Erdos 975 #691
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Erdos 975 #691
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Paul-Lez
reviewed
Sep 29, 2025
Co-authored-by: Paul Lezeau <[email protected]>
Paul-Lez
reviewed
Oct 14, 2025
Co-authored-by: Paul Lezeau <[email protected]>
Paul-Lez
reviewed
Oct 23, 2025
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| /- Auxiliary definition for nonnegative irreducible polynomials over $\mathbb{Z}$ on $\mathbb{N}$. -/ | ||
| def Erdos975NonnegIrredPoly (f : ℕ → ℕ) (g : ℤ[X]) : Prop := | ||
| Irreducible g ∧ ∃ N : ℕ, ∀ n ≥ N, f n = g.eval ↑n ∧ 0 ≤ f n |
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I would be tempted to remove this condition, and just do everything in terms of the polynomial g (and g.eval when you want to use the corresponding function!)
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| /-- | ||
| 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 | ||
| $\sum_{n \le x} \tau(f(n)) \approx c \cdot x \log x$? | ||
| -/ | ||
| @[category research open, AMS 11] | ||
| theorem erdos_975 (f : ℕ → ℕ) (g : ℤ[X]) (hf : Erdos975NonnegIrredPoly f g) : | ||
| (∃ (c : ℝ), 0 < c ∧ Erdos975Asymptotic f c) ↔ answer(sorry) := by | ||
| sorry |
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Here, I would recommend moving the quantification "inside". That way, replacing sorry by false will correspond to the negation of the statement.
| @[category research solved, AMS 11] | ||
| theorem erdos_975.variant.quadratic (f : ℕ → ℕ) (g : ℤ[X]) (hf : Erdos975NonnegIrredPoly f g) | ||
| (hg_degree : g.degree = 2) : | ||
| ∃ (c : ℝ), 0 < c ∧ Erdos975Asymptotic f c := by |
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Here you could possibly use answer(sorry) as a placeholder for c (and then later replace sorry by the actual closed form!)
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There might be a better way to deal with nonnegative irreducible polynomials, and feel free to give a comment on it.