The Harmonies of Nature (Revised)

The lesson here is that counting is related to the symmetry of what one is counting; the harmonies of nature, its resonant tones, are related to counting; and the counting as expressed in the resonant or main frequencies of a system will tell you about its geometry (or shape).

Mathematics demands precise statements of claims, or of what is to be proven, just because slight deviations might not be the case or at least not proven by the current means. You might be able to prove something weaker for a less precise statement, but often you want the strongest statement possible. In effect, an identity may have a very sharp profile. Moreover, a claim may well have several different ways of being proven, in effect it is an identity in a manifold presentation of profiles, so to speak.

So when I am tempted to see more general lessons from mathematical work, it is surely the case that by being more general I am likely to be sacrificing what can be actually proven,  but at the same time I am getting a hint at what is really going on in this model.

To preview: Where a function, say one that packages the prime numbers, crosses the horizontal axis (its "zeros") is deeply related to those primes, much as where a polynomial crosses the horizontal axis (its solutions) is related to the polynomial’s coefficients. And, in general, processes of counting are related to a symmetry of a system. Finally, you can "hear the shape of a drum," namely, listening for its resonant frequencies will tell you about its shape. Put differently, resonances are related to geometry.

In number theory, there is a way of packaging the prime numbers into something that looks like a function, called the Riemann zeta function, denoted by the Greek letter ζ, or more generally an L-function. It turns out that the zeros of that function, that is the z's such that ζ(z)=0 or L(z)=0, can be shown to have a nice relationship to the prime numbers (as we might hope for since the function, zeta or L is defined by its packaging the primes or other such numbers). Studying zeta's or L's zeros may be easier than studying the primes directly. Moreover, it is possible to connect zeta (and often also L) to another function θ, a theta function (a cousin of the sines and cosines), which has deep symmetries and is rather easier to study than is zeta or L. Theta is automorphic in that changing its dependent variation, the z in theta(z), leads to theta in a well prescribed way, and in the case of theta there is a multiplicative factor, a "modulus."

Zeta and L, is a harmonic series, that is it is a sum of terms g(n)/n^s, for all n or all primes (that is how it packages those numbers), and for the simplest case, zeta itself, g(n)=1. g(n) may come from counting things (enumerating the primes, for zeta) or from a matrix system’s traces (that is, “the characters of a group representation” of its symmetries) or from counting the number of solution to an equation modulo a prime number. There is as well Fourier series (sometimes called harmonic analysis), the sum of h(n)×exp 2πins, f(s), and g(n)= h(n)!! The counting function, L, is nicely related to a function f (or sometimes theta) that exhibits remarkable symmetries.

Last, it may be possible to express a system’s behavior in terms of a matrix (as in quantum mechanics as matrix mechanics). If that is so, the matrix's eigenvalues, in effect its resonant tones or its spectrum of masses of particles, will tell you about its geometry in terms of area, perimeter, etc.

Technically: zeros and primes are interwoven in the zeta function; a theta function is related to zeta, and more generally in the Langlands Program, the automorphic (representation) is related to the combinatoric or motivic or representation theoretic; and, in various "trace formulas," the trace of an operator (the sum of its resonant tones) is related to the geometry of that operator (and in its own way, the Weyl asymptotics say that you can hear the shape of a drum by counting up the resonant tones of the drum).