Ramanujan τ-function

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import FormalConjecturesUtil

/-!
# Ramanujan τ-function

There are two conjectures related to the Ramanujan τ-function:

- Ramanujan-Petersson conjecture: For every prime `p`, the absolute value of the
  Ramanujan τ-function at `p` is bounded by `2 * p^(11/2)`.
- Lehmer's conjecture: The Ramanujan τ-function is never zero for any positive integer `n`.

*References:*
- [Ramanujan-Petersson conjecture](https://en.wikipedia.org/wiki/Ramanujan%E2%80%93Petersson_conjecture)
- [Lehmer's conjecture](https://en.wikipedia.org/wiki/Ramanujan_tau_function#Conjectures_on_the_tau_function)
-/

namespace RamanujanTau

open PowerSeries PowerSeries.WithPiTopology

noncomputable def Δ : PowerSeries ℤ := X * ∏' (n : ℕ+), (1 - X ^ (n : ℕ)) ^ 24

noncomputable def τ (n : ℕ) : ℤ := PowerSeries.coeff n Δ


@[category API, AMS 11]
lemma multipliable : Multipliable fun n : ℕ+ ↦ ((1 - X ^ (n : ℕ)) ^ 24 : PowerSeries ℤ) := by
  sorry

@[category test, AMS 11]
lemma τ_zero : τ 0 = 0 := by simp [τ, Δ]

@[category test, AMS 11]
lemma τ_one : τ 1 = 1 := by
  obtain ⟨i, hi⟩ := by simpa using ((continuous_constantCoeff ℤ).tendsto _).comp multipliable.hasProd
  simp [τ, Δ, hi i]

@[category test, AMS 11]
lemma τ_two : τ 2 = -24 := by
  sorry


/-- The Ramanujan-Petersson conjecture: $|\tau(p)| \le 2 p^{11/2}$ for primes $p$. -/
@[category research solved, AMS 11]
theorem ramanujan_petersson : ∀ p : ℕ, Prime p → abs (τ p) ≤ 2 * (p : ℝ) ^ ((11 : ℝ) / 2) := by
  sorry

/-- Lehmer's conjecture: $\tau(n) \ne 0$ for all $n > 0$. -/
@[category research open, AMS 11]
theorem lehmer_ramanujan_tau : ∀ n > 0, τ n ≠ 0 := by
  sorry

end RamanujanTau

• Formalization source ↗

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