Mean values of divisors of forms $n^2+Nm^2$

TL;DR

This paper evaluates the asymptotic behavior of the sum of divisor counts over quadratic forms of the type n^2 + N m^2 for various N, extending previous results for N=1 and providing new insights into divisor sums of quadratic forms.

Contribution

It extends earlier work by providing asymptotic formulas for divisor sums over quadratic forms n^2 + N m^2 for multiple N values, generalizing previous results.

Findings

Asymptotic formulas for S_N(x) for various N

Extension of Gafurov and Yu's results from N=1 to general N

New insights into divisor sums of quadratic forms

Abstract

Let $N$ be any fixed positive integer and define \begin{align*} S_N(x)=\sum_{m, n \leq x}d(n^2+Nm^2), \end{align*} where $d(n)$ is the divisor function. We evaluate asymptotically $S_N(x)$ for several $N$, extending earlier works of Gafurov and Yu on the case $N=1$.