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

For $\sigma >1$, unique prime factorization and absolute convergence give the Euler productEvery factor is nonzero and the product converges to a nonzero limit. Equivalently, the absolutely convergent identityprovides a reciprocal. This proves the Euler-product nonvanishing of the Riemann zeta function.