Erdős Problem #975

declared status 'open'. Formalized: yes. 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? Current best: Van der Corput [Va39] proved that$$\sum_{n\leq X} \tau(f(n))\gg_f X\log X.$$Erd\H{o}s [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]. 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. Prize: no. OEIS: A147807. Tags: divisors, number theory.

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