---
tag: 00RA98X2CP
kind: statement
role: theorem
name: Divergence of the sum of the reciprocals of the primes
msc: [11N05]
uses: [0SN0MAEKYX]
sources:
  - url: https://en.wikipedia.org/wiki/Divergence_of_the_sum_of_the_reciprocals_of_the_primes
    locator: lead section
same_as:
  wikidata: Q1343972
  wikipedia: 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|0SN0MAEKYX]], 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|0292EPPPT0]], 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 $4$.