Problem Statement
Let $f\in \mathbb{Z}[x]$ be an irreducible non-constant polynomial such that $f(n)\geq 1$ for all large $n\in\mathbb{N}$. Does there exist a constant $c=c(f)>0$ such that\[\sum_{n\leq X} \tau(f(n))\sim cX\log X,\]where $\tau$ is the divisor function?
Categories:
Number Theory Divisors Polynomials
Progress
Van der Corput [Va39] proved that\[\sum_{n\leq X} \tau(f(n))\gg_f X\log X.\]Erdős [Er52b] proved using elementary methods that\[\sum_{n\leq X} \tau(f(n))\ll_f X\log X.\]Such an asymptotic formula is known whenever $f$ is an irreducible quadratic, as proved by Hooley [Ho63]. The form of $c$ depends on $f$ in a complicated fashion (see the work of McKee [Mc95], [Mc97], and [Mc99] for expressions for various types of quadratic $f$). For example,\[\sum_{n\leq x}\tau(n^2+1)=\frac{3}{\pi}x\log x+O(x).\]Tao has a blog post on this topic.Source: erdosproblems.com/975 | Last verified: January 19, 2026