# Convergent series of primes [closed]

If $f(n)$ is a strictly increasing elementary function from the reals to the reals, and $p(n)$ is the $n$'th prime number. Is there any $f(n)$ such that $\sum_{n=1}^\infty\frac{1}{f(p(n))}$ is algebraic, or has a closed form expression in terms of elementary functions?

If not, can we prove the sum to be transcendental?

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## closed as no longer relevant by David Loeffler, Felipe Voloch, Gerry Myerson, Kevin O'Bryant, S. Carnahan♦Nov 3 '11 at 15:55

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en.wikipedia.org/wiki/Elementary_function I am willing to relax all conditions, if it leads to any nontrivial results. – user1708 Nov 3 '11 at 8:43
Simul-posted to m.se, math.stackexchange.com/questions/78541/… so voting to close here. – Gerry Myerson Nov 3 '11 at 12:56
No, you should give people a chance to give you a satisfactory answer on math.stackexchange. If, after a week or so, you aren't happy with what you've seen there, you could come back here, give a full report on what was posted there, and then open a meta-thread to ask for re-opening. – Gerry Myerson Nov 4 '11 at 11:10
@Charles, there are some very good people at m.se. I reckon it's a pity OP chose to post it at two places at the same time without telling either place it was being posted to the other. – Gerry Myerson Nov 5 '11 at 10:21
If people on m.se "don't even understand the question," then it is your job to improve your exposition. Go back, do some editing, meet them halfway. – Gerry Myerson Nov 5 '11 at 10:23