Is n/ln(n) always less than pi(n)?

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Question:

The Prime Number Theorem says that as pi(n) approaches infinity, the formula n/ln(n) becomes a better approximation to pi(n) [where pi(n) is the number of primes less than or equal to some number n]. Does this necessarily mean that pi(n) will always and infallibly be greater than n/ln(n)? To me, this conclusion seems obvious, though I wouldn't know how to express its obviousness in rigorous mathematical terms.

Even more to the point, is it ever the case that pi(n + x)/((n + x)/ln(n + x)) is more than pi(n)/(n/ln(n)) [where x is some positive integer]?

I guess where the addition of "x" would introduce another prime would be such a case.

Hi!

The PNT says that pi(x) is asymptotically x/ln x which is a simple expression to calculate, but it is really not a good
approximation to pi(x). A better one is given by x/(ln x - 1), which is always larger than x/ln x, so we can expect
that x/ln x is too small (at least as long as x is sufficiently large).

HTH.

I should add that the logarithmic integral (http://en.wikipedia.org/wiki/Logarithmic_integral_function) is an even better approximation, and that pi(x) - Li(x) has been proven to change sign infinitely often.