# 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

*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

In the case of the polynomial \(f(n)=n^2+1\), we can give an improvement to Corollary 3 from [3].

## Corollary 3A

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

## 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

## 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

This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

## References

- 1.Frolenkov, D.A., Soundararajan, K.: A generalization of Pólya–Vinogradov inequality. Ramanujan J.
**31**(3), 271–279 (2013)MathSciNetCrossRefMATHGoogle Scholar - 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.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.Moser, L., MacLeod, R.A.: The error term for the squarefree integers. Can. Math. Bull.
**9**, 303–306 (1966)MathSciNetCrossRefMATHGoogle Scholar - 5.Pomerance, C.: Remarks on the Pólya–Vinogradov inequality. Integers
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