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

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Number of divisors Quadratic polynomial Character sums 

Mathematics Subject Classification

Primary 11N56 Secondary 11D09 

1 Correction to: Monatsh Math

Let \(\tau (n)\) denote the number of positive divisors of the integer n. In [3], we provided an explicit upper bound for the sum \(\sum _{n=1}^N\tau \left( n^2+2bn+c\right) \) 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 \(\delta :=b^2-c\) is nonzero and square-free, and \(\delta \not \equiv 1\pmod 4\). Assume also that for \(n\ge 1\) the function f(n) is nonnegative. Then for any \(N\ge 1\) satisfying \(f(N)\ge f(1)\), and \(X:=\sqrt{f(N)}\), we have the inequality
$$\begin{aligned} \sum _{n=1}^N \tau (n^2+2bn+c)&\le \frac{2}{\zeta (2)}L(1,\chi ) N\log X \\&\quad + \left( 2.332L(1,\chi )+\frac{4M_\delta }{\zeta (2)}\right) N+\frac{2M_\delta }{\zeta (2)} X\\&\quad +4\sqrt{3}\left( 1-\frac{1}{\zeta (2)}\right) M_\delta \frac{N}{\sqrt{X}}\\&\quad +2\sqrt{3}\left( 1-\frac{1}{\zeta (2)}\right) M_\delta \sqrt{X} \end{aligned}$$
where \(\chi (n)=\left( \frac{4\delta }{n}\right) \) for the Kronecker symbol \(\left( \frac{.}{.}\right) \) and
$$\begin{aligned} M_\delta =\left\{ \begin{array}{ll} \frac{4}{\pi ^2}\delta ^{1/2}\log {4\delta }+1.8934\delta ^{1/2}+1.668, &{}\quad \mathrm{if }\,\, \delta >0;\\ &{} \\ \frac{1}{\pi }|\delta |^{1/2}\log {4|\delta |}+1.6408|\delta |^{1/2}+ 1.0285,&{}\quad \mathrm{if }\,\, \delta <0. \end{array}\right. \end{aligned}$$

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 \(N\ge 1\), we have
$$\begin{aligned} \sum _{n\le N}\tau (n^2+1)<\frac{3}{\pi }N\log N +3.0475 N+1.3586 \sqrt{N}. \end{aligned}$$

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 \(4.7\cdot 10^{58}\) in [2] and \(3.713\cdot 10^{58}\) in [3] to the following bound.

Corollary 4A

There are at most \(3.677\cdot 10^{58}\) \(D(-1)\)-quadruples.

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 \(N\ge 1\) we have
$$\begin{aligned} \sum _{n\le N}\mu ^2(n)=\frac{N}{\zeta (2)}+E_1(N), \end{aligned}$$
where \(|E_1(N)|\le \sqrt{3}\left( 1-\frac{1}{\zeta (2)}\right) \sqrt{N}<0.6793\sqrt{N}\).


This is inequality (10) from Moser and MacLeod [4]. \(\square \)

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].

Lemma 3A

$$\begin{aligned} M_\chi :=\max _{L,P}\left| \sum _{n=L}^P \chi (n)\right| \end{aligned}$$
for a primitive character \(\chi \) to the modulus \(q>1\). Then
$$\begin{aligned} M_\chi \le \left\{ \begin{array}{ll} \frac{2}{\pi ^2}q^{1/2}\log {q}+0.9467q^{1/2} +1.668\,, &{} \chi \,\, \mathrm{even};\\ &{} \\ \frac{1}{2\pi }q^{1/2}\log {q}+0.8204q^{1/2}+1.0285,&{}\chi \,\, \mathrm{odd}. \end{array}\right. \end{aligned}$$


Both inequalities for \(M_\chi \) 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 \(1\le L\le q\) and \(L=\left[ \pi ^2/4\sqrt{q}+9.15\right] \) for \(\chi \) even, \(L=\left[ \pi \sqrt{q}+9.15\right] \) for \(\chi \) odd. Thus both inequalities for \(M_\chi \) hold when \(q>25\).

Then we have a look of the maximal possible values of \(M_\chi \) when \(q\le 25\) 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 \(q\le 25\) is larger than the maximal value of \(M_\chi \) for any primitive Dirichlet character \(\chi \) of modulus q. This proves the lemma. \(\square \)



The author thanks Olivier Bordellès and Dmitry Frolenkov for their comments on [3] which led to the improvements in this addendum. The author is also very grateful to Leo Goldmakher for kindly providing the data used in the proof of Lemma 3A.

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  1. 1.
    Frolenkov, D.A., Soundararajan, K.: A generalization of Pólya–Vinogradov inequality. Ramanujan J. 31(3), 271–279 (2013)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Lapkova, K.: Explicit upper bound for an average number of divisors of quadratic polynomials. Arch. Math. (Basel) 106(3), 247–256 (2016)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Lapkova, K.: Explicit upper bound for the average number of divisors of irreducible quadratic polynomials. Monatsh. Math. (2017).
  4. 4.
    Moser, L., MacLeod, R.A.: The error term for the squarefree integers. Can. Math. Bull. 9, 303–306 (1966)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Pomerance, C.: Remarks on the Pólya–Vinogradov inequality. Integers 11, 531–542 (2011)MathSciNetCrossRefMATHGoogle Scholar

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© The Author(s) 2018

Authors and Affiliations

  1. 1.Institute of Analysis and Number TheoryGraz University of TechnologyGrazAustria

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