"It could easily be shown..." Probability and Pi and the Riemann Zeta Function

 Rushed re-edit , Fingers crossed

Beware of articles that begin, "It could easily be shown..." It is like arm wrestling a two year old, if you win, so what, and if you LOSE??? Yow....
I know this, and yet, I still proceed foolishly to read them. The one currently on my mind was a "Note on Pi" by R. Chartes in the March 1904 Philosophical Magazine, my current old document of choice. It pointed out that ICEBS "that if two numbers are written down at random, the probability that they will be prime to each other is 6/pi²."
Here it is from Wolfram Mathworld:

This is the reciprocal of the famous answer to the Basel problem evaluate by Euler.

The fact that the probability that two random numbers are relatively prime was equal to this value was discovered by M. Cesaro and J. J. Sylvester in the same year, 1883. Sylvester gives a proof in a footnote to a paper I found in his collected works (page 602)

The original image from Sylvester seems to have dissappeared into the abyss since 2009, so I share a proof I found on the Physics Harvard edu websight,

Ok, yeah, that is sort of easy, and I should have figured it out...The proof is easy to extend to the probability that three numbers are relatively prime is the reciprocal of the sum of the reciprocal of the cubes (if that seems hard to read, try to write it). More simply, the probability is the reciprocal of ζ(3)=

113+123+133+143+⋯

 Let  a,b and c be integers chosen at random

$<br>$

The probability that a, b, and c have no common divisor:

Pr(⊥(a,b,c))=1ζ(3)$Pr(\perp (𝑎,𝑏,𝑐))={\displaystyle \frac{1}{𝜁\left(3\right)}}$

where ζ$𝜁$ denotes the zeta function:

ζ(3)=113+123+133+143+⋯

the decimal value is approximately 
1ζ(3)

1ζ(3)=0⋅831907372580707…

 Strangely, the discovery (by Sylvester) is nested in work he was doing with Farey Fractions.  

If you haven't been exposed to Farey Fractions, a quick share from Wolfram's Mathworld

The Farey sequence  for any positive integer  is the set of irreducible rational numbers  with  and  arranged in increasing order. The first few are

			(1)
			(2)
			(3)
			(4)
			(5)

(OEIS A006842 and A006843). Except for , each  has an odd number of terms and the middle term is always 1/2.

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The image at the top shows a pair of coordinate axes with a point (x,y) painter black if GCF(m,n)=1, and white otherwise.

For students, it should be made clear that these methods apply to random numbers up to a very large n. They should understand that these are the probabilities as the range of the random selections approach infinity.  

They should also realize that if one of the numbers is prime, the pair will always be co-prime.  As the numbers get very large, the percentage of the primes gets smaller.  A good general rule for the number of primes that are equal to or less than n approaches x/ ln (n) as n approaches infinity. For smaller numbers they should be helped to realize that this limit usually under-counts the number of primes.  For n=100, for instance, 100/ln(100) =21.7... but there are actually 25 prime numbers less than 100. As n gets larger the ratio of primes to predicted primes grows much closer to 1.  (A great use of spreadsheets to let them explore on their own.  The top graph below shows this error.

I think one of the really nice things that can be done with younger students studying common factors and slope (can I say in Alg I?) is to show them that the greatest common factor of m and n is the number of lattice points on the line from (0,0) to (m,n)....[not counting (0,0)] Here is a graph of the segments to (4,10) and (12,3)