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Erdős #975 · number theory / divisors

Let f∈Z[x]f\in \mathbb{Z}[x] be an irreducible non-constant polynomial such that f(n)≥1f(n)\geq 1 for all large n∈Nn\in\mathbb{N}. Does there exist a constant c=c(f)>0c=c(f)>0 such that

∑n≤Xτ(f(n))∼cXlog⁡X,\sum_{n\leq X} \tau(f(n))\sim cX\log X,

w…

MSC 11 · Number theory

Inferred from the community field; finer classification has not been reviewed.

Source reports OPENAs of 2026-09-08Source & evidence ↓
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Problem statement

Let f∈Z[x]f\in \mathbb{Z}[x] be an irreducible non-constant polynomial such that f(n)≥1f(n)\geq 1 for all large n∈Nn\in\mathbb{N}. Does there exist a constant c=c(f)>0c=c(f)>0 such that

∑n≤Xτ(f(n))∼cXlog⁡X,\sum_{n\leq X} \tau(f(n))\sim cX\log X,

where τ\tau is the divisor function?

Lean 4

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Formal statement

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Background

Van der Corput [Va39] proved that

∑n≤Xτ(f(n))≫fXlog⁡X.\sum_{n\leq X} \tau(f(n))\gg_f X\log X.

Erdős [Er52b] proved using elementary methods that

∑n≤Xτ(f(n))≪fXlog⁡X.\sum_{n\leq X} \tau(f(n))\ll_f X\log X.

Such an asymptotic formula is known whenever ff is an irreducible quadratic, as proved by Hooley [Ho63]. The form of cc depends on ff in a complicated fashion (see the work of McKee [Mc95], [Mc97], and [Mc99] for expressions for various types of quadratic ff). For example,

∑n≤xτ(n2+1)=3πxlog⁡x+O(x).\sum_{n\leq x}\tau(n^2+1)=\frac{3}{\pi}x\log x+O(x).

Tao has a blog post on this topic.

Progress & source assessment

The source labels this entry OPEN in the 2026-09-08 snapshot. Editorial progress notes are reproduced as source commentary, not independently verified results. Prize amounts and Lean badges at the source are not local awards or verification.

References
  1. [1]Thomas Bloom (ed.), Erdős Problems #975 link
  2. [2]Upstream Lean 4 statement for Erdős #975 linkExternal statement reference. Not compiled or verified by this platform; upstream files may contain open proof obligations.
  3. [3]Formal Conjectures — Erdős Problem 975 link

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