Divergence of the sum of the reciprocals of the primes

The sum of the reciprocals of all prime numbers diverges; that is:

This was proved by Leonhard Euler in 1737, and strengthens Euclid's 3rd-century-BC result that there are infinitely many prime numbers.

There are a variety of proofs of Euler's result, including a lower bound for the partial sums stating that

for all natural numbers n. The double natural logarithm (ln ln) indicates that the divergence might be very slow, which is indeed the case. See Meissel–Mertens constant.

First, we describe how Euler originally discovered the result. He was considering the harmonic series

He had already used the following "product formula" to show the existence of infinitely many primes.

Here, p refers to the set of all primes.

Such infinite products are today called Euler products. The product above is a reflection of the fundamental theorem of arithmetic. Euler noted that if there were only a finite number of primes, then the product on the right would clearly converge, contradicting the divergence of the harmonic series.

Euler considered the above product formula and proceeded to make a sequence of audacious leaps of logic. First, he took the natural logarithm of each side, then he used the Taylor series expansion for ln x as well as the sum of a converging series:

for a fixed constant K < 1. Then he invoked the relation

which he explained, for instance in a later 1748 work, by setting x = 1 in the Taylor series expansion

This allowed him to conclude that

It is almost certain that Euler meant that the sum of the reciprocals of the primes less than n is asymptotic to ln ln n as n approaches infinity. It turns out this is indeed the case, and a more precise version of this fact was rigorously proved by Franz Mertens in 1874. Thus Euler obtained a correct result by questionable means.

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