Explicit upper bound for an average number of divisors of quadratic polynomials

TL;DR

This paper derives an explicit near-optimal 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 divisor sums of quadratic polynomials and enhances the known maximum size of D(-1)-quadruples.

Findings

Derived an explicit upper bound close to optimal for divisor sums of quadratic polynomials.

Improved the maximum possible number of D(-1)-quadruples.

Applied the bound to advance results in Diophantine problem classifications.

Abstract

Consider the divisor sum $\sum_{n\leq N}\tau(n^2+2bn+c)$ for integers $b$ and $c$ which satisfy certain extra conditions. For this average sum we obtain an explicit upper bound, which is close to the optimal. As an application we improve the maximal possible number of $D(-1)$-quadruples.