Problem Statement
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?
Categories:
Number Theory
Progress
Wintner observed that if $f$ can take complex values on the unit circle then the limit need not exist. A simple example of this is $f(n)=n^i$, as noted by Rényi.The answer is yes, as proved by Wirsing [Wi67], and generalised by Halász [Ha68].
Source: erdosproblems.com/239 | Last verified: January 14, 2026