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

Correction to: Monatsh Math 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 $\delta \not\equiv 1(mod4)$. 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

$$\begin{array}{cc}\hfill \sum _{n=1}^N\tau (n^2+2bn+c)& \le \frac{2}{\zeta (2)}L(1,\chi )NlogX\hfill \\ \hfill & +\left(2.332L(1,\chi )+\frac{4M_{\delta }}{\zeta (2)}\right)N+\frac{2M_{\delta }}{\zeta (2)}X\hfill \\ \hfill & +4\sqrt{3}\left(1-\frac{1}{\zeta (2)}\right)M_{\delta }\frac{N}{\sqrt{X}}\hfill \\ \hfill & +2\sqrt{3}\left(1-\frac{1}{\zeta (2)}\right)M_{\delta }\sqrt{X}\hfill \end{array}$$

where $\chi (n)=\left(\frac{4\delta }{n}\right)$ for the Kronecker symbol $\left(\frac{.}{.}\right)$ and

$$\begin{array}{c}\hfill M_{\delta }=\left\{\begin{array}{cc}\frac{4}{{\pi }^2}{\delta }^{1/2}log4\delta +1.8934{\delta }^{1/2}+1.668,\hfill & \mathrm{if}\delta >0;\hfill \\ \hfill & \hfill \\ \frac{1}{\pi }{|\delta |}^{1/2}log4|\delta |+1.6408{|\delta |}^{1/2}+1.0285,\hfill & \mathrm{if}\delta <0.\hfill \end{array}\right.\end{array}$$

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

$$\begin{array}{c}\hfill \sum _{n\le N}\tau (n^2+1)<\frac{3}{\pi }NlogN+3.0475N+1.3586\sqrt{N}.\end{array}$$

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.

Corollary 4A

There are at most $3.677·{10}^{58}$ $D(-1)$-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

$$\begin{array}{c}\hfill \sum _{n\le N}{\mu }^2(n)=\frac{N}{\zeta (2)}+E_1(N),\end{array}$$

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

$$\begin{array}{c}\hfill M_{\chi }:=\underset{L,P}{max}\left|\sum _{n=L}^P\chi (n)\right|\end{array}$$

for a primitive character $\chi$ to the modulus $q>1$. Then

$$\begin{array}{c}\hfill M_{\chi }\le \left\{\begin{array}{cc}\frac{2}{{\pi }^2}{q}^{1/2}logq+0.9467q^{1/2}+1.668,\hfill & \chi \mathrm{even};\hfill \\ \hfill & \hfill \\ \frac{1}{2\pi }{q}^{1/2}logq+0.8204q^{1/2}+1.0285,\hfill & \chi \mathrm{odd}.\hfill \end{array}\right.\end{array}$$

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. $\square$

Funding

This work was supported by a Hertha Firnberg grant of the Austrian Science Fund (FWF) [T846-N35].