Geometric / physical / probabilistic interpretations of Riemann zeta($n>1$)? 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
$$\operatorname{vol}(\mathrm{SL}_2(\mathbb{R})/\mathrm{SL}_2(\mathbb{Z}))=\zeta(2).$$
2) In "Quantum Gauge Theories in Two Dimensions," Edward Witten derives
$$\operatorname{vol}(\mathcal M)=\frac{2}{(\sqrt{2}\:\pi)^{2g-2}}\zeta(2g-2)$$
from a volume form for the moduli space $\mathcal M$ 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 $RP^2$, he derives  
$$\operatorname{vol}(\mathcal M) = \frac{2(1-2^{1-(2g-2+2k+r)})}{(\sqrt{2}\:\pi)^{2g-2+2k+r}} \zeta(2g-2+2k+r).$$ 
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
$$j^{*}=2\pi\:3!\zeta(4)\:\frac{(kT)^4}{c^2 h^3}.$$
(In $n$-dimensional space, it's proportional to $n!\zeta(n+1)$, 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
$$2\pi\:2!\zeta(3)\:\frac{(kT)^3}{c^2 h^3}.$$
(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 $p_n(z)$ of MO-Q111165 seem formally related to the Chern classes $c_k(V)$ of a direct (infinite) sum of line bundles $\:\:\:\: V=L_1\oplus L_2\oplus \cdots\:.$ :
With $x_{i}=c_1(L_i)$, the first Chern classes, 
$$p_k(z)=k!\:c_{k}(V)=k!\:e_{k}(x_{1},x_{2}, \ldots),$$
where $e_k$ are elementary symmetric polynomials. The $\zeta(n)$ can be identified as the power sums of the first Chern classes, and then, for example,
$$3!\:c_3(V)=p_3(z)=(z+\gamma)^3-3\zeta(2)(z+\gamma)+2\zeta(3)$$
$$4!\:c_4(V)=p_4(z)=(z+\gamma)^4-6\zeta(2)(z+\gamma)^2+8\zeta(3)(z+\gamma)+3[\zeta^2(2)-2\zeta(4)].$$
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.
 A: Hint: A literally enlightning geometric presentation of the Basel-problem at YouTube is the video

