For $\sigma>1$, unique prime factorization and absolute convergence give the <Euler product>
$$
\zeta(s)=\prod_p(1-p^{-s})^{-1}.
$$
Every factor is nonzero and the product converges to a nonzero limit. Equivalently, the absolutely convergent identity
$$
\frac1{\zeta(s)}=\sum_{n=1}^\infty\frac{\mu(n)}{n^s}
$$
provides a reciprocal. This proves the <Euler-product nonvanishing of the Riemann zeta function>.

Solution

Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 150 / 3 / a / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2021/iii/paper-150/3/a/solution