Newsgroups: sci.math
From: haoyuep@aol.com (Dan Hoey)
Date: Thu, 15 Apr 2004 18:04:02 +0000 (UTC)
Subject: Re: Continuous prime number function

On 14 Apr 2004, Peter Webb wrote:

>What we need is a continuous function that has f^n(x) defined for all
>0<=x<=1 and all n, f^n(0)=0 for all n, but isn't just f(x)=0. I
>recall from my calculus days of 30 years ago that such functions
>exist.

Good on you for remembering.  An example is  f(x)= exp(-x^2).  But
this isn't _quite_ enough for Marc Moore's problem, because you also
need  f^n(1)=0  so that the pieces together.  For this, use
  f(x)=exp(-x^2)(ke - exp(-(1-x)^2))
which interpolates between  f(0)=0  and  f(1)=k  with all derivatives
vanishing at the endpoints.

It almost seems a shame to make the derivatives zero, since they
really only have to match up, and this way the function is bumpier
than it has to be.  Still, if the twin prime conjecture is
true, the derivative will have to hit each point of  (0,1/2]
infinitely often any way we slice it.  Maybe I should ask how
low we can keep  lim sup f''(x).  There's probably a good
lower bound based on the prime constellation conjecture.

Dan Hoey
haoyuep@aol.com