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import FormalConjecturesUtilNumerator of $\zeta(4n)/\zeta(2n)^2$ (with $a(0)=2$ instead of $-2$)
The ratio $\zeta(4n)/\zeta(2n)^2$ for $n \ge 1$ is the rational number $$ Q_n = -2 \frac{B_{4n}}{B_{2n}^2 \binom{4n}{2n}} $$ where $B_k$ is the $k$-th Bernoulli number. The sequence $a(n)$ is the numerator of $Q_n$, with $a(0)$ defined as $2$.
References:
namespace OeisA114362open scoped Nat Realopen Filteropen Complex
The primary defining sequence a.
Numerator of $\zeta(4n)/\zeta(2n)^2$ (with $a(0)=2$ instead of $-2$).
noncomputable def a (n : ℕ) : ℕ :=
if n = 0 then
2
else
let b4n : ℚ := bernoulli (4 * n)
let b2n : ℚ := bernoulli (2 * n)
let binomQn : ℚ := ↑(Nat.choose (4 * n) (2 * n))
let qN : ℚ := -2 * b4n / (b2n * b2n * binomQn)
qN.num.natAbs@[category test, AMS 11]
theorem a_0 : a 0 = 2 := ⊢ a 0 = 2
All goals completed! 🐙@[category test, AMS 11]
theorem a_1 : a 1 = 2 := ⊢ a 1 = 2
⊢ (-(2 * bernoulli 4) / (6⁻¹ * 6⁻¹ * ↑(Nat.choose 4 2))).num.natAbs = 2
All goals completed! 🐙⊢ (-(2 * (1 - ∑ k ∈ Finset.range 8, ↑(Nat.choose 8 k) / (↑8 - ↑k + 1) * bernoulli' k)) / (7 / 90)).num = 6 ∨
(-(2 * (1 - ∑ k ∈ Finset.range 8, ↑(Nat.choose 8 k) / (↑8 - ↑k + 1) * bernoulli' k)) / (7 / 90)).num = -6
α:∀ (n : ℕ), ∑ k ∈ Finset.range n, ↑(n.choose k) * bernoulli' k = ↑n⊢ (-(2 * (1 - ∑ k ∈ Finset.range 8, ↑(Nat.choose 8 k) / (↑8 - ↑k + 1) * bernoulli' k)) / (7 / 90)).num = 6 ∨
(-(2 * (1 - ∑ k ∈ Finset.range 8, ↑(Nat.choose 8 k) / (↑8 - ↑k + 1) * bernoulli' k)) / (7 / 90)).num = -6
All goals completed! 🐙⊢ ((-(1 - ∑ k ∈ Finset.range 12, ↑(Nat.choose 12 k) / (↑12 - ↑k + 1) * bernoulli' k) +
-(1 - ∑ k ∈ Finset.range 12, ↑(Nat.choose 12 k) / (↑12 - ↑k + 1) * bernoulli' k)) /
((1 - ∑ k ∈ Finset.range 6, ↑(Nat.choose 6 k) / (↑6 - ↑k + 1) * bernoulli' k) *
(1 - ∑ k ∈ Finset.range 6, ↑(Nat.choose 6 k) / (↑6 - ↑k + 1) * bernoulli' k) *
924)).num.natAbs =
691
this:∀ (n : ℕ), ∑ k ∈ Finset.range n, ↑(n.choose k) * bernoulli' k = ↑n⊢ ((-(1 - ∑ k ∈ Finset.range 12, ↑(Nat.choose 12 k) / (↑12 - ↑k + 1) * bernoulli' k) +
-(1 - ∑ k ∈ Finset.range 12, ↑(Nat.choose 12 k) / (↑12 - ↑k + 1) * bernoulli' k)) /
((1 - ∑ k ∈ Finset.range 6, ↑(Nat.choose 6 k) / (↑6 - ↑k + 1) * bernoulli' k) *
(1 - ∑ k ∈ Finset.range 6, ↑(Nat.choose 6 k) / (↑6 - ↑k + 1) * bernoulli' k) *
924)).num.natAbs =
691
this:∀ (n : ℕ), ∑ k ∈ Finset.range n, ↑(n.choose k) * bernoulli' k = ↑nR:∀ (M : ℕ), ↑(M + 1) = ∑ x ∈ Finset.range M, ↑((M + 1).choose x) * bernoulli' x + ↑((M + 1).choose M) * bernoulli' M⊢ ((-(1 - ∑ k ∈ Finset.range 12, ↑(Nat.choose 12 k) / (↑12 - ↑k + 1) * bernoulli' k) +
-(1 - ∑ k ∈ Finset.range 12, ↑(Nat.choose 12 k) / (↑12 - ↑k + 1) * bernoulli' k)) /
((1 - ∑ k ∈ Finset.range 6, ↑(Nat.choose 6 k) / (↑6 - ↑k + 1) * bernoulli' k) *
(1 - ∑ k ∈ Finset.range 6, ↑(Nat.choose 6 k) / (↑6 - ↑k + 1) * bernoulli' k) *
924)).num.natAbs =
691
All goals completed! 🐙Conjecture: if an integer $n > 1$ is odd, then $\zeta(2n)/\zeta(n)^2$ is irrational. Cf. W. Kohnen (link) and my conjecture in A348829. - Thomas Ordowski, Jan 05 2022
@[category research open, AMS 11]
theorem conjecture1 (n : ℕ) (hn_gt_one : 1 < n) (hn_odd : Odd n) :
Irrational ((riemannZeta (2 * n : ℂ) / (riemannZeta (n : ℂ)) ^ 2).re) := n:ℕhn_gt_one:1 < nhn_odd:Odd n⊢ Irrational (riemannZeta (2 * ↑n) / riemannZeta ↑n ^ 2).re
All goals completed! 🐙
t n is used in the second conjecture.
noncomputable def t (n : ℕ) : ℝ :=
(riemannZeta (2 * (n : ℂ))).re / ((riemannZeta (n : ℂ)).re ^ 2)Conjecture: $\frac{1 - t(n)}{1 + t(n)} = \frac{1}{2^n} + \frac{1}{3^n} + \frac{1}{5^n} + \frac{1}{7^n} + O(\frac{1}{11^n})$, where $t(n) = \zeta(2n)/\zeta(n)^2$. Cf. A348829. - Thomas Ordowski, Nov 13 2022
@[category research solved, AMS 11, formal_proof using formal_conjectures at
"https://github.com/chy4pro/formal-conjectures/blob/872759d0b464254d868f107fe7f9cf762900f57d/FormalConjectures/OEIS/114362.lean#L646"]
theorem conjecture2 :
(fun n : ℕ => (1 - t n) / (1 + t n) -
(1 / (2:ℝ)^n + 1 / (3:ℝ)^n + 1 / (5:ℝ)^n + 1 / (7:ℝ)^n))
=O[atTop] (fun n : ℕ => 1 / (11:ℝ)^n) := ⊢ (fun n ↦ (1 - t n) / (1 + t n) - (1 / 2 ^ n + 1 / 3 ^ n + 1 / 5 ^ n + 1 / 7 ^ n)) =O[atTop] fun n ↦ 1 / 11 ^ n
All goals completed! 🐙end OeisA114362