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?
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}]]
Best Answer
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 $A=6/\pi^2$.