Sums of Higher Divisor Functions with Almost Equal Variables

Abstract

Let k, ℓ ≥ 3 be integers, and let τ_k(n) denote the k-th divisor function. In this paper, we apply the circle method to obtain an asymptotic formula for the sum

\sum _{|m_i-X|Y}{\tau }_k(m_1^2+m_2^2+\cdots +m_{\ell }^2)(i=1,2,\dots ,\ell )

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\sum_{|m_{i}-X|\leq Y}\ \ \tau_{k}(m_{1}^{2}+m_{2}^{2}+\cdots+m_{\ell}^{2})\ \ (i=1,2,\ldots,\ell)$$\end{document}

with

Y=X^{1-\frac{8}{5(k+1)}+20\epsilon }

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$Y = {X^{1 - {8 \over {5(k + 1)}} + 20\varepsilon }}$$\end{document}

tending to infinity. Previously, only the case k = 2 was considered.

Keywords

Higher divisor functions / almost equal variables / circle method

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Peking University