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

[This corrects the article DOI: 10.1007/s00605-017-1061-y.].


Mathematics Subject Classification Primary 11N56; Secondary 11D09
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 D(−1)-quadruples.
The aim of this addendum is to announce improvements in the results from [3] . We start with sharpening of Theorem 2 [3]. Theorem 2A Let f (n) = n 2 +2bn +c for integers b and c, such that the discriminant δ := b 2 − c is nonzero and square-free, and δ ≡ 1 (mod 4). Assume also that for n ≥ 1 the function f (n) is nonnegative. Then for any N ≥ 1 satisfying f (N ) ≥ f (1), In the case of the polynomial f (n) = n 2 + 1, we can give an improvement to Corollary 3 from [3].

Corollary 3A For any integer
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 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.

Lemma 2A For any integer
Proof This is inequality (10) from Moser and MacLeod [4].
The following numerically explicit Pólya-Vinogradov inequality is essentially proven by Frolenkov and Soundararajan [1], though it was not formulated explicitly. It supersedes the main result of Pomerance [5], which was formulated as Lemma 3 in [3].
Proof Both inequalities for M χ are shown to hold by Frolenkov and Soundararajan in the course of the proof of their Theorem 2 [1] as long as a certain parameter L satisfies 1 ≤ L ≤ q and L = π 2 /4 √ q + 9.15 for χ even, L = π √ q + 9.15 for χ odd. Thus both inequalities for M χ hold when q > 25.
Then we have a look of the maximal possible values of M χ when q ≤ 25 from a data sheet, provided by Leo Goldmakher. It represents the same computations of Bober and Goldmakher used by Pomerance [5]. We see that the right-hand side of the bounds of Frolenkov-Soundararajan for any q ≤ 25 is larger than the maximal value of M χ for any primitive Dirichlet character χ of modulus q. This proves the lemma.