Theorems · Theorem · number theory

EulerProduct.eulerProduct_hasProd_mulIndicator

∀ {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 → HasProd ({p | Nat.Prime p}.mulIndicator fun p => ∑' (e : ℕ), f (p ^ e)) (∑' (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 : ℕ, if p.Prime then ∑' e, f (p ^ e) else 1 = ∑' n, f n. This version is stated using HasProd and Set.mulIndicator.

Defined in

Mathlib.NumberTheory.EulerProduct.Basic

Cited by

2 results in Mathlib

Foundations

Depth 172 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NormedCommRingCompleteSpace