mixed power form, including explicit error bounds and constants depending on k.

Findings

Asymptotic formula for alS_k(x) with main terms involving x^{3/2+1/k} and al log x

Explicit constants C_1(k) and C_2(k) depending on k

Error term with explicit elta_k depending on k

Abstract

Let $d(n)$ denote the Dirichlet divisor function. Define \begin{equation*} \mathcal{S}_{k}(x)=\sum_{\substack{1\leqslant n_1,n_2,n_3 \leqslant x^{1/2} \\ 1\leqslant n_4\leqslant x^{1/k} }} d(n_1^2+n_2^2+n_3^2+n_4^k), \qquad 3\leqslant k\in \mathbb{N}. \end{equation*} In this paper, we establish an asymptotic formula of $\mathcal{S}_k(x)$ and prove that \begin{equation*} \mathcal{S}_k(x)=C_1(k)x^{3/2+1/k}\log x+C_2(k)x^{3/2+1/k}+O(x^{3/2+1/k-\delta_k+\varepsilon}), \end{equation*} where $C_1(k),\,C_2(k)$ are two constants depending only on $k,$ with $\delta_3=\frac{19}{60},\,\delta_4=\frac{5}{24},\,\delta_5=\frac{19}{140},\,\delta_6=\frac{25}{192},\, \delta_7=\frac{457}{4032},\,\delta_k=\frac{1}{k+2}+\frac{1}{2k^2(k-1)}$ for $k\geqslant8.$