Totient graph sublines

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I.N. Galidakis

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Nov 8, 2007, 5:30:32 PM11/8/07
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I was looking at the Wiki page for Euler's Totient function phi:

http://en.wikipedia.org/wiki/Euler's_totient_function

Wiki has to the right a graph of phi(n).

I decided to find out what were the major "trends" or approximate "accumulation
lines" in that graph, so I coded the graph in Maple. By "accident" I found that
certain functions fit almost exactly the points below the highest line (which
is, I suppose, y=x-1, since phi(p)=p-1 for p prime).

Here is the graph:
http://misc.virtualcomposer2000.com/phi.gif

The colors are as follows:

1) dark blue: f(x)=x/2
2) cyan: f(x) in {x/3,2*x/3}
3) yellow: f(x) in {x/5,2*x/5,3*x/5,4*x/5}
4) magenta: f(x) in {x/7,2*x/7,3*x/7,4*x/7,5*x/7,6*x/7}
5) green: f(x) in
{x/11,2*x/11,3*x/11,4*x/11,5*x/11,6*x/11,7*x/11,8*x/11,9*x/11,10*x/11}

Not all functions are associated with accumulation lines, but at least one
function from each set 1)-5) seems to follow an accumulation line.

The question is, can this coincidence be formulated as a number theoretical
conjecture?

My meager number theory memory (it's been at least 20 years since I had any)
tells me that since the "densest" lines are the line f(x)=x-1 and f(x)=x/2,
"most" of the naturals are such that either n=prime or such that approximately
half of the numbers k in {1,2,...,n-1} are such that GCD(k,n)=1 (Top line and
dark blue line)

Continuing, it looks like the next "most abundant" groups are the cyan lines
(the numbers such that GCD(k,n)=1 for approximately 1/3 of the numbers k in
{1,2,...,n-1}, which correspond to the functions x/3 and 2*x/3.

Yellow has a match for 2*x/5 and 4*x/5, but the rest of the positions look sort
of sparse.

Magenta also has several matches.

Can somebody give me a number theoretic formulation of what I am seeing?

Many thanks,
--
I.N. Galidakis

Gerry Myerson

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Nov 8, 2007, 7:02:33 PM11/8/07
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In article <1194561033.884249@athprx04>,
"I.N. Galidakis" <morp...@olympus.mons> wrote:

As you noticed, if p is prime then phi(p) = p - 1, so there will be
points just below y = x for each prime. But the primes are few and far
between once you get to large values of x, so in a sense those points
are not really very dense at all. But you also get close to y = x if
n = p q with p and q both large primes. In fact, if n is any number all
of whose prime factors are large then phi(n) will be relatively close to
n.

If n = 2 p or n = 4 p or n = 8 p or ... where p is an odd prime then
phi(n) is just less than n / 2, so that gives you points near y = x / 2.
Numbers of the form 3 p, 9 p, 27 p, etc., give points near y = 2 x / 3.
Numbers 6 p, 12 p, 18 p, 24 p, 36 p, 48 p, 54 p, 72 p, etc., give
points near y = x / 3.

--
Gerry Myerson (ge...@maths.mq.edi.ai) (i -> u for email)

I.N. Galidakis

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Nov 8, 2007, 8:46:14 PM11/8/07
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Thanks.
--
I.N. Galidakis

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