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 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/