# Explicit upper bound for the average number of divisors of irreducible quadratic polynomials

## TL;DR

This paper derives an explicit upper bound for the average number of divisors of quadratic polynomials and applies it to improve bounds on certain Diophantine quadruples.

## Contribution

It provides a new explicit upper bound for the divisor sum of quadratic polynomials and applies this to enhance bounds on D(-1)-quadruples.

## Key findings

- Established an asymptotic formula for the divisor sum
- Provided an explicit upper bound matching the main term
- Improved the maximal possible number of D(-1)-quadruples

## Abstract

Consider the divisor sum $\sum_{n\leq N}\tau(n^2+2bn+c)$ for integers $b$ and $c$. We extract an asymptotic formula for the average divisor sum in a convenient form, and provide an explicit upper bound for this sum with the correct main term. As an application we give an improvement of the maximal possible number of $D(-1)$-quadruples.