Divisor sums of irreducible polynomial values: is $\sum_{n\le X}\tau(f(n))\sim cX\log X$? (Erdős #975)
Statement
Let $f\in\mathbb{Z}[x]$ be a non-constant irreducible polynomial with $f(n)\geq 1$ for all large $n\in\mathbb{N}$, and let $\tau$ denote the divisor function. Is there a constant $c=c(f)>0$ such that $$\sum_{n\leq X}\tau(f(n))\sim c\,X\log X\qquad(X\to\infty)?$$ In other words, does the average number of divisors of $f(n)$ obey a clean asymptotic law with a leading constant depending only on $f$?
Acceptance. FULLY RESOLVES: a complete proof (machine-checkable in Lean/Coq preferred, otherwise a full written proof with all analytic steps) that, for every admissible irreducible non-constant $f$ — or, as a decisive first breakthrough, for one specific irreducible cubic $f$ — one has $\sum_{n\leq X}\tau(f(n))\sim c(f)\,X\log X$ with an explicitly described constant $c(f)>0$. ADVANCES: establish the asymptotic for a genuinely new class of $f$ beyond the irreducible quadratics already handled by Hooley [Ho63] (e.g. any single irreducible cubic), with full proof; OR improve the known bounds toward the asymptotic strictly beyond the background (e.g. a two-sided estimate $\sum_{n\leq X}\tau(f(n))=(c+o(1))X\log X$ on a restricted range, or sharper error terms in the quadratic case), with proof; OR provide reproducible high-precision numerical estimates of $c(f)$ for named higher-degree $f$, with error bars and the computation code. Deliver the proof file, the improved-bound proof, or the computation code plus estimated constants.
Background
Posed by Erdős [Er65b]. Listed as open on erdosproblems.com/975 (fetched 2026-07-13, status 'open'), tagged 'number theory | divisors | polynomials'; related sequence OEIS A147807. The correct order of magnitude is fully known: van der Corput [Va39] proved $\sum_{n\leq X}\tau(f(n))\gg_f X\log X$, and Erdős [Er52b] proved the matching upper bound $\sum_{n\leq X}\tau(f(n))\ll_f X\log X$ by elementary methods, so the sum is $\asymp_f X\log X$. The asymptotic itself (existence of $c$) is known for every irreducible quadratic $f$, by Hooley [Ho63]; the constant $c$ then depends on $f$ in a complicated arithmetic way, worked out for various quadratics by McKee [Mc95],[Mc97],[Mc99]. A concrete example is $\sum_{n\leq x}\tau(n^2+1)=\tfrac{3}{\pi}x\log x+O(x)$. The case of degree $\geq 3$ is completely open — no single irreducible cubic is known to admit such an asymptotic. Terence Tao has a 2011 blog post surveying this 'Erdős divisor bound' problem. The statement is formalised in Lean (google-deepmind/formal-conjectures). Attacker's tool: numerically compute $\sum_{n\leq X}\tau(f(n))/(X\log X)$ for specific irreducible cubics (e.g. $x^3+2$) to estimate and stress-test the conjectural constant $c$, alongside sieve-theoretic and Hooley-type analytic techniques (the $\Delta$-function, the hyperbola method) aimed at pushing existence of the asymptotic beyond quadratics.
References
| Ref | Source | Type |
|---|---|---|
| REF-01 | Erdős Problem #975 (T. F. Bloom) | website |
| REF-02 | OEIS A147807 — related divisor-function sequence (site-suggested) | website |
| REF-03 | T. Tao — blog post on the Erdős divisor bound (2011) | website |
| REF-04 | Lean 4 formalisation (google-deepmind/formal-conjectures) of Erdős #975 | website |
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