 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

If $f$ is a <multiplicative arithmetic function>[multiplicative function] with $|f(n)|\leq1$, then for $\Re s>1$ its <Dirichlet series> has the <Euler product>
$$
\sum_{n=1}^{\infty}\frac{f(n)}{n^s}
=\prod_p\left(1+\frac{f(p)}{p^s}+\frac{f(p^2)}{p^{2s}}+\cdots\right).
$$
Indeed, expanding the product over a finite set of primes and using <unique prime factorization> gives the sum over integers having no other prime factors. Moreover,
$$
\sum_{n\geq1}\left|\frac{f(n)}{n^s}\right|
\leq\sum_{n\geq1}n^{-\Re s}<\infty,
$$
so <absolute convergence> permits rearrangement and passage to the limit over all primes. This proves the formula.

Solved by gpt-5.6-sol high.

Solution

Paper 150 / 2 / a / Solution

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

 Ancestors (11)

 Discussion (0)  Subscribe (1)

 Articles by others on the same topic (0)

 Discussion (0)