On the average sum of the $k$-th divisor function over values of quadratic polynomials

Kostadinka Lapkova, Nian Hong Zhou

Research output: Contribution to journalArticleResearch

Abstract

Let $F({\bf x})\in\mathbb{Z}[x_1,x_2,\dots,x_n]$ be a quadratic polynomial in $n\geq 3$ variables with a nonsingular quadratic part. Using the circle method we derive an asymptotic formula for the sum $$ \Sigma_{k,F}(X; {\mathcal{B}})=\sum_{{\bf x}\in X\mathcal{B}\cap\mathbb{Z}^{n}}\tau_{k}\left(F({\bf x})\right), $$ for $X$ tending to infinity, where $\mathcal{B}\subset\mathbb{R}^n$ is an $n$-dimensional box such that $\min\limits_{{\bf x}\in X\mathcal{B}}F({\bf x})\ge 0$ for all sufficiently large $X$, and $\tau_{k}(\cdot)$ is the $k$-th divisor function for any integer $k\ge 2$.
Original languageEnglish
JournalThe Ramanujan Journal
Publication statusPublished - 17 Sep 2019

Keywords

  • math.NT
  • Primary: 11P55, Secondary: 11L07, 11N37

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