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Claim. Let $b$ and $c$ be integers with $\Delta=b^2-4c>0$ not a perfect square. Then

$$\sum_{n\le x}d(n^2+bn+c)=\lambda\,x\log x+O(x),\qquad \lambda=\frac{12H^*(\Delta)\log\epsilon_\Delta}{\pi^2\sqrt\Delta},$$

where $d$ is the divisor function, $H^*(\Delta)$ is a weighted class number and $\epsilon_\Delta$ is the fundamental unit of discriminant $\Delta$. The zbMATH review of the paper (Zbl 0923.11132) states the theorem as the determination of $\lambda$ for $\Delta>0$, completing the case $\Delta<0$ of [[problems/polynomials/E0975/claims/1995_05_01_mckee|McKee's 1995 paper]]; the formula for $\lambda$ is as Lapkova (arXiv:1704.02498, (1.3)) restates it. Because the automorphism groups of the forms of positive discriminant are infinite, the paper defines $H^*(\Delta)$ through the index of a subgroup of the automorphism group rather than its order; for $\Delta<0$ this agrees with the earlier definition except at $\Delta=-3$ and $-4$. In the notation of Problem 975, for $f(x)=x^2+bx+c$ with $\Delta>0$ not a square, which is irreducible and positive for all large $n$,

$$\sum_{n\le X}\tau(f(n))\sim c(f)\,X\log X,\qquad c(f)=\lambda>0.$$

Covers. The monic quadratics $x^2+bx+c$ with $b^2-4c$ positive and not a perfect square. For even $b$ the polynomial is a translate of $x^2+a$ with $a<0$ and $-a$ not a square, already covered by Hooley's theorem; the new instances are those with odd $b$, such as $x^2+x-1$. Not covered: the non-monic quadratics of positive discriminant, and every degree three or more, where the problem is open.

Depends on. No page of this wiki.

Acceptance. J. McKee, The average number of divisors of an irreducible quadratic polynomial, Math. Proc. Cambridge Philos. Soc. 126 (1999), no. 1, 17--22, a refereed journal (refereed); the record dates the issue to January 1999 without a day, so the page is dated to the first day of that month. The curator of erdosproblems.com, Thomas Bloom, credits this paper as [Mc99] in the problem's commentary, but the site labels the problem OPEN, so that credit is not acceptance of this partial claim.