Erdős problem 975 · problems.science

For an irreducible polynomial $f \in \mathbb{Z}[x]$ with $f(n) \ge 1$ for sufficiently large $n$ , does there exists a constant $c = c(f) > 0$ such that $\sum_{n \le x} \tau(f(n)) \approx c \cdot x \log x$ ?

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FormalConjectures/ErdosProblems/

975.lean

Retained formal statement5 of 6

When $f$ is an irreducible quadratic polynomial, the question is answered first by Hooley [Ho63]. More compact expression of the constant in terms of Hurwitz class numbers (when $a = 1$ ) is given by McKey in [Mc95], [Mc97], [Mc99].

TODO: formalize Hurwitz class numbers and the expression of the constant in terms of them.

FormalConjectures/ErdosProblems/975.leanErdos975.erdos_975.variants.quadratic7 linesExact file

1∀ (f : Polynomial ℤ),2  Irreducible f →3    (∀ᶠ (n : ℕ) in Filter.atTop, 1 ≤ Polynomial.eval (↑n) f) →4      f.degree = 2 →5        ∀ (c : ℝ),6          c = sorry →7            0 < c ∧ Filter.Tendsto (fun x => Erdos975.Erdos975Sum f x / (x * Real.log x)) Filter.atTop (nhds c)

SolvedStatement only, no proof

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