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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?
#239
:
[Er61]
[Er73]
number theory
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]
.
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