Mertens' Proof of Mertens' Theorem

We present a self-contained elementary and detailed exposition of Mertens’ own proof of his theorem on the divergence of the series of the reciprocals of the primes and compare it with the modern proofs. His proof contains explicit numerical error estimates and computations of the Mertens “constant,” both of which are missing from modern expositions. In as much as his strategy and techniques are quite different from today’s methods, his original methods deserve to be better known.