SOLVED
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?
Wintner observed that if $f$ can take complex values on the unit circle then the limit need not exist. The answer is yes, as proved by Wirsing
[Wi67], and generalised by Halász
[Ha68].
