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Does anyone happen to have at hand a short, elegant proof that demonstrates that there do (or do not) exist one or more algebraically representable prime number generating functions?

After having thought about this for not a trivial amount of time I have found myself unable to make any leeway, mainly because I can't relate algebraic representable-ness to the definition or really any properties of prime numbers.

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What do you mean by "algebraically representable"? – Qiaochu Yuan Jun 1 '12 at 3:30
A reasonable version might be: there is a nonconstant bivariate polynomial $P(z,w)$ such that for some sequence $p_n$ of distinct primes, $P(n,p_n) = 0$. – Robert Israel Jun 1 '12 at 19:04
I guess that wasn't precise. By 'algebraically representable' I meant: en.wikipedia.org/wiki/Algebraic_function – enthdegree Jun 3 '12 at 0:24

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Try http://en.wikipedia.org/wiki/Prime_generating_function

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$$\pi(x)=\sum_{n\le x}\left|\operatorname{sgn}\prod_{k=2}^n\prod_{l=2}^n(n-kl)\right|$$ is a representation that can be linked fairly directly to the definition of primes. – anon Jun 1 '12 at 1:54
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@anon I'm not sure that can be called "algebraic." – Thomas Andrews Jun 1 '12 at 2:12

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