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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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4 
 en.wikipedia.org/wiki/Elementary_function I am willing to relax all conditions, if it leads to any nontrivial results. – HINOSXZ Nov 3 2011 at 8:43
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 Simul-posted to m.se, math.stackexchange.com/questions/78541/… so voting to close here. – Gerry Myerson Nov 3 2011 at 12:56
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 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 2011 at 11:10
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 @Gerry: I agree with HINOSXZ. This seems like a very hard question, certainly not something I expect math.se to be able to answer. It's a pity it was closed here. – Charles Nov 4 2011 at 18:51
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 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 2011 at 10:23
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closed as no longer relevant by David Loeffler, Felipe Voloch, Gerry Myerson, Kevin O'Bryant, S. Carnahan♦ Nov 3 2011 at 15:55

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