Some interesting facts on relative primeness here:
So there is an n-dimensional way of picturing whether any n integers a_1, a_2, ..., a_n are coprime - if you have a clear line of sight from the origin (0,0,...,0) to the point (a_1,a_2,...,a_n). And the probability of coprimality is just the probability that there's a clear line of sight to a randomly chosen point with integer coordinates. Naively, you might suppose that each such point has infinitely many points of the form (k.a_1,k.a_2,...,k.a_n) "behind" it - again, as viewed from the origin - and so that there are infinitely many number sets with gcd > 1 for each individual number set with gcd = 1, suggesting that the probability is infinitesimal, i.e. zero. But I suppose we are talking about asymptotic behavior - the fraction of points within a certain distance from the origin which are visible from it, as the distance increases without bound. Evidently it's one of those tricky infinite subsets where the order in which you count things matters...
- 6/pi^2 "is the fractional number of lattice points visible from the origin" - which is the geometric interpretation of coprimality.
- The probability that n positive integers are coprime is 1/zeta(n), where zeta() is the famous Riemann zeta function. (Here's a proof for n=2.)
Anyway, back to pi. This all started with the observation that pi^2 shows up in connection with a 4-dimensional sphere. Could there be some systematic connection between n-spheres and n-coprimality? Before proceeding, let me quote Mathworld again:Unfortunately, geometers and topologists adopt incompatible conventions for the meaning of "n-sphere," with geometers referring to the number of coordinates in the underlying space... and topologists referring to the dimension of the surface itself... ("Hypersphere")In other words, for a geometer, a circle is a 2-sphere, but for a topologist, it's a 1-sphere. Up above, Michael Gurvich is using the naming conventions of a topologist; this table of area and volume formulae uses the geometer's convention.
As the "Hypersphere" article describes, the general formula for hyper-surface area is 2.pi^(n/2)/Gamma(n/2), where the gamma function is a generalization of the factorial function n!. Now, as it happens the gamma function also shows up in connection with the zeta function. The zeta function, which is what we need to get at those coprimality probabilities, is fundamentally harder to calculate than the gamma function, even just at integer values. We can calculate Gamma(1/2), Gamma(1), Gamma(3/2),... - i.e. all the gammas we need for the area formula - but for the zeta function, we only have a formula for even integers. zeta(3) has also been studied, but in general the Riemann zeta for odd integers seems to be very poorly understood.
But back to the search for a connection. Is there perhaps some geometric construction, associated with an n-dimensional hypersphere, with which the zeta function can in turn be associated? Well, a start might be made by embedding the n-dimensional lattice of points with integer coordinates, through the inverse of an n-dimensional stereographic projection. The "line of sight" then follows a "great circle" in the (n-1)-dimensional hypersurface. The link would be complete, if the formula for hyper-surface area played a part in the derivation of the formula for fraction of points visible, with the gamma function directly carrying over from one to the other...
-- (August 14, 2004) on Pi