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Is this series known to converge, and if so, what does it converge to (if known)?

Where $p_n$ is prime number n, and $p_1 = 2$,

$$\sum\limits_{n=1}^{\infty} \frac{1}{p_n}$$

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A cultural note: Euler was (as far as I know) the first person to observe that this series diverges (and without assuming a priori that it was an infinite series), thus obtaining a new proof of the infinitude of the primes. This was a precursor to Dirichlet's work in which he proved the infinitude of primes in arithmetic progressions, and then Riemann, Hadamard, and de la Vallee Poussin's work leading to the prime number theorem. – Matt E Dec 31 '10 at 4:28
Unfortunately, the same technique does not apply for other problems. For instance, the sum of reciprocals of the twin primes converges and the existence of infinitely many twin primes remains open. – lhf Dec 31 '10 at 6:57

2 Answers 2

up vote 8 down vote accepted

No, it does not converge. See this: Proof of divergence of sum of reciprocals of primes.

In fact it is known that $$\sum_{p \le x} \frac{1}{p} = \log \log x + A + \mathcal{O}(\frac{1}{\log^2 x})$$

Related: Proving $\sum\limits_{p \leq x} \frac{1}{\sqrt{p}} \geq \frac{1}{2}\log{x} -\log{\log{x}}$

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I would like to note that this implies that according the Müntz-Szász Theorem that every continuous function in $[0,1]$ is a uniform limit of polynomials whose exponents are prime numbers!

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Could that implication possibly have any applications? – KCd Dec 31 '10 at 22:09
@KCd: I don't know, I'm more into analysis and there (at least the part I do) prime numbers are not that important, maybe it can be useful in analytic number theory. – Jonas Teuwen Jan 1 '11 at 15:13

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