Mertens' Proof of Mertens' Theorem

Abstract

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.

A modern proof of the Grossehilfsatz 2 added to Mertens' original proof. New biographical material on Mertens and new numerical table on the accuracy of the error term. Many improvements of details

Cited by in corpus (9)

• Some heuristics on the gaps between consecutive primes
• Exact Formulas for the Generalized Sum-of-Divisors Functions
• A Totient Function Inequality
• A Divisor Function Inequality
• Oscillations in Mertens Theorems and Other Finite Sums and Products
• The origin of the logarithmic integral in the prime number theorem
• Squarefree Integers And Extreme Values Of Some Arithmetic Functions
• Errata and Addenda to Mathematical Constants
• Divisor and Totient Functions Estimates