erdos_975

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 $\sum_{n \le x} \tau(f(n)) \approx c \cdot x \log x$? Note that it is unclear whether the polynomial should have integer coefficients or merely be integer-valued. We assume the former.

theorem erdos_975 : answer(sorry) ↔
    ∀ f : ℤ[X], f.natDegree ≠ 0 → Irreducible f → (∀ᶠ n in atTop, 1 ≤ f.eval n) →
    ∃ c > (0 : ℝ), Tendsto (fun x ↦ Erdos975Sum f x / (x * log x)) atTop (𝓝 c)

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