Divergence of the sum of the reciprocals of the primes

Statement

The series $\sum_{p} \frac{1}{p}$, where $p$ runs over the prime numbers, diverges. That is, the partial sums $\sum_{p \le x} \frac{1}{p}$ are not bounded as $x \to \infty$.

Remarks

The result implies Euclid's theorem, since a finite sum is bounded. The partial sums grow like $\ln \ln x$. More precisely, $\sum_{p \le x} \frac{1}{p} - \ln \ln x$ has a finite limit as $x \to \infty$. In contrast, the sum of $1 / n^2$ over all positive integers $n$ converges. So the primes are not as rare as the squares.

History

Euler proved the result in 1737. Erdős gave an elementary proof by upper and lower estimates.

Explanations

L1 human

The sum $\frac{1}{2} + \frac{1}{3} + \frac{1}{5} + \frac{1}{7} + \frac{1}{11} + \cdots$ becomes larger than each number that you choose, if you take enough terms. It grows very slowly: the terms for the primes up to $10^9$ still give a sum less than $4$.