From: Bob Silverman <bobs@rsa.com>
Subject: Re: Euler's phi function
Date: Thu, 02 Dec 1999 20:56:24 GMT
Newsgroups: sci.math
Keywords: lower bounds for Euler's phi function

In article <8267h7$fu2$1@nnrp1.deja.com>,
  ray steiner <steiner@math.bgsu.edu> wrote:
>  It is known that if n is a composite
> natural number then
> phi(n) <= n - sqrt(n)--a good
> upper bound for phi(n).
> Is there a good lower bound for
> phi(n)?


Let n = 2 * 3 * 5 * 7 * .....p

phi(n) = n(1-1/2)(1-1/3)(1-1/5)....

This is asymptotically  n * exp(-gamma)/log(p) from Mertens' Thm.
Since p ~ log(n)  we get the lower bound of n * exp(gamma)/loglog n.
gamma is Euler's constant.


--
Bob Silverman
"You can lead a horse's ass to knowledge, but you can't make him think"


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