Geometric / physical / probabilistic interpretations of Riemann zeta(n>1)? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T17:22:00Z http://mathoverflow.net/feeds/question/112062 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/112062/geometric-physical-probabilistic-interpretations-of-riemann-zetan1 Geometric / physical / probabilistic interpretations of Riemann zeta(n>1)? Tom Copeland 2012-11-11T04:25:51Z 2012-11-17T14:39:48Z <p><strong>What are some physical, geometric, or probabilistic interpretations of the values of the Riemann zeta function at the positive integers greater than one?</strong></p> <p>I've found some examples:</p> <p>1) In <a href="http://mathoverflow.net/questions/111339/simple-tamagawa-number-calculations" rel="nofollow">MO-Q111339</a> on a Tamagawa number, GH states</p> <p>$$\mathrm{vol}(\mathrm{SL}_2(\mathbb{R})/\mathrm{SL}_2(\mathbb{Z}))=\zeta(2).$$</p> <p>2) In "Quantum Gauge Theories in Two Dimensions," Edward Witten derives</p> <p>$$\mathrm{vol}(\mathcal M)=\frac{2}{(\sqrt{2}\:\pi)^{2g-2}}\zeta(2g-2)$$</p> <p>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 </p> <p>$$\mathrm{vol}(\mathcal M)=\frac{2(1-2^{1-(2g-2+2k+r)})}{(\sqrt{2}\:\pi)^{2g-2+2k+r}} \zeta(2g-2+2k+r).$$ </p> <p>3) In Wikipedia on the <a href="http://en.wikipedia.org/wiki/Stefan-Boltzmann_law" rel="nofollow">Stefan-Boltzmann law</a>, the black body irradiance (total energy radiated per unit surface area of a black body per unit time) is given as</p> <p>$$j^{*}=2\pi\:3!\zeta(4)\:\frac{(kT)^{4}}{c^{2}h^{3}}.$$</p> <p>(In n-dimensional space, it's proportional to $n!\zeta(n+1)$, and <a href="http://en.wikipedia.org/wiki/Plancks_law" rel="nofollow">Planck's law</a> for the electromagnetic energy density inside the 3-D black body has an extra factor of $4/c$.)</p> <p>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</p> <p>$$2\pi\:2!\zeta(3)\:\frac{(kT)^{3}}{c^{2}h^{3}}.$$</p> <p>(And the density of photons inside the body has an extra factor of $4/c$.)</p> <p><strong>Motivation</strong>: I'm motivated not only by general interest, but also by <a href="http://mathoverflow.net/questions/111165/riemann-zeta-function-at-positive-integers-and-an-appell-sequence-of-polynomials" rel="nofollow">MO-Q111165</a> and <a href="http://mathoverflow.net/questions/111770/cycling-through-the-zeta-garden-zeta-functions-for-graphs-cycle-index-polynomia" rel="nofollow">MO-Q111770</a>. Determinants (volumes) of adjacency matrices and, therefore, the <a href="http://en.wikipedia.org/wiki/Cycle_index" rel="nofollow">cycle index polynomials</a> (CIPs) for the symmetric group pop up in statistical physics, e.g., in <a href="http://arxiv.org/abs/1104.4323" rel="nofollow">Potts q-color field theory and scaling random cluster model</a>, and the CIPS can be "rescaled" to obtain the <a href="http://en.wikipedia.org/wiki/Complete_Bell_polynomials" rel="nofollow">complete Bell polynomials</a> (<a href="https://oeis.org/A036040" rel="nofollow">OEIS-A036040</a>) which are related to the <a href="http://en.wikipedia.org/wiki/Cumulants" rel="nofollow">cumulant</a> expansion polynomials (<a href="https://oeis.org/A127671" rel="nofollow">OEIS-A127671</a>), both of which are related to statistical correlations and their diagrammatics (see references in <a href="https://oeis.org/A036040" rel="nofollow">OEIS-A036040</a>).</p> <p>5) The $p_n(z)$ of <a href="http://mathoverflow.net/questions/111165/riemann-zeta-function-at-positive-integers-and-an-appell-sequence-of-polynomials" rel="nofollow">MO-Q111165</a> seem formally related to the <a href="http://en.wikipedia.org/wiki/Chern_class" rel="nofollow">Chern classes</a> $c_{k}(V)$ of a direct (infinite) sum of line bundles $\:\:\:\: V=L_{1}\oplus L_2\oplus ...\:.$ :</p> <p>With $x_{i}=c_{1}(L_i)$, the first Chern classes, </p> <p>$$p_k(z)=k!\:c_{k}(V)=k!