FormalConjectures.Wikipedia.RamanujanTau

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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

Lehmer's conjecture


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 declaration uses 'sorry'multipliable : Multipliable fun n : ℕ+ ↦ ((1 - X ^ (n : ℕ)) ^ 24 : PowerSeries ℤ) := by⊢ Multipliable fun n => (1 - X ^ ↑n) ^ 24
  sorryAll goals completed! 🐙


@[category test, AMS 11]
lemma τ_zero : τ 0 = 0 := by⊢ τ 0 = 0 simp [τ, Δ]All goals completed! 🐙


@[category test, AMS 11]
lemma τ_one : τ 1 = 1 := by⊢ τ 1 = 1
  obtain ⟨i, hi⟩ := by⊢ ?m.7 simpa using ((continuous_constantCoeff ℤ).tendsto _).comp multipliable.hasProdAll goals completed! 🐙
  simp [τ, Δ, hi i]All goals completed! 🐙


@[category test, AMS 11]
lemma declaration uses 'sorry'τ_two : τ 2 = -24 := by⊢ τ 2 = -24
  sorryAll goals completed! 🐙

The Ramanujan-Petersson conjecture: $|\tau(p)| \le 2 p^{11/2}$ for primes $p$.

@[category research solved, AMS 11]
theorem declaration uses 'sorry'ramanujan_petersson : ∀ p : ℕ, Prime p → abs (τ p) ≤ 2 * (p : ℝ) ^ ((11 : ℝ) / 2) := by⊢ ∀ (p : ℕ), Prime p → ↑|τ p| ≤ 2 * ↑p ^ (11 / 2)
  sorryAll goals completed! 🐙

Lehmer's conjecture: $\tau(n) \ne 0$ for all $n > 0$.

@[category research open, AMS 11]
theorem declaration uses 'sorry'lehmer_ramanujan_tau : ∀ n > 0, τ n ≠ 0 := by⊢ ∀ n > 0, τ n ≠ 0
  sorryAll goals completed! 🐙


end RamanujanTau