Note on the number of divisors of reducible quadratic polynomials

TL;DR

This paper revisits the asymptotic analysis of divisor sums related to reducible quadratic polynomials, providing an alternative proof and refining the formula by adding extra terms.

Contribution

It offers a new proof approach for the divisor sum asymptotics and extends the existing formula with additional terms, improving precision.

Findings

Reproves Lapkova's asymptotic formula using a different method.

Provides an extended asymptotic formula with more terms.

Establishes a relationship between two divisor sums for quadratic polynomials.

Abstract

In a recent paper, Lapkova uses a Tauberian theorem to derive the asymptotic formula for the divisor sum $\sum_{n \leq x} d( n (n+v))$ where $v$ is a fixed integer and $d(n)$ denotes the number of divisors of $n$. We reprove her result by following a suggestion of Hooley, namely investigating the relationship between this sum and the well-known sum $\sum_{n \leq x} d( n ) d (n+v)$. As such, we are able to furnish additional terms in the asymptotic formula.