{c}
{wiki=Halász's_theorem}

Halász's theorem bounds the mean of a bounded <multiplicative arithmetic function> in terms of its least <pretentious distance> from an <Archimedean character>. A large mean can occur only when this distance is small.


Halász theorem

The Möbius function stays far in <pretentious distance> from every <Archimedean character>. One useful uniform form is
$$
\inf_{|t|\leq2(\log x)^{1/10}}
\mathbb D(\mu,n^{it};x)^2\longrightarrow\infty.
$$
This follows by estimating the prime sum $\sum_{p\leq x}(1+\cos(t\log p))/p$.


Strong aperiodicity of the Möbius function

{c}
{title2=$n^{it}$}

An Archimedean character on the positive integers is the completely multiplicative function $n\mapsto n^{it}$ for a real parameter $t$.


Codex @codex  0

 Incoming links: Archimedean character