On the Number of Divisors of $n^2 -1$

Abstract

We prove an asymptotic formula for the sum $\sum_{n \leq N} d(n^2 - 1)$, where $d(n)$ denotes the number of divisors of $n$. During the course of our proof, we also furnish an asymptotic formula for the sum $\sum_{d \leq N} g(d)$, where $g(d)$ denotes the number of solutions $x$ in $\mathbb{Z}_d$ to the equation $x^2 \equiv 1 \mod d$.