Divergence of the sum of the reciprocals of the primes explained

The sum of the reciprocals of all prime numbers diverges; that is: $\sum_\frac1p = \frac12 + \frac13 + \frac15 + \frac17 + \frac1 + \frac1 + \frac1 + \cdots = \infty$

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 $\sum_\frac1p \ge \log \log (n+1) - \log\frac6$ for all natural numbers . The double natural logarithm indicates that the divergence might be very slow, which is indeed the case. See Meissel–Mertens constant.

The harmonic series

First, we describe how Euler originally discovered the result. He was considering the harmonic series $\sum_^\infty \frac = 1 + \frac + \frac + \frac + \cdots = \infty$

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

$\sum_^\infty \frac = \prod_ \left(1+\frac+\frac+\cdots \right) = \prod_ \frac$

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 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 as well as the sum of a converging series:

$\begin \log \left(\sum_^\infty \frac\right) & = \log\left(\prod_p \frac\right) = -\sum_p \log \left(1-\frac\right) \\[5pt] & = \sum_p \left(\frac + \frac + \frac + \cdots \right) \\[5pt] & = \sum_\frac + \frac\sum_p \frac + \frac\sum_p \frac + \frac\sum_p \frac+ \cdots \\[5pt] & = A + \frac B+ \frac C+ \frac D + \cdots \\[5pt] & = A + K\end$

for a fixed constant . Then he invoked the relation

$\sum_^\infty\frac = \log\infty,$

which he explained, for instance in a later 1748 work,^[2] by setting in the Taylor series expansion

$\log\left(\frac1\right)=\sum_^\infty\fracn.$

This allowed him to conclude that $A = \frac + \frac + \frac + \frac + \frac + \cdots = \log \log \infty.$

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.^[3] 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

$\sum_^\infty \frac 1 < \frac12 \qquad(1)$

For a positive integer, let denote the set of those in which are not divisible by any prime greater than (or equivalently all which are a product of powers of primes). 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:

Every in can be written as with positive integers and, where is square-free. Since only the primes can show up (with exponent 1) in the prime factorization of , there are at most different possibilities for . Furthermore, there are at most possible values for . This gives us the upper estimate

|M_x| \le 2^k\sqrt \qquad(2)

Lower estimate:

The remaining numbers in the set difference are all divisible by a prime greater than . Let denote the set of those in

Notes and References

Leonhard. Euler. Leonhard Euler. Variae observationes circa series infinitas. Various observations concerning infinite series. Commentarii Academiae Scientiarum Petropolitanae. 9. 1737. 160–188.
Book: Euler, Leonhard. Leonhard Euler. Introductio in analysin infinitorum. Tomus Primus. Introduction to Infinite Analysis. Volume I. Bousquet. Lausanne. 1748. p. 228, ex. 1.
Franz Mertens . Mertens . F. . 1874 . Ein Beitrag zur analytischen Zahlentheorie . . 78 . 46–62 .