Packaging Spectra (as in Partition Functions and L/$ζ$-functions) to Reveal Symmetries (Reciprocity) in Nature and in Numbers

Abstract: In statistical mechanics one packages the possible energies of a system into a partition function. In number theory, and elsewhere in mathematics, one packages the spectrum of a phenomenon, say the prime numbers, into a $\zeta$-function or more generally into an L-function. These packaging functions have symmetries and properties not at all apparent from the energies or the primes themselves, often exhibiting scaling symmetries for example. One might be able to understand those symmetries and compute the packaging function independently of the actual packaging. And so one finds a way of putting together objects into a package, and ways of discerning symmetries of that package independent of the actual mode of packaging. This is a recurrent theme of the Langlands Program as well. Packaging is also found in Weyl's asymptotics and "hearing the shape of a drum" (Kac), the Schwinger Greens function in quantum electrodynamics packaging the Feynman sum of histories, and more generally in the Selberg trace formula.