What are some geometric / physical / probabilistic interpretations of the Riemann zeta function at integer arguments n ≤ 1?

Question

Introduction: This is slightly edited and generalised version of a question I asked on the Physics Stack Exchange website. This question has a twin brother asked here on MO, only now we consider values of the Riemann zeta function $\zeta(n)$ where the function diverges: $n \leq 1$. I am especially interested in the physical interpretations, but I value geometrical/probabilistic or other interpretations highly as well. Recently, I also learned that some divergent series have a combinatorial interpretation as well. See this post on "The Everything Seminar". I am curious about such interpretations of divergent series as well.

Body: For my bachelor's thesis, I am investigating Divergent Series. Apart from the mathematical theory behind them (which I find fascinating), I am also interested in their applications in physics. Currently, I am studying the divergent series that arise when considering the Riemann zeta function at negative arguments. The Riemann zeta function can be analytically continued. By doing this, finite constants can be assigned to the divergent series. For $n \geq 1$, we have the formula:

$$ \zeta(-n) = - \frac{B_{n+1}}{n+1} . $$

This formula can be used to find:

• $\zeta(-1) = \sum_{n=1}^{\infty} n = - \frac{1}{12} . $ This identity is used in Bosonic String Theory to find the so-called "critical dimension" $d = 26$. For more info, one can consult the relevant wikipedia page.
• $\zeta(-3) = \sum_{n=1}^{\infty} n^3 = - \frac{1}{120} $ . This identity is used in the calculation of the energy per area between metallic plates that arises in the Casimir Effect.

Furthermore, the sum $\sum_{n=0}^{\infty}2^n $ converges to $-1$ in the 2-adic number system. I guess this could allow a geometric interpretation of this divergent sum, to a certain extent.

My first question is: do more of these values of the Riemann zeta function at negative arguments arise in physics/geometry/probability theory? If so: which ones, and in what context?

Furthermore, I consider summing powers of the Riemann zeta function at negative arguments. I try to do this by means of Faulhaber's formula. Let's say, for example, we want to compute the sum of $$p = \Big( \sum_{k=1}^{\infty} k \Big)^3 . $$ If we set $a = 1 + 2 + 3 + \dots + n = \frac{n(n+1)}{2} $, then from Faulhaber's formula we find that $$\frac{4a^3 - a^2}{3} = 1^5 + 2^5 + 3^5 + \dots + n^5 , $$ from which we can deduce that $$ p = a^3 = \frac{ 3 \cdot \sum_{k=1}^{\infty} k^5 + a^2 }{4} .$$ Since we can also sum the divergent series arising from the Riemann zeta function at negative arguments by means of Ramanujan Summation (which produces that same results as analytic continuation) and the Ramanujan Summation method is linear, we find that the Ramanujan ($R$) or regularised sum of $p$ amounts to $$R(p) = R(a^3) = \frac{3}{4} R\Big(\sum_{k=1}^{\infty} k^5\Big) + \frac{1}{4} R(a^2) . $$ Again, we know from Faulhaber's Formula that $a^2 = \sum_{k=1}^{\infty} k^3 $ , so $R(a^2) = R(\zeta(-3)) = - \frac{1}{120} $, so $$R(p) = \frac{3}{4} \Big(- \frac{1}{252} \Big) + \frac{ - ( \frac{1}{120} )} {4} = - \frac{17}{3360} . $$

My second (bunch of) question(s) is: Do powers of these zeta values at negative arguments arise in physics/probability theory/geometry? If so, how? Are they summed in a manner similar to process I just described, or in a different manner? Of the latter is the case, which other summation method is used? Do powers of divergent series arise in physics in general? If so: which ones, and in what context?

My third and last (bunch of) question(s) is: which other divergent series arise in physics/probability theory/geometry (not just considering (powers of) the Riemann zeta function at negative arguments) ? I know there are whole books on renormalisation and/or regularisation in physics. However, for the sake of my bachelor's thesis I would like to know some concrete examples of divergent series that arise in physics which I can study. It would also be nice if you could mention some divergent series which have defied summation by any summation method that physicists (or mathematicians) currently employ. Please also indicate as to how these divergent series arise in physics, or how they can be geometrically/probabilistically/combinatorially interpreted.

Answer 1

Let $g \geq 1$ be an integer. Let $\mathcal{M}_{g,1}$ be the moduli space of genus g Riemann surfaces with one marked point. It is an orbifold (each point comes with an automorphism group). Let $\chi(\mathcal{M}_{g,1})$ be the orbifold Euler characteristic of $\mathcal{M}_{g,1}$ (which takes into acount the automorphism groups). Then Harer and Zagier have shown : $\chi(\mathcal{M}_{g,1})= \zeta(1-2g)$.

Example : $\chi(\mathcal{M}_{1,1})=\zeta(-1)=-1/12$, this is easy to see directly and is in some sense the same -1/12 that the one appearing in the derivation of the critical dimension of the bosonic string theory.

Addition: as mentionned in the question, values of Riemann zeta function at negative integers are almost the same as Bernoulli numbers, so this mathoverflow question should be relevant :

Why do Bernoulli numbers arise everywhere?

In particular, as $\frac{x}{1-e^{-x}} = \sum_{n=0}^{\infty} (-1)^{n-1} \frac{\zeta(1-n)}{(n-1)!}x^{n}$, these values are deeply related to the Todd class and so to the various uses of the Riemann-Roch theorem (or more general index theorems).

Example : the degree 2 part of the Todd class is $\frac{1}{12}(c_{1}^{2}+c_{2})$ (this expression is well-known for example in the theory of surfaces: Noether's formula). In some sense, this $1/12$ is the same as the one appearing in string theory : Belavin and Knizhnik have shown how the Polyakov measure on the moduli space of genus g curves ($g \geq 2$) is related to a well-known result of Mumford on relations between some cohomology classes on the moduli of curves. These classes are first Chern classes of some line bundles coming from the universal curve. The Mumford's proof is to use the relative Riemann-Roch theorem for the morphism from the universal curve to the moduli space. As one is interested by $c_1$ on the moduli space and as one integrates along the fibers of the morphism which are curves, the relevant part of the Todd class is the one of degree 2 in the Chern classes, hence the apparition of the $1/12$.