• Register
PhysicsOverflow is a next-generation academic platform for physicists and astronomers, including a community peer review system and a postgraduate-level discussion forum analogous to MathOverflow.

Welcome to PhysicsOverflow! PhysicsOverflow is an open platform for community peer review and graduate-level Physics discussion.

Please help promote PhysicsOverflow ads elsewhere if you like it.


New printer friendly PO pages!

Migration to Bielefeld University was successful!

Please vote for this year's PhysicsOverflow ads!

Please do help out in categorising submissions. Submit a paper to PhysicsOverflow!

... see more

Tools for paper authors

Submit paper
Claim Paper Authorship

Tools for SE users

Search User
Reclaim SE Account
Request Account Merger
Nativise imported posts
Claim post (deleted users)
Import SE post

Users whose questions have been imported from Physics Stack Exchange, Theoretical Physics Stack Exchange, or any other Stack Exchange site are kindly requested to reclaim their account and not to register as a new user.

Public \(\beta\) tools

Report a bug with a feature
Request a new functionality
404 page design
Send feedback


(propose a free ad)

Site Statistics

204 submissions , 162 unreviewed
5,026 questions , 2,180 unanswered
5,344 answers , 22,686 comments
1,470 users with positive rep
815 active unimported users
More ...

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

+ 5 like - 0 dislike

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 $$R(p) = \frac{3}{4} \Big(- \frac{1}{252} \Big) + \frac{ ( \frac{1}{120} )} {4} = - \frac{1}{1120} . $$

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.

This post imported from StackExchange at 2014-04-07 13:22 (UCT), posted by SE-user Max Muller
asked Mar 30, 2014 in Mathematics by Max Muller (115 points) [ no revision ]
retagged Apr 7, 2014

1 Answer

+ 7 like - 0 dislike

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.

This post imported from StackExchange at 2014-04-07 13:22 (UCT), posted by SE-user user25309
answered Mar 31, 2014 by user25309 (70 points) [ no revision ]
Interesting. But which sense is "some sense" here? Do you have more details on this hint?

This post imported from StackExchange at 2014-04-07 13:22 (UCT), posted by SE-user Urs Schreiber

Your answer

Please use answers only to (at least partly) answer questions. To comment, discuss, or ask for clarification, leave a comment instead.
To mask links under text, please type your text, highlight it, and click the "link" button. You can then enter your link URL.
Please consult the FAQ for as to how to format your post.
This is the answer box; if you want to write a comment instead, please use the 'add comment' button.
Live preview (may slow down editor)   Preview
Your name to display (optional):
Privacy: Your email address will only be used for sending these notifications.
Anti-spam verification:
If you are a human please identify the position of the character covered by the symbol $\varnothing$ in the following word:
Then drag the red bullet below over the corresponding character of our banner. When you drop it there, the bullet changes to green (on slow internet connections after a few seconds).
To avoid this verification in future, please log in or register.

user contributions licensed under cc by-sa 3.0 with attribution required

Your rights