\:e_{k}(x_{1},x_{2}, ...),$$</p> <p>where $e_{k}$ are <a href="http://en.wikipedia.org/wiki/Newton%2527s_identities#Expressing_power_sums_in_terms_of_elementary_symmetric_polynomials" rel="nofollow">elementary symmetric polynomials</a>. The $\zeta(n)$ can be identified as the power sums of the first Chern classes, and then, for example,</p> <p>$$3!\:c_{3}(V)=p_3(z)=(z+\gamma)^3-3\zeta(2)(z+\gamma)+2\zeta(3)$$ $$4!\:c_{4}(V)=p_4(z)=(z+\gamma)^4-6\zeta(2)(z+\gamma)^2+8\zeta(3)(z+\gamma)+3[\zeta^2(2)-2\zeta(4)].$$</p> <p><strong>Update (Nov. 16, 2012)</strong>: Just found the sequence in a thesis by R. Lu, "<a href="http://digital.library.adelaide.edu.au/dspace/bitstream/2440/50479/1/02whole.pdf" rel="nofollow">Regularized Equivariant Euler Classes and Gamma Functions</a>," which discusses the relationship to Chern and Pontrjagin classes. </p> <p>See also "<a href="http://arxiv.org/abs/1101.1647" rel="nofollow">An integral lift of the Gamma-genus</a>" and "<a href="http://arxiv.org/abs/math/0306151" rel="nofollow">The motivic Thom isomorphism</a>" by Jack Morava and "<a href="http://arxiv.org/abs/0806.0107" rel="nofollow">Hodge theoretic aspects of mirror symmetry</a>" by L. Katzarkov, M. Kontsevich, and T. Pantev.</p> http://mathoverflow.net/questions/112062/geometric-physical-probabilistic-interpretations-of-riemann-zetan1/112124#112124 Answer by Tom Copeland for Geometric / physical / probabilistic interpretations of Riemann zeta(n>1)? Tom Copeland 2012-11-11T22:49:25Z 2012-11-17T14:39:48Z <p><strong>Elaborating on Nash's comment</strong>:</p> <p>Oliver, special case of <a href="http://en.wikipedia.org/wiki/Zipf%2527s_law" rel="nofollow">Zipf's law</a>, right? That leads to the <a href="http://en.wikipedia.org/wiki/Zipf-Mandelbrot_law" rel="nofollow">Zipfâ€“Mandelbrot law</a> 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 $$(1-2^{1-s})\zeta(s)=\sum_{n=0}^{\infty } \frac{1}{2^{n+1}} \sum_{k=0}^{n}(-1)^k \binom{n}{k}\frac{1}{(k+1)^s}$$ $$=\eta(s)=\int_{0}^{\infty }\frac{1}{\exp(x)+1}\frac{x^{s-1}}{(s-1)!}dx$$</p> <p>where $\eta(s)$ is the <a href="http://en.wikipedia.org/wiki/Dirichlet_eta_function" rel="nofollow">Dirichlet eta function</a>, and so the Klein bottle manifolds seem connected to fermions and <a href="http://en.wikipedia.org/wiki/Fermi-Dirac_statistics" rel="nofollow">Fermi-Dirac statistics</a> (as apropos <a href="http://en.wikipedia.org/wiki/M%C3%B6bius_strip" rel="nofollow">Möbius twists</a>), whereas the orientable Riemann manifolds seem related to bosons and Bose-Einstein statistics. </p> <p><strong>And, Alan Gut</strong> in "<a href="http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.66.3284&amp;rep=rep1&amp;type=pdf" rel="nofollow">Some remarks on the zeta distribution</a>" defines the random variable $U$ with probability mass function (choose your favorite $\sigma= 2, 3, ...$) </p> <p>$$P(U_\sigma)=\frac{1}{\zeta(\sigma)n^\sigma}$$</p> <p>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. "</p> <p>He shows how the moments and cumulants of the distribution (related to OEIS <a href="https://oeis.org/A036040" rel="nofollow">A036040</a> and <a href="https://oeis.org/A127671" rel="nofollow">A127671</a>) 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. </p> <p><strong>On a tangent</strong>, the zeta values can be used to translate the Gamma-genus: </p> <p>With $$R_z = z+\gamma + \sum_{n=1}^{\infty } (-1)^n\zeta (n+1)(d/dz)^n,$$ </p> <p>then $$\displaystyle \exp(\omega\:R_z)\frac{e^{(t\:z)}}{t!}=\exp{(\omega\:d/dt)}\frac{e^{(t\:z)}}{t!}=\frac{e^{((t+\omega)\:z)}}{(t+\omega)!}$$</p> http://mathoverflow.net/questions/112062/geometric-physical-probabilistic-interpretations-of-riemann-zetan1/112132#112132 Answer by Jonah Sinick for Geometric / physical / probabilistic interpretations of Riemann zeta(n>1)? Jonah Sinick 2012-11-11T23:47:17Z 2012-11-11T23:47:17Z <p>I wrote an article about this very subject titled <a href="http://mathisbeauty.org/ZetaValuesinGeometryandTopology1016.pdf" rel="nofollow">Zeta Values in Geometry and Topology</a> 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.