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

Abstract

Consider the divisor sum 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 -quadruples.

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

“…The improvements announced above are achieved by using more powerful explicit estimates than the ones used in [3]. More precisely, the results are obtained when instead of Lemma 2 and Lemma 3 from [3] we plug in the proof the following stronger results.…”

“…More precisely, the results are obtained when instead of Lemma 2 and Lemma 3 from [3] we plug in the proof the following stronger results.…”

“…Let τ (n) denote the number of positive divisors of the integer n. In [3], we provided an explicit upper bound for the sum N n=1 τ n 2 + 2bn + c under certain conditions on the discriminant, and we gave an application for the maximal possible number of…”

“…Just as in [2,3], we have an application of the latter inequality in estimating the maximal possible number of D(−1)-quadruples, whereas it is conjectured there are none. We can reduce this number from 4.7 · 10 58 in [2] and 3.713 · 10 58 in [3] to the following bound.…”

“…The aim of this addendum is to announce improvements in the results from [3] . We start with sharpening of Theorem 2 [3].…”