Proof that the sum of the reciprocals of the primes diverges

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, the product is taken over all primes p; in this article, a sum or product taken over p always represents a sum or product taken over a specified set of primes, unless noted otherwise.)

Such infinite products are today called Euler products. The product above is a reflection of the fundamental theorem of arithmetic. Of course, the above "equation" is not necessary because the harmonic series is known (by other means) to diverge.

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. (In modern language, we now say that the existence of infinitely many primes is reflected by the fact that the Riemann zeta function has a simple pole at s = 1.)

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:

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