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Claim. Let $b$ and $c$ be integers with $\Delta=b^2-4c<0$, and let $H^*(\Delta)$ be the Hurwitz class number. Then

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

where $d$ is the divisor function; this is the theorem as the zbMATH review of the paper (Zbl 0841.11050) states it. The proof writes the number of roots of $n^2+bn+c\equiv0\pmod k$ as a weighted count of proper representations of $k$ by reduced binary quadratic forms of discriminant $\Delta$. In the notation of Problem 975, for $f(x)=x^2+bx+c$ with $\Delta<0$, which is irreducible and positive at every integer,

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

Covers. The monic quadratics $x^2+bx+c$ with $b^2-4c<0$. For even $b$ the polynomial is a translate of $x^2+a$ with $a>0$, already covered by Hooley's theorem; the new instances are those with odd $b$, such as $x^2+x+1$. Not covered: the monic quadratics of positive non-square discriminant, which are McKee's 1999 theorem; the non-monic quadratics; and every degree three or more, where the problem is open.

Depends on. No page of this wiki.

Acceptance. J. McKee, On the average number of divisors of quadratic polynomials, Math. Proc. Cambridge Philos. Soc. 117 (1995), no. 3, 389--392, a refereed journal (refereed); the record dates the issue to May 1995 without a day, so the page is dated to the first day of that month. The curator of erdosproblems.com, Thomas Bloom, cites this paper as [Mc95] in the problem's commentary, but the site labels the problem OPEN, so that citation is not acceptance of this partial claim.