Past exam of the mathematics course of the University of Cambridge / 2019 / ii / Paper 4 / 1I (Number Theory)

For $x\ge 2$, expand the finite Euler product as a product of geometric series:Every integer $1\le n\le x$ has all its prime factors at most $x$, soThe harmonic series diverges, and therefore

Suppose instead that ${\sum }_{p}1/p$ converged. For $0<t\le 1/2$,Since ${\sum }_{p}1/p^2$ converges, this would makeconverge to a finite limit, contradicting the divergence of $P(x)$. Hence the Euler proof that the sum of reciprocals of primes diverges gives