Generating function for the nth prime Is there a generating function for nth prime that is easy to deal with? i.e. is there a simple closed form for the series $p_1x + p_2x^2 + ...$ or of the form $\sum_{n = 1}^\infty x^{p_n}$
 A: How about a generating function of the form
$$
\left(1-p_1^{-s}\right)\left(1-p_2^{-s}\right)\left(1-p_3^{-s}\right)\cdots
= \frac{1}{\zeta(s)},\qquad \mathrm{Re}\;s > 1
$$
A: No, at least there is none known which would give you what you seem to be looking for.
Otherwise it would be so well-known that I think you would know it as well! --
Though if you are actually happy with much weaker results in this direction, I suggest
that you specify this in the question.
A: In 1983, Mr. A. Venugopalan published an explicit formula for generating the (n+1)th prime number in  the Proceedings of the Indian Academy of Science (Math.Sei.),Vol.92, No.1,September 1983,pp.49–52 . This is the first formula which generated ONLY prime numbers. He also published other formulas, namely, formula for twin primes, formula for number of primes and formula for number of twin primes. The amazing thing of the fourth formula is, it gives  a promising step to  solve the Twin Prime Conjecture as he  gave a formula for number of twin primes not exceeding any given integer. If one can prove the solution to the equation is infinity or not by substituting the given number as infinity still remains open.
The heading of this paper is, "Formula for primes, twinprimes, number of primes and number of twinprimes" . You may be able to get a reprint by searching on Google.
