riemannZeta_residue_one — Mathlib

English

The zeta function has a simple pole at s = 1 with residue 1; equivalently, (s−1) ζ(s) → 1 as s → 1 (excluding s = 1).

Русский

У дзеты Римана простой полюс в s = 1 c остатку равен 1; то есть (s−1) ζ(s) → 1 при s → 1 (кроме точки 1).

LaTeX

$$$\\\\lim_{s\\to 1} (s-1) \\\\riemannZeta(s) = 1$ (in an appropriate punctured neighborhood)$$

Lean4

/-- The residue of `ζ(s)` at `s = 1` is equal to 1. -/
theorem riemannZeta_residue_one : Tendsto (fun s ↦ (s - 1) * riemannZeta s) (𝓝[≠] 1) (𝓝 1) := by
  exact hurwitzZetaEven_residue_one 0

Похожие утверждения