Let $f:\mathbb{N}\to \{-1,1\}$ be a multiplicative function. Is it true that
\[ \lim_{N\to \infty}\frac{1}{N}\sum_{n\leq N}f(n)\] always exists?
The answer is yes, as proved by Wirsing [Wi67], and generalised by Halász [Ha68].
Actions
Submit a Proof
Have a proof attempt? Submit it for zero-trust verification.
theorem erdos_239 :
answer(True) ↔ ∀ f : ℕ → ℝ,
(∀ n ≥ 1, f n = 1 ∨ f n = -1) ∧
(∀ m n, m.Coprime n → f (m * n) = f m * f n) ∧
f 1 = 1 →
∃ L, Tendsto (fun N ↦ (∑ n ∈ Finset.Icc 1 N, f n) / N) atTop (𝓝 L)