The stunning geometry behind this surprising equation by 3Blue1Brown

uploaded a short time ago. It needs a little physics, a little mathematics and a lot of creativity to derive this famous result completely different than Euler's proof. I also think it is pedagogically valuable and could be presented in many classes.
A: Elaborating on Nash's comment:
Oliver, special case of Zipf's law, right? That leads to the Zipf–Mandelbrot law that has a probability mass function of $$f(k;N,1,s)=\displaystyle\frac{\frac{1}{(k+1)^s}}{\sum_{i=1}^{N}\frac{1}{(i+1)^s}}$$ and then back to $\mathrm{vol}(\mathcal M)$ for the Klein bottles and particle statistics through $$(1-2^{1-s})\zeta(s)=\sum_{n=0}^{\infty }  \frac{1}{2^{n+1}}  \sum_{k=0}^{n}(-1)^k \binom{n}{k}\frac{1}{(k+1)^s}$$ 
$$=\eta(s)=\int_{0}^{\infty }\frac{1}{\exp(x)+1}\frac{x^{s-1}}{(s-1)!}dx$$
where $\eta(s)$ is the Dirichlet eta function, and so the Klein bottle manifolds seem connected to fermions and Fermi-Dirac statistics (as apropos Möbius twists), whereas the orientable Riemann manifolds seem related to bosons and Bose-Einstein statistics. 
And, Alan Gut in "Some remarks on the zeta distribution" defines the random variable $U$ with probability mass function (choose your favorite $\sigma= 2, 3, ...$) 
$$P(U_\sigma)=\frac{1}{\zeta(\sigma)n^\sigma}$$
and says, "The main point is that, for $\sigma>1$, one can view the normalized zeta function
$\varphi_{\sigma}(t)=\frac{\zeta(\sigma\:+\:i\:t)) }{\zeta(\sigma)}$ as the characteristic function of, as it turns out, a compound Poisson distribution. "
He shows how the moments and cumulants of the distribution (related to OEIS A036040 and A127671) given as functions of $\zeta(\sigma)$ and its derivatives are related to the von Mangoldt and Moebius functions and re-derives (and extends) an identity due to Selberg. 
On a tangent, the zeta values can be used to translate the Gamma-genus: 
With
$$R_z = z+\gamma + \sum_{n=1}^{\infty } (-1)^n\zeta (n+1)(d/dz)^n,$$ 
then
$$\displaystyle \exp(\omega\:R_z)\frac{e^{(t\:z)}}{t!}=\exp{(\omega\:d/dt)}\frac{e^{(t\:z)}}{t!}=\frac{e^{((t+\omega)\:z)}}{(t+\omega)!}$$
A: The probability that a ``random" number is $n$-free (AKA does not have any integer $b$ so that $b^n$ divides it) is given by $1/\zeta(n)$ 
A: There is a relation to the famous Veneziano amplitudes of nascent string theory (cf. Kholodenko, New Strings for old Veneziano amplitudes) through the factorials via the Euler beta function integral
$$ B(s,\alpha) = \left.\frac{(s-1)!(\alpha-1)!}{(s+\alpha-1)!} x^{s+\alpha-1} \right|_{x=1} = \int_0^x t^{s-1} \; (x-t)^{\alpha-1} \; dt |_{x=1}.$$
This can be morphed into the core Riemann-Liouville fractional integroderivative of fractional calculus (analytically continued) 
$$ D_x^{-s} x^{\alpha-1} \Big|_{x=1} = \left.\int_0^x \frac{t^{s-1}}{(s-1)!} \; (x-t)^{\alpha-1} \; dt \right|_{x=1}= \left. \frac{(\alpha-1)!}{(s+\alpha-1)!} x^{s+\alpha-1} \right|_{x=1}\; , $$
which can be expressed in terms of the infinitesimal generator 
$$ R_x = -\log(x) + \psi(1 + xD_x) = -(\log(x) + \gamma) - \sum_{n \ge 1} (-1)^n \zeta(n+1) \; (xD_x)^n \; ,$$ 
incorporating $\zeta(n>1)$ using well-known formulas for the digamma, or Psi, function $\psi(\beta)$ with $ \gamma =- \frac{d\beta!}{d\beta} |_{\beta=0}= \psi(1)$, the Euler-Mascheroni constant, as
$$ D_x^{-s} x^{\alpha-1} = e^{-sR_x} x^{\alpha-1} \; .$$
$$$$
$$$$
For easy reference, from the paper, Veneziano's 4-particle scattering amplitude is proportional to
$$A(s,t,u) = V(s,t) + V(s,u) + V(t,u) \; ,$$
where $V(s,t) = B(-\alpha(s),-\alpha(t))$, in which the arguments of beta are Regge trajectories.
A: 
The partition function $Z(T)$ [of primon gases] is given by the Riemann zeta function:
$${\displaystyle Z(T):=\sum _{n=1}^{\infty }\exp \left({\frac {-E_{n}}{k_{B}T}}\right)=\sum _{n=1}^{\infty }\exp \left({\frac {-E_{0}\log n}{k_{B}T}}\right)=\sum _{n=1}^{\infty }{\frac {1}{n^{s}}}=\zeta (s)}
$$
  with $s = E_0/k_BT$ where $k_B$ is Boltzmann's constant and $T$ is the absolute temperature.
The divergence of the zeta function at $s = 1$ corresponds to the divergence of the partition function at a Hagedorn temperature of $T_H = E_0/k_B$.

So choose your temperature in units of $T_H$...
A: Bourgade, Fujita & Yor shows to get Zeta functions from Cauchy Random Variables for even values and the $\chi_4$ L-functions for odd values.  For some reason they always come in this pair.  
This proof is simplified by Luigi Pace for $\zeta(2)$.  The Cauchy Random variable is 
$$ p_X (x) = \frac{2}{1+x^2}$$
when we look at the ration of two such random variables $Y = X/X'$.
$$ p_Y(y) = \frac{4}{\pi^2} \frac{\log y}{y^2-1}$$
Then observe $\mathbb{P}(Y \geq 1) = \mathbb{P}(X < X') = \frac{1}{2}$.  So they compute
$$ \sum_{k=0}^\infty \frac{1}{(2k+1)^2}= \int_0^1 \frac{-\log y}{1 - y^2} = \mathbb{P}(Y \geq 1)= \frac{\pi^2}{8}$$

I learned through a blog a proof using 2D Brownian motion at least for the case $\zeta(2)$.


