Convergent series of primes - MathOverflow [closed] most recent 30 from http://mathoverflow.net 2013-06-19T21:53:22Z http://mathoverflow.net/feeds/question/79904 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/79904/convergent-series-of-primes Convergent series of primes HINOSXZ 2011-11-03T08:11:53Z 2011-11-05T01:59:33Z <p>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 <a href="http://en.wikipedia.org/wiki/Closed-form_expression" rel="nofollow">closed form</a> expression in terms of <a href="http://en.wikipedia.org/wiki/Elementary_function" rel="nofollow">elementary functions</a>?</p> <p>If not, can we prove the sum to be transcendental?</p>