Is there an approximation for the maximum difference between P(n) and n x ln(n) as a function of n, where P(n) denotes the nth prime number?
In other words, given D(n) = Max(|P(n) - n x ln(n)|), is there a known formula for D(n)?
Thanks
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$\begingroup$ Take a look at the first two pages of arxiv.org/abs/1203.5413 $\endgroup$– blablerCommented Jun 26, 2013 at 22:36
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1 Answer
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There are several good inequalities for this difference, e.g., $$ n \log n + n(\log \log n -1)<P(n) < n \log n + n\log \log n $$ for all $n\ge 6$, which can be derived from the prime number theorem. Better estimates are given (among others) in the thesis of Dusart - see http://www.unilim.fr/laco/rapports/1998/R1998_06.pdf.
answered Jun 26, 2013 at 19:49
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$\begingroup$ Thank you! Is there a similar approximation for P2(n) which denotes the nth number with exactly 2 prime factors? And more generally, for Pk(n)? $\endgroup$ Commented Jun 29, 2013 at 18:35