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I was playing around with a few of the number theory functions in Mathematica when I found an interesting relationship between some of them. Below I have plotted points with coordinates $x=\dfrac{n\cdot\mu(n)}{\sigma(n)}$ and $y=\dfrac{n\cdot\mu(n)}{\phi(n)}$ for $n$ from 1 to 400,000, where $\sigma$ represents the sum of the divisors function, $\phi$ represents the totient function, and $\mu$ represents the Moebius function. I've also plotted a bit of the curve $y = \frac{9}{10x^{1.5}}$.

It seems to me there is some kind of approximate relationship being shown here. Unfortunately, I know basically nothing about this subject, aside from the basic definitions of the functions I'm using.

Can anyone provide any insight about what is going on in my example? More generally, how well explored are these kinds of relationships$-$is there a great deal of theory behind all of this?

a relationship between the totient function and divisor sigma

The code to make this diagram in Mathematica is:

Show[ParallelMap[{(#/DivisorSigma[1, #])*
  MoebiusMu[#], (#/EulerPhi[#])*MoebiusMu[#]} &, Range[400000]] //
   ListPlot[#, PlotRange -> All,PlotStyle -> {Black, PointSize[0.005]}] &,   
   Plot[0.9/x^1.5, {x, 0.1, 1}], {x, -1, 1}]]
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1 Answer

up vote 7 down vote accepted

Theorem 329 of Hardy and Wright, An Introduction to the Theory of Numbers, says there is a positive constant $A$ such that $$A\lt{\sigma(n)\phi(n)\over n^2}\lt1$$

In a footnote, they show that $A=6/\pi^2$.

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Thanks. I find these kinds of facts remarkable. Do you know what are the prerequisites for Hardy and Wright? –  JOwen Feb 7 '12 at 23:10
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"The book is written for mathematicians, but it does not demand any great mathematical knowledge or technique. In the first eighteen chapters we assume nothing that is not commonly taught in schools, and any intelligent university student should find them comparatively easy reading. The last six are more difficult, and in them we presuppose a little more, but nothing beyond the content of the simpler university courses." My opinion is that the main prerequisite is that elusive quality, "mathematical maturity". –  Gerry Myerson Feb 8 '12 at 0:01
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