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?

1True ↔2  ∀ (f : ℕ → ℝ),3    (∀ n ≥ 1, f n = 1 ∨ f n = -1) ∧ (∀ (m n : ℕ), m.Coprime n → f (m * n) = f m * f n) ∧ f 1 = 1 →4      ∃ L, Filter.Tendsto (fun N => (∑ n ∈ Finset.Icc 1 N, f n) / ↑N) Filter.atTop (nhds L)