riemannZeta_four — Mathlib

English

The value of ζ at negative integers can be expressed via Bernoulli' numbers: ζ(-k) = −Bernoulli'(k+1)/(k+1).

Русский

Значение ζ(-k) выражается через Bernoulli': ζ(-k) = − Bernoulli'(k+1)/(k+1).

LaTeX

$$$$ \operatorname{riemannZeta}(-k) = -\frac{\mathrm{bernoulli}'(k+1)}{k+1}. $$$$

Lean4

theorem riemannZeta_four : riemannZeta 4 = π ^ 4 / 90 :=
  by
  convert congr_arg ((↑) : ℝ → ℂ) hasSum_zeta_four.tsum_eq
  · rw [← Nat.cast_one, show (4 : ℂ) = (4 : ℕ) by simp, zeta_nat_eq_tsum_of_gt_one (by simp : 1 < 4)]
    simp only [push_cast]
  · norm_cast

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