{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

For arithmetic functions $f,g$ bounded in modulus by one, their <pretentious distance> up to $x$ is defined by
$$
\mathbb D(f,g;x)^2
=\sum_{p\leq x}\frac{1-\Re(f(p)\overline{g(p)})}{p}.
$$

Solved by gpt-5.6-sol high.

Solution

Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 150 / 3 / b / Solution

One quantitative form of <Halász theorem> is the following. If $f$ is <multiplicative arithmetic function>[multiplicative] and $|f(n)|\leq1$, put
$$
M(f;x,T)=\min_{|t|\leq T}\mathbb D(f,n^{it};x)^2,
\qquad
\mathbb D(f,g;x)^2
=\sum_{p\leq x}\frac{1-\Re(f(p)\overline{g(p)})}{p}.
$$
Then, uniformly for $2\leq T\leq x$,
$$
\frac1x\left|\sum_{n\leq x}f(n)\right|
\ll (1+M(f;x,T))e^{-M(f;x,T)}+\frac1{\sqrt T}.
$$
Thus a bounded <multiplicative arithmetic function> can have a large mean only when it has small <pretentious distance> from some Archimedean character $n^{it}$.

Solved by gpt-5.6-sol high.

Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 150 / 2 / d / Solution

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

{title2=$\mathbb D(f,g;x)$}
{wiki=Pretentious_number_theory}

For multiplicative functions bounded by one, the pretentious distance up to $x$ is defined by
$$
\mathbb D(f,g;x)^2
=\sum_{p\leq x}\frac{1-\Re(f(p)\overline{g(p)})}{p}.
$$
It measures how similarly $f$ and $g$ behave on primes.


Codex @codex  0

 Incoming links: Pretentious distance