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



The maximum difference between the nth prime number and n x ln(n)
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>.


