of the Riemann zeta function as a function of a complex variable. Mertens' proof is in that respect remarkable. Indeed, with modern notation it yields whereas the prime number theorem (in its simplest form, without error estimate), can be shown to imply In 1909 Edmund Landau, by using the best version of the prime number theorem then at his disposition, proved that holds; in particular the error term is smaller than for any fixed integer k. A simple summation by parts exploiting the strongest form known of the prime number theorem improves this to for some . Similarly a partial summation shows that is implied by the PNT. Mertens' third theorem and sieve theory An estimate of the probability of () having no factor is given by This is closely related to Mertens' third theorem which gives an asymptotic approximation of Proof The main step in the proof of Mertens' second theorem is where the last equality needs which follows from . Thus, we have proved that . Since the sum over prime powers with converges, this implies . A partial summation yields . References Further reading Yaglom and Yaglom Challenging mathematical problems with elementary solutions Vol 2, problems 171, 173, 174 External links Varun Rajkumar, π(x) and the Sieve of Eratosthenes Mathematical series Summability theory Theorems about prime numbers