Suppose that $f: \mathbb{C} \to \mathbb{C}$ is an analytic function on the neighbourhood of the unit disk. This
  function maps the unit disk  to  with boundary  where . A two dimensional brownian motion 
  started at $f(0)$ takes on average time
  $$ \mathbb{E}[\tau] = \sum_{k \geq 1} |a_k|^2 $$
  to exit domain $f(\mathbb{D})$ where $f(z) = \sum_{k \geq 0} a_k z^k$ and $\tau = \inf \{ t > 0: B_t \in \partial f(\mathbb{D}) \}$  is the hitting time of the boundary . 

You can get $\zeta(2)$ by considering Brownian motion on the strip $\{ x+iy: |x| < \pi/2 \}$ and evaluating the left and right sides.  The Brownian motion exit time is $\tau = \pi^2/4$ and $$f(z) = \log\left(\frac{1-z}{1+z}\right) = -2\left(z + \frac{z^3}{3} + \frac{z^5}{5}  + \dots \right)$$ maps the strip to the unit disk.  
This style is traced to the arXiv article by Greg Markowsky.


Also check out this paper by Noam Elkies who relates them to Alternating permutations.  One can show:

\begin{eqnarray*}
\sum_{k=0}^\infty \frac{1}{(2k+1)^2} &=& \sum_{k= 0}^\infty \int_0^1 \int_0^1 (xy)^{2k}dx\, dy \\\\
&=&  \int_0^1 \int_0^1 \left( \sum_{k= 0}^\infty(xy)^{2k} \right)dx \, dy  = \int_0^1 \int_0^1 \frac{ dx \, dy}{1 - (xy)^2}
\end{eqnarray*}
Then he does the strange Calabi substitution:
\[  x = \frac{\sin u}{\cos v} ,y = \frac{\sin v }{\cos u} \]
and recovers a calculus identity:
\[  \int_0^1 \int_0^1 \frac{ dx \, dy}{1 - (xy)^2} = \int_{u+v < \pi/2} 1 \, du \, dv = \frac{\pi^2}{8} \]
This proof is extended to higher dimensions in Elkies' paper.

You can then study the transform $T: L^2[0,\pi/2] \to L^2[0,\pi/2]$, the characteristic function of a triangle.

\[ (Tf)(x)=\int_0^{\pi/2 -x} f(t) \, dt  \]
and ask when does $Tf = \lambda f$.  The spectrum of this operator is
\[ \lambda = \frac{1}{4k+1} , f_\lambda(x) = cos (4k+1)u \]
Then one can take the trace of $T^n$ and compare to the volume of a polytope:
\begin{eqnarray}  \sum_{k=-\infty}^\infty \frac{1}{(4k+1)^k}&=& \sum_\lambda \langle f |T^n | f \rangle \\\\
& =& \mathrm{Vol}\bigg(\{0 < x_1 > x_2 < x_3 > \dots < x_{n-1} > x_n > \frac{\pi}{2}\}\bigg) \end{eqnarray}
The volume of this polytope can be expressed in terms of alternating permutations.  
I first learned of this iterated integral idea in Stanley's survey on Alternating Permutations, but also in some papers by Chebikin on Parking Functions, this seems to be an example of a chain polytope.


