What are some physical, geometric, or probabilistic interpretations of the values of the Riemann zeta function at the positive integers greater than one?
I've found some examples:
1) In MO-Q111339 on a Tamagawa number, GH states
vol(SL2(R)/SL2(Z))=ζ(2).
2) In "Quantum Gauge Theories in Two Dimensions," Edward Witten derives
vol(M)=2(√2π)2g−2ζ(2g−2)
from a volume form for the moduli space M of flat connections on a gauge group (G=SU(2)) bundle over a compact two-dimensional manifold, a Riemann surface of genus g, and, for a connected sum of an orientable surface of genus g with k Klein bottles and r copies of the projective plane RP2, he derives
vol(M)=2(1−21−(2g−2+2k+r))(√2π)2g−2+2k+rζ(2g−2+2k+r).
3) In Wikipedia on the Stefan-Boltzmann law, the black body irradiance (total energy radiated per unit surface area of a black body per unit time) is given as
j∗=2π3!ζ(4)(kT)4c2h3.
(In n-dimensional space, it's proportional to n!ζ(n+1), and Planck's law for the electromagnetic energy density inside the 3-D black body has an extra factor of 4/c.)
4) In "Feynman's Sunshine Numbers," David Broadhurst gives the rate per unit surface area at which a black body at temperature T emits photons as
2π2!ζ(3)(kT)3c2h3.
(And the density of photons inside the body has an extra factor of 4/c.)
Motivation: I'm motivated not only by general interest, but also by MO-Q111165 and MO-Q111770. Determinants (volumes) of adjacency matrices and, therefore, the cycle index polynomials (CIPs) for the symmetric group pop up in statistical physics, e.g., in Potts q-color field theory and scaling random cluster model, and the CIPS can be "rescaled" to obtain the complete Bell polynomials (OEIS-A036040) which are related to the cumulant expansion polynomials (OEIS-A127671), both of which are related to statistical correlations and their diagrammatics (see references in OEIS-A036040).
5) The pn(z) of MO-Q111165 seem formally related to the Chern classes ck(V) of a direct (infinite) sum of line bundles V=L1⊕L2⊕.... :
With xi=c1(Li), the first Chern classes,
pk(z)=k!ck(V)=k!ek(x1,x2,...),
where ek are elementary symmetric polynomials. The ζ(n) can be identified as the power sums of the first Chern classes, and then, for example,
3!c3(V)=p3(z)=(z+γ)3−3ζ(2)(z+γ)+2ζ(3)
4!c4(V)=p4(z)=(z+γ)4−6ζ(2)(z+γ)2+8ζ(3)(z+γ)+3[ζ2(2)−2ζ(4)].
Update (Nov. 16, 2012): Just found the sequence in a thesis by R. Lu, "Regularized Equivariant Euler Classes and Gamma Functions," which discusses the relationship to Chern and Pontrjagin classes.
See also "An integral lift of the Gamma-genus" and "The motivic Thom isomorphism" by Jack Morava and "Hodge theoretic aspects of mirror symmetry" by L. Katzarkov, M. Kontsevich, and T. Pantev.
This post imported from StackExchange at 2014-04-07 13:24 (UCT), posted by SE-user Tom Copeland