Numerator of $\zeta(4n)/\zeta(2n)^2$ (with $a(0)=2$ instead of $-2$)

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import FormalConjecturesUtil

/-!
# Numerator 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:*
- [A114362](https://oeis.org/A114362)
-/

namespace OeisA114362

open scoped Nat Real
open Filter
open 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 := by
  congr

@[category test, AMS 11]
theorem a_1 : a 1 = 2 := by
  simp_all [a]
  norm_num only [bernoulli_eq_bernoulli'_of_ne_one, bernoulli'_four, bernoulli'_two, Nat.choose]

@[category test, AMS 11]
theorem a_2 : a 2 = 6 := by
  delta a
  norm_num +decide
    [bernoulli_eq_bernoulli'_of_ne_one, bernoulli'_eq_zero_of_odd, Int.natAbs_eq_iff, Nat.choose]
  rw [bernoulli'_def]
  have α := sum_bernoulli'
  norm_num only
    [←eq_sub_of_add_eq' (α _ ▸ Finset.sum_range_succ _ _).symm ▸ mul_div_cancel_left₀ _,
      Finset.sum_range_succ, or_false, or_true, Nat.choose]

@[category test, AMS 11]
theorem a_3 : a 3 = 691 := by
  delta and a
  norm_num [bernoulli_eq_bernoulli'_of_ne_one, two_mul, Nat.cast_choose]
  rw [bernoulli'_def, bernoulli'_def]
  have := sum_bernoulli'
  have R M := this (M+1) ▸ Finset.sum_range_succ _ _
  norm_num only
    [Nat.choose, ←sub_eq_of_eq_add' (R _) ▸ mul_div_cancel_left₀ _, Finset.sum_range_succ]

/--
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) := by
  sorry

/-- `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) := by
  sorry

end OeisA114362

• Formalization source ↗

On-site verification needs an executable Lean target. None is attached yet.

A source attachment or link is not a verified proof. The statement’s correspondence with the original question needs human review.