Theorems · Theorem · number theory

EulerProduct.eulerProduct

∀ {R : Type u_1} [inst : NormedCommRing R] {f : ℕ → R} [CompleteSpace R],
  f 1 = 1 →
    (∀ {m n : ℕ}, m.Coprime n → f (m * n) = f m * f n) →
      (Summable fun x => ‖f x‖) →
        f 0 = 0 →
          Filter.Tendsto (fun n => ∏ p ∈ n.primesBelow, ∑' (e : ℕ), f (p ^ e)) Filter.atTop (nhds (∑' (n : ℕ), f n))

The Euler Product for multiplicative (on coprime arguments) functions. If f : ℕ → R, where R is a complete normed commutative ring, f 0 = 0, f 1 = 1, f is multiplicative on coprime arguments, and ‖f ·‖ is summable, then ∏' p : {p : ℕ | p.Prime}, ∑' e, f (p ^ e) = ∑' n, f n. This is a version using convergence of finite partial products.

Defined in

Mathlib.NumberTheory.EulerProduct.Basic

Cited by

1 results in Mathlib

Foundations

Depth 173 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NormedCommRingCompleteSpace