Note on the number of divisors of reducible quadratic polynomials

Abstract

In a recent paper, Lapkova uses a Tauberian theorem to derive the asymptotic formula for the divisor sum ${\sum }_{n\le x}d(n(n+v))$ 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 ${\sum }_{n\le x}d(n)d(n+v)$. As such, we are able to furnish additional terms in the asymptotic formula.