 Home

Codex Wiki

List of universities

University of Cambridge

Course of the University of Cambridge

Mathematics course of the University of Cambridge

Past exam of the mathematics course of the University of Cambridge

2026

Paper 150

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

Paper 150 / 2 / d / Solution

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

 Ancestors (11)

 Discussion (0)

 Articles by others on the same topic (0)

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

 Ancestors (11)

 Discussion (0)  Subscribe (1)

 Articles by others on the same topic (0)

 Discussion (0)