</p> http://mathoverflow.net/questions/112062/geometric-physical-probabilistic-interpretations-of-riemann-zetan1/112147#112147 Answer by John Mangual for Geometric / physical / probabilistic interpretations of Riemann zeta(n>1)? John Mangual 2012-11-12T05:48:07Z 2012-11-12T13:17:56Z <p>Bourgade, Fujita &amp; Yor shows to get <a href="http://ecp.ejpecp.org/article/view/1244" rel="nofollow">Zeta functions from Cauchy Random Variables</a> for even values and the $\chi_4$ L-functions for odd values. For some reason they always come in this pair. </p> <p>This proof is simplified by <a href="http://www.jstor.org/stable/10.4169/amer.math.monthly.118.10.954" rel="nofollow">Luigi Pace</a> for $\zeta(2)$. The Cauchy Random variable is $$p_X (x) = \frac{2}{1+x^2}$$</p> <p>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 $\mathbb{P}(Y \geq 1) = \mathbb{P}(X &lt; X') = \frac{1}{2}$. So they compute $$\sum_{k=0}^\infty \frac{1}{(2k+1)^2}= \int_0^1 \frac{-\log y}{1 - y^2} = \mathbb{P}(Y \geq 1)= \frac{\pi^2}{8}$$</p> <p><hr> I learned through a blog a proof using <a href="http://linbaba.wordpress.com/2012/06/24/zeta-two-probabilistic-proof/" rel="nofollow">2D Brownian motion</a> at least for the case $\zeta(2)$.</p> <blockquote> <p>Suppose that $f: \mathbb{C} \to \mathbb{C}$ is an analytic function on the neighbourhood of the unit disk. This<br> 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 . </p> </blockquote> <p>You can get $\zeta(2)$ by considering Brownian motion on the strip $\{ x+iy: |x| &lt; \pi/2 \}$ and evaluating the left and right sides. The Brownian motion exit time is $\tau = \pi^2/4$ and $$f(z) = \log(\frac{1-z}{1+z}) = -2\left(z + \frac{z^3}{3} + \frac{z^5}{5} + \dots \right)$$ maps the strip to the unit disk. </p> <p>This style is traced to the arXiv article by <a href="http://arxiv.org/abs/1108.1188" rel="nofollow">Greg Markowsky</a>. <hr> <hr> Also check out this paper by Noam Elkies who relates them to <a href="http://mathdl.maa.org/images/upload_library/22/Ford/Elkies561-573.pdf" rel="nofollow">Alternating permutations</a>. One can show:</p> <p>\begin{eqnarray*} \sum_{k=0}^\infty \frac{1}{(2k+1)^2} &amp;=&amp; \sum_{k= 0}^\infty \int_0^1 \int_0^1 (xy)^{2k}dx\, dy \\ &amp;=&amp; \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 <a href="http://www.staff.science.uu.nl/~kolk0101/Publications/calabi.pdf" rel="nofollow">Calabi substitution</a>: $x = \frac{\sin u}{\cos v} ,y = \frac{\sin v }{\cos u}$</p> <p>and recovers a calculus identity: $\int_0^1 \int_0^1 \frac{ dx \, dy}{1 - (xy)^2} = \int_{u+v &lt; \pi/2} 1 \, du \, dv = \frac{\pi^2}{8}$</p> <p>This proof is extended to higher dimensions in Elkies' paper.</p> <p><hr> You can then study the transform $T: L^2[0,\pi/2] \to L^2[0,\pi/2]$, the characteristic function of a triangle.</p> <p>$(Tf)(x)=\int_0^{\pi/2 -x} f(t) \, dt$</p> <p>and ask when does $Tf = \lambda f$. The spectrum of this operator is</p> <p>$\lambda = \frac{1}{4k+1} , f_\lambda(x) = cos (4k+1)u$</p> <p>Then one can take the trace of $T^n$ and compare to the volume of a polytope:</p> <p>\begin{eqnarray} \sum_{k=-\infty}^\infty \frac{1}{(4k+1)^k}&amp;=&amp; \sum_\lambda \langle f |T^n | f \rangle \\ &amp; =&amp; \mathrm{Vol}\bigg(\{0 &lt; x_1 > x_2 &lt; x_3 > \dots &lt; x_{n-1} > x_n > \frac{\pi}{2}\}\bigg) \end{eqnarray} The volume of this polytope can be expressed in terms of alternating permutations. </p> <p>I first learned of this iterated integral idea in Stanley's <a href="http://arxiv.org/abs/0912.4240" rel="nofollow">survey on Alternating Permutations</a>, but also in some papers by Chebikin on <a href="http://arxiv.org/abs/0806.0440" rel="nofollow">Parking Functions</a>, this seems to be an example of a <em>chain polytope</em>. <hr> <hr> 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 $K / \mathbb{Q}$.</p>