Number theory Analytic number theory Theorem

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.

Reading path

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Divisibility definition
Prime number definition
Divergence of the sum of the reciprocals of the primes statement

Tag	`00RA98X2CP`
URI	`m2://00RA98X2CP`
Kind	statement, theorem
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Uses	Prime number
Same as	Wikidata Q1343972 Wikipedia Divergence of the sum of the reciprocals of the primes

Divergence of the sum of the reciprocals of the primes

Statement

Remarks

History

Reading path

Further reading

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