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

Open Access
Correction

## 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.

## References

1. 1.
Frolenkov, D.A., Soundararajan, K.: A generalization of Pólya–Vinogradov inequality. Ramanujan J. 31(3), 271–279 (2013)
2. 2.
Lapkova, K.: Explicit upper bound for an average number of divisors of quadratic polynomials. Arch. Math. (Basel) 106(3), 247–256 (2016)
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)
5. 5.
Pomerance, C.: Remarks on the Pólya–Vinogradov inequality. Integers 11, 531–542 (2011)