Given the Prime number theorem with the error term : $π(x) = li(x) + O(x.e^{−c\sqrt{log(x)}}))$ , how do you go about obtaining an expression for the nth prime?
Any help would be much appreciated
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If we take $x=p_n$ in the prime number theorem we obtain $n=\pi(p_n)=\frac{p_n}{\log(p_n)}+$ O-Term, so that $p_n=n\log(n)+$ O-Term. For a discussion see Estimate of $n$th prime. For better estimates on the $n$-prime see the article by Dusard.
Dietrich Burde
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Wouldn't that be n=π(p_n)=li(p_n)+ O-Term? how do i go from there? – Maria Mar 04 '15 at 22:02
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We have $li(p_n)=p_n/\log(p_n)+$ O-Term, so this goes as before. – Dietrich Burde Mar 04 '15 at 22:20