Past exam of the mathematics course of the University of Cambridge / 2006 / ii / Paper 4 / 1H / Solution

Expand the finite Euler product into convergent geometric series. Every positive integer $n\le x$ has all its prime factors at most $x$, soThe harmonic series diverges, hence $P(x)\to 0$. If the sum of reciprocal primes were finite, then the inequality $-\log (1-t)\le 2t$ for $0\le t\le 1/2$ would giveuniformly in $x$, contradicting $P(x)\to 0$. Thus the sum of prime reciprocals diverges. Unique factorization justifies the expansion; no estimate for the distribution of primes is required. This is the Euler proof that the sum of reciprocals of primes diverges.