Divergence of The Sum of The Reciprocals of The Primes

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 indicates that the divergence might be very slow, which is indeed the case, see Meissel–Mertens constant.