What other L-functions can take neat values like $L(k) \in \mathbb{Q}\pi^k$ where $k \in \mathbb{Z}$ ? Possibly need an algebraic extension $K / \mathbb{Q}$.
A: Here is a generalization of GH's answer concerning Tamagawa numbers:
Reference Paul Garett's notes: http://www.math.umn.edu/~garrett/m/v/volumes.pdf
$$ \operatorname{vol}(SL(n,\mathbb{Z}) \backslash SL(n, \mathbb{R})) = \zeta(2) \zeta(3) \zeta(4) \zeta(5) \cdots  \zeta(n).$$
$$\operatorname{vol}(Sp(n,\mathbb{Z})\backslash Sp(n, \mathbb{R})) = \zeta(2) \zeta(4) \zeta(6) \zeta(8) \cdots \zeta(2n)$$
This works for number field and global function fields analogous. There you have to replace Riemann zeta by the Dedekind zeta or Hasse-Weil zeta function and additionally root numbers show up.
A: Here's a physical interpretation of ζ(2):  Suppose you're stuck in stopped traffic on a long road, and that you're standing just in front of your car. Suppose further that all the cars behind you are blowing their horns at you.  Then the noise you hear is ζ(2) times louder than the sound of the first car's horn.  Of course, there's a similar visual interpretation, maybe more realistic, using the illumination given by a long string of Christmas lights.
A: I wrote an article about this very subject titled Zeta Values in Geometry and Topology three years ago. My thinking on the points in the article has evolved, in particular, I'm fairly convinced that Questions 0.1-0.4 aren't fruitful lines of inquiry. Still, the material therein is fascinating to me.
A: This isn't about the Riemann zeta function, but instead has to do with the zeta function of a number field $K$. But I found it unexpected. Let $\zeta_K(s)=\sum_{\mathfrak a\ne0} N\mathfrak a^{-s}$ be the usual zeta function, where we sum over all non-zero integral ideals, and let $\zeta_K^{\text{prin}}(s) = \sum_{(\alpha)\ne0} N\alpha^{-s}$ be the same, except now we sum only over principal ideals. Then the probability that an elliptic curve over $K$ has a global minimal Weierstrass equation is more-or-less equal to $$\frac{\zeta_K^{\text{prin}}(10)}{\zeta_K(10)},$$ where the probability is counted by taking elliptic curve $y^2=x^3+Ax+B$ with $A$ and $B$ in appropriate boxes. See:


*

*The density of elliptic curves having a global minimal Weierstrass
equation,  Ebru Bekyel, Journal of Number Theory, Volume 109,
Issue 1, November 2004, Pages 41–58 doi:10.1016/j.jnt.2004.06.003
A: You asked for a geometric interpretation. Let $\gamma_\pm(a)$ be the two geometric shapes described 
by the transcendental equation $e^{-y}\pm~e^{-\large x^a}=1$. Then their areas for $~a=\dfrac1n~$ are 
$$
\begin{align}
A_+&=-\int_0^\infty\ln\Big(1-e^{-\large x^a}\Big)dx=\Gamma(1+n)\cdot\zeta(1+n),\\
A_-&=-\int_0^\infty\ln\Big(1+e^{-\large x^a}\Big)dx=\Gamma(1+n)\cdot\zeta(1+n)\cdot(1-2^{-n}).
\end{align}
$$ 
Since the various polynomial curves, such as circles $(x^2+y^2=r^2)$, ellipses & hyperbolae 
$\bigg(\dfrac xa\bigg)^2\pm\bigg(\dfrac yb\bigg)^2=1$, superellipses $\bigg(\dfrac xa\bigg)^n+\bigg(\dfrac yb\bigg)^m=1$, etc., have been studied for 
centuries, it seemed only natural to expand this line of thought by inquiring about the properties 
of exponential ones, such as $a^{\large x^n}\pm b^{\large y^m}=1$.
A: $$\zeta(3) = \int_0^1\int_0^1\int_0^1 \frac{1}{1-xyz}dxdydz $$
is Apéry's constant, found in calculations of the electron's gyromagnetic ratio and according to its Wikipedia entry:
The reciprocal of $\zeta(3)$ is the probability that any three positive integers, chosen at random, will be relatively prime (in the sense that as N goes to infinity, the probability that three positive integers less than N chosen uniformly at random will be relatively prime approaches this value).

Apery's constant also appears in "Determinants of Laplacians, the Ray-Singer Torsion on Lens Spaces
and the Riemann zeta function" by Charles Nash and D. J. O’ Connor in evaluations of the Ray-Singer torsion in certain lens spaces in the expressions for volume elements on the discrete moduli spaces associated with the Laplacians.
A: 
From Strange Numbers Found in Particle Collisions
by K. Hartnett 
Periods and amplitudes were presented together for the first time in 1994 by Kreimer and David Broadhurst, a physicist at the Open University in England, with a paper following in 1995. The work led mathematicians to speculate that all amplitudes were periods of mixed Tate motives — a special kind of motive named after John Tate, emeritus professor at Harvard University, in which all the periods are multiple values of one of the most influential constructions in number theory, the Riemann zeta function. In the situation with an electron-positron pair going in and a muon-antimuon pair coming out, the main part of the amplitude comes out as six times the Riemann zeta function evaluated at three.
A: The sequence $a(2k)=-2(2i\pi)^{-2k}\zeta(2k),a(1)=\frac{-1}2$ is the convolutive inverse of the sequence $\frac1{(k+1)!},k\ge 0$ 
