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

TL;DR

This paper derives an asymptotic formula for the sum of the number of divisors of n^2 - 1 up to N, and also for the sum of solutions to x^2 ≡ 1 mod d, advancing understanding of divisor sums and modular solutions.

Contribution

It provides new asymptotic formulas for divisor sums involving quadratic expressions and solutions to quadratic congruences, extending previous results in number theory.

Findings

Asymptotic formula for ∑_{n ≤ N} d(n^2 - 1)

Asymptotic formula for ∑_{d ≤ N} g(d)

Enhanced understanding of divisor functions and quadratic congruences

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 $x^2 \equiv 1 \mod d$.