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

{c}

Absolute convergence of the <Euler product> gives $\zeta(s)\ne0$ for $\Re s>1$. Equivalently,
$$
\frac1{\zeta(s)}=\sum_{n\geq1}\frac{\mu(n)}{n^s}
$$
converges absolutely there.

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