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

Abstract

Consider the divisor sum ${\sum }_{n\le N}\tau (n^2+2bn+c)$ 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.

arXiv commentv1: Improves the result in arXiv:1509.08243, 11 pages; v2: Lemma 2 and Lemma 3 in stronger versions, thus improved results altogether. Lemma 3 gives an effective Polya-Vinogradov inequality implicit in a work of Frolenkov-Soundararajan v3: misprint corrected