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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 $\sum_{n = 1}^\infty x^{p_n}$

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Mayank Pandey, you are truly an optimist –  John Wiltshire-Gordon Nov 4 '13 at 18:05
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The short answer is 'no'. Relevant reading material: en.wikipedia.org/wiki/Formula_for_primes Did you research this question before asking it? –  user43208 Nov 4 '13 at 18:09
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2 Answers

How about a generating function of the form $$ \left(1-p_1^{-s}\right)\left(1-p_2^{-s}\right)\left(1-p_3^{-s}\right)\cdots = \frac{1}{\zeta(s)},\qquad \mathrm{Re}\;s > 1 $$

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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.

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