$\sum_{p\le n} 1/p = \ln(\ln n) + O(1)$: An Exposition

Abstract

It is well known that $\sum_{p\le n} 1/p =\ln(\ln(n)) + O(1)$ where $p$ goes over the primes. We give several known proofs of this. We first present a a proof that $\ge \ln(\ln(n)) + O(1)$. This is based on Euler's proof that $\sum_p 1/p$ diverges. We then present three proofs that $\sum_{p\le n} 1/p \le \ln(\ln(n)) + O(1)$ The first one, due to Mertens, does not use the prime number theorem. The second and third one do use the prime number theorem and hence are shorter.