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

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Keywords

Number of divisors Quadratic polynomial Character sums 

Mathematics Subject Classification

Primary 11N56 Secondary 11D09 

1 Correction to: Monatsh Math  https://doi.org/10.1007/s00605-017-1061-y

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 \(|E_1(N)|\le \sqrt{3}\left( 1-\frac{1}{\zeta (2)}\right) \sqrt{N}<0.6793\sqrt{N}\).

Proof

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

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

Proof

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. \(\square \)

Notes

Acknowledgements

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.

Open Access

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References

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    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).  https://doi.org/10.1007/s00605-017-1061-y
  4. 4.
    Moser, L., MacLeod, R.A.: The error term for the squarefree integers. Can. Math. Bull. 9, 303–306 (1966)MathSciNetCrossRefMATHGoogle Scholar
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    Pomerance, C.: Remarks on the Pólya–Vinogradov inequality. Integers 11, 531–542 (2011)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© The Author(s) 2018

Authors and Affiliations

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

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