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.

Solution

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

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2026/iii/paper-150/2/d/solution