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 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. The double natural logarithm indicates that the divergence might be very slow, which is indeed the case. For instance, the sum of the reciprocals of all prime numbers up to a certain integer does not exceed 3 for the first time until 5195977 and does not exceed 4 for the first time until approximately 1.8 quintillion.

The harmonic series

First, we will 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 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.

Proofs

Euler's proof

Euler's proof works by first taking the natural logarithm of each side, then using the Taylor series expansion for as well as the sum of a converging series:
for a fixed constant. Then, by using the following relation:
of which, as shown in a later 1748 work, the right hand side can be obtained by setting in the Taylor series expansion
Thus,
It is almost certain that Euler meant that the sum of the reciprocals of the primes less than is asymptotic to as 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.

Erdős's proof by upper and lower estimates

The following proof by contradiction comes from Paul Erdős.
Let denote the th prime number. Assume that the sum of the reciprocals of the primes converges.
Then there exists a smallest positive integer such that
For a positive integer, let denote the set of those in which are not divisible by any prime greater than . We will now derive an upper and a lower estimate for, the number of elements in. For large, these bounds will turn out to be contradictory.
;Upper estimate:
;Lower estimate:

Proof that the series exhibits log-log growth

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. The proof is due to Ivan Niven, adapted from the product expansion idea of Euler. In the following, a sum or product taken over always represents a sum or product taken over a specified set of primes.
The proof rests upon the following four inequalities:
  • Every positive integer can be uniquely expressed as the product of a square-free integer and a square as a consequence of the fundamental theorem of arithmetic. Start with where the βs are 0 or 1. Factor out one copy of all the primes whose β is 1, leaving a product of primes to even powers, itself a square. Relabeling: where the first factor, a product of primes to the first power, is square free. Inverting all the s gives the inequality
To see this, note that and That is, is one of the summands in the expanded product. And since is one of the summands of, every summand is represented in one of the terms of when multiplied out. The inequality follows.
Combining all these inequalities, we see that
Dividing through by and taking the natural logarithm of both sides gives
as desired. Q.E.D.
Using
, the above constant can be improved to ; in fact it turns out that
where is the Meissel–Mertens constant.

Proof from Dusart's inequality

From Dusart's inequality, we get
Then
by the integral test for convergence. This shows that the series on the left diverges.

Geometric and harmonic-series proof

The following proof is modified from James A. Clarkson.
Define the k-th tail
Then for, the expansion of contains at least one term for each reciprocal of a positive integer with exactly prime factors only from the set. It follows that the geometric series contains at least one term for each reciprocal of a positive integer not divisible by any. But since always satisfies this criterion,
by the divergence of the harmonic series. This shows that for all, and since the tails of a convergent series must themselves converge to zero, this proves divergence.

Partial sums

While the partial sums of the reciprocals of the primes eventually exceed any integer value, they never equal an integer.
One proof is by induction: The first partial sum is, which has the form. If the th partial sum has the form, then the st sum is
as the st prime is odd; since this sum also has an form, this partial sum cannot be an integer, and the induction continues.
Another proof rewrites the expression for the sum of the first reciprocals of primes in terms of the least common denominator, which is the product of all these primes. Then each of these primes divides all but one of the numerator terms and hence does not divide the numerator itself; but each prime does divide the denominator. Thus the expression is irreducible and is non-integer.