From: shallit@graceland.uwaterloo.ca (Jeffrey Shallit)
Subject: Re: Prime Number Theorem - basic question
Date: 4 Sep 2000 23:27:18 GMT
Newsgroups: sci.math
Summary: [missing]

In article <20000903181319.29838.00000982@ng-md1.aol.com>,
MAppell917 <mappell917@aol.com> wrote:
>Hi,
>
>I have a question about the prime number theorem.  One good estimate for the
>total number of primes less than a number N is:
>
>N/ln(N)  where ln is the natural log.
>
>My question is for a large N does N/lnN always under estimate or over estimate
>the number of primes.  I'm looking for a formula that ALWAYS under estimates
>the true number of primes less than a large N.  If it only under estimates the
>number of primes when N is large that's fine but I want something that
>continues to under estimate the number of primes as N gets larger and larger. 
>Also, how large does N have to be?  
>
>Thanks.
>
>Mike

Try looking at Rosser and Schoenfeld,
Approximate formulas for some functions of prime numbers.
Ill. J. Math.  6 (1962), 64-94.

They prove, among other things, that
	pi(x) > x/(log x) for x > 17.

Jeffrey Shallit, Computer Science, University of Waterloo,
Waterloo, Ontario  N2L 3G1 Canada shallit@graceland.uwaterloo.ca
URL = http://www.math.uwaterloo.ca/~shallit/