Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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

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

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

share|improve this question
3  
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
2  
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 $\sum\limits_{p \leq x} \frac{1}{\sqrt{p}} \geq \frac{1}{2}\log{x} -\log{\log{x}}$

share|improve this answer

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!

share|improve this answer
    
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

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.