Let $f:\mathbb{N}\to \{-1,1\}$ be a multiplicative function. Is it true that$$\underset{N\to \infty }{\lim }\frac{1}{N}\sum _{n\le N}f(n)$$always exists?

AMS 11 · solved (literature)

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)