# On the average number of divisors of reducible quadratic polynomials

## TL;DR

This paper derives an asymptotic formula for the average number of divisors of reducible quadratic polynomials, revealing that the main term's coefficient is independent of the discriminant when it is a perfect square.

## Contribution

It provides a new asymptotic formula for divisor sums of reducible quadratic polynomials and establishes effective upper bounds with a consistent main term.

## Key findings

- Main term coefficient is independent of the discriminant for perfect squares.
- Asymptotic formula for divisor sums of reducible quadratics.
- Effective upper bounds matching the main term.

## Abstract

We give an asymptotic formula for the divisor sum $\sum_{c<n\leq N}\tau\left((n-b)(n-c)\right)$ for integers $b<c$ of the same parity. Interestingly, the coefficient of the main term does not depend on the discriminant as long as it is a full square. We also provide effective upper bounds of the average divisor sum for some of the reducible quadratic polynomials considered before, with the same main term as in the asymptotic formula.