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,[1] and strengthens Euclid's 3rd-century-BC result that there are infinitely many prime numbers and Nicole Oresme's 14th-century proof of the divergence of the sum of the reciprocals of the integers (harmonic series).

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 (log log) indicates that the divergence might be very slow, which is indeed the case.

First, we will describe how Euler originally discovered the result.

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

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's proof works by first taking the natural logarithm of each side, then using the Taylor series expansion for log x as well as the sum of a converging series:

of which, as shown in a later 1748 work,[2] the right hand side can be obtained by setting x = 1 in the Taylor series expansion

It is almost certain that Euler meant that the sum of the reciprocals of the primes less than n is asymptotic to log log 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.

[3] Thus Euler obtained a correct result by questionable means.

The following proof by contradiction comes from Paul Erdős.

Let pi denote the ith prime number.

Assume that the sum of the reciprocals of the primes converges.

Then there exists a smallest positive integer k such that

For a positive integer x, let Mx denote the set of those n in {1, 2, ..., x} which are not divisible by any prime greater than pk (or equivalently all n ≤ x which are a product of powers of primes pi ≤ pk).

We will now derive an upper and a lower estimate for |Mx|, the number of elements in Mx.

For large x, these bounds will turn out to be contradictory.

This produces a contradiction: when x ≥ 22k + 2, the estimates (2) and (3) cannot both hold, because ⁠x/2⁠ ≥ 2k√x.

Here is another proof that actually gives a lower estimate for the partial sums; in particular, it shows that these sums grow at least as fast as log log n. The proof is due to Ivan Niven,[4] adapted from the product expansion idea of Euler.

is represented in one of the terms of AB when multiplied out.

Dividing through by ⁠5/3⁠ and taking the natural logarithm of both sides gives

This shows that the series on the left diverges.

prime factors (counting multiplicities) only from the set

contains at least one term for each reciprocal of a positive integer not divisible by any

One proof[6] is by induction: The first partial sum is ⁠1/2⁠, which has the form ⁠odd/even⁠.

as the (n + 1)st prime pn + 1 is odd; since this sum also has an ⁠odd/even⁠ form, this partial sum cannot be an integer (because 2 divides the denominator but not the numerator), and the induction continues.

Another proof rewrites the expression for the sum of the first n reciprocals of primes (or indeed the sum of the reciprocals of any finite set of primes) in terms of the least common denominator, which is the product of all these primes.