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

TL;DR

This paper reviews multiple proofs of the asymptotic behavior of the sum of reciprocals of primes, demonstrating that it approximates ln(ln n) with bounded error, highlighting classical and modern approaches.

Contribution

It provides a comprehensive exposition of various known proofs for the prime reciprocal sum's asymptotic estimate, including those independent of and reliant on the prime number theorem.

Findings

Sum of 1/p over primes p up to n behaves like ln(ln n) + O(1)

Multiple proofs confirm the divergence and asymptotic estimate of the sum

Different proof techniques, including Euler's and Mertens' methods, are presented.

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.