I attended a talk this morning on Ray-Singer torsion, in which Rafael Siejakowski introduced [zeta function regularization](http://en.wikipedia.org/wiki/Zeta_function_regularization) in a compelling way. The goal is to define the determinant of a positive self-adjoint operator $A$ with "pure point spectrum" $0>\lambda\_1>\lambda\_2>\cdots$. The definition of the determinant is $\exp(-\zeta\_A^\prime(0))$ where $\zeta\_A$ is the zeta function $\zeta\_A(s)=\sum\_{i=1}^\infty (-\lambda\_i)^{-s}$. This sum diverges in general- but it converges for values of $s$ with large enough real part, and we define it for other values of $s$ (including zero) by analytic continuation.


Why should this be related to the determinant? Well, in the finite dimensional case (the motivating case is when $A$ is the `combinatorial Laplacian'), then $\zeta\_A(s)=\sum\_{i=1}^N (-\lambda\_i)^{-s}$ is a finite sum. In this case:


$\zeta^\prime\_A(s)=\sum\_{i=1}^N -\ln (-\lambda\_i)(-\lambda\_i)^{-s}$


and


$\zeta^\prime\_A(0)=-\ln \prod\_{i=1}^N(-\lambda\_i)=-\ln \det A$.


This looks to me like an ad-hoc trick, indicating that I don't understand what is actually going on.


The equation $\det(A)=\exp(-\zeta\_A^\prime(0))$ (in the finite dimensional case an equation, in the infinite dimensional case a definition) equates two familiar mathematical quantities:


1. The determinant, which I can think of as a volume, as an action on a highest exterior power, or maybe most evocatively as the signed sum of weights of non-intersecting paths in a graph between "source" vertexes $a\_1,\ldots,a\_n$ and "sink" vertices $b\_1,\ldots,b\_n$. See [this blog post](http://qchu.wordpress.com/2009/11/17/the-lindstrom-gessel-viennot-lemma/).
2. The [Riemann zeta function](http://en.wikipedia.org/wiki/Riemann_zeta_function), which I don't understand conceptually almost as well, but which is heavily studied and so is clearly important and natural.



> 
> **Question**: Is there a conceptual (hand-wavy is fine) explanation for zeta function regularization, and for how this expression in the zeta function is capturing the idea of a "determinant"? How is the derivation which I wrote above more than an ad-hoc trick? Is there a sense in which the derivative of a zeta function at zero heuristically calculates a signed sum of weights of non-intersecting paths, or something like that? 
> 



Understanding zeta function regularization
Not a complete answer. First, here is an alternate derivation of the result in the finite-dimensional case which might be more enlightening. If $A$ is positive self-adjoint, we can write $A = \exp(L)$ for some self-adjoint $L$. This lets us define
$$A^s = \exp(sL)$$


for all real $s$. The trace
$$\text{tr}(A^s) = \sum\_{i=1}^n \lambda\_i^s = \zeta\_A(s)$$


is then the zeta function associated to $A$ (I am getting rid of all of the minus signs). Now, for small $\epsilon$ we can write
$$A^{s+\epsilon} = A^s A^{\epsilon} = A^s (1 + \epsilon L + O(\epsilon^2))$$


so it follows that
$$\zeta\_A(0)' = \text{tr}(L).$$


But **Jacobi's identity** $\det \exp M = \exp \text{tr } M$ gives
$$\det A = \det \exp L = \exp \text{tr } L = \exp \zeta\_A(0)'$$


and we conclude. 




---


So what conceptual significance can we attach to the above? Well, it seems to me like we should think of the map $s \mapsto A^s$ as a representation of the Lie group $\mathbb{R}$, so the zeta function is the character of the corresponding representation. The derivative of the zeta function at zero gives the trace of the infinitesimal generator of this representation, $L$, which generates the abelian Lie algebra $\mathbb{R}$. And this is connected to the determinant of $A$ by Jacobi's identity.


So I think most of what needs explanation is Jacobi's identity. I freely admit that I do not have a good conceptual explanation of Jacobi's identity (beyond the fact that it's obvious for diagonalizable matrices). In [these](http://qchu.wordpress.com/2009/08/23/newtons-sums-necklace-congruences-and-zeta-functions/) [two](http://qchu.wordpress.com/2009/11/04/newtons-sums-necklace-congruences-and-zeta-functions-ii/) blog posts I attempted to meander towards a combinatorial proof of Jacobi's identity in the form
$$\det (I - At)^{-1} = \exp \text{tr } \log (I - At)^{-1}$$


(where $A$ was the adjacency matrix of a graph) but didn't quite succeed. There is a combinatorial proof of Jacobi's identity in the literature [due to Foata](http://www.sciencedirect.com/science/article/pii/S016750600870699X) but I haven't gone through it in detail. 


