# Do Shintani zeta functions satisfy a functional equation?

Probably my questions are known or evident to the experts but I'm a bit puzzled. First of all there seem to be two kinds of zeta functions that go under the name of Shintani zeta functions.

First, there are zeta functions $\zeta^{SS}$ associated with so called prehomogenous vector spaces going back to important work by Sato and Shintani (see the original article or this book by Yukie) and then, second, zeta functions $\zeta^S$ that appeared in Shintani's work on special values of Dedekind zeta functions of totally real number fields at negative integers (see Shintani's article or Neukirch's book for example).

1) I'm mainly interested in the question if it is known (or expected) if the latter zeta functions $\zeta^S$ satisfy functional equations. (From what I understand the $\zeta^{SS}$ satisfy functional equations or are expected to satisfy in case it is not proven).

Let me just note that one can write Shintani zeta functions in the following form $$\Gamma(s)^n \zeta^S(s,z,x) = \int_0^\infty \cdots\int_0^\infty \sum_{z_1,\dots , z_n=0}^\infty e^{-\sum_{i=1}^n t_i L_i(z+x)}(t_1\cdots t_n)^{s-1} dt_1\cdots dt_n$$ where the $L_i(x)$ are linear forms, i.e. essentially we could say that we're looking at multivariable theta-like functions and Mellin transforms thereof. So the question can be rephrased in asking whether these theta-like functions occurring in the above integral satisfy a functional equation/theta inversion formula. (Note that these theta-like functions do in general not come from symplectic structures, i.e. they are not related to abelian varieties (at least as far as I see)).

2) But next to this question I'm also extremely interested in the relationship of these two kinds of zeta functions. In which cases do the two constructions agree? Is there anything known?

Thank you very much in advance!

EDIT: OK, so I could speak a bit with one of the absolute authorities in this field and I learned, that

1) one shouldn't expect functional equations for single functions $\zeta^S$ but rather for certain finite linear combinations and

2) one shouldn't expect relations between the two notions of "zeta" functions.

This doesn't destroy the applications I had in mind with my question but I have to rethink the question and will try to give a better and less naive version of it soon. Thank you so far very much for your helpful comments!

• Regarding "there seem to be two kinds of zeta functions that go under the name of Shintani zeta functions": I can definitively confirm this. I know about the prehomogeneous ones, but unfortunately not about the ones you asked about. AFAIK the only connection is that Shintani worked on both of them. But if there is some further connection I would be very interested to hear about it! – Frank Thorne Dec 12 '11 at 19:43
• @Bora: My problem is that the expression $\prod_{i=1}^n L_i(x)^{-s}$ is not theta-like. To me a theta-like function is an infinite series such as $\sum_{n\in\mathbb{Z}} e^{-n^2 x}$ whose Mellin-transform is a zeta-like function. – GH from MO Dec 12 '11 at 20:01
• @Bora and GH : probably one needs to take an infinite sum (over $x$) of the expression Bora has written. One can also introduce denominators in the integrand. – François Brunault Dec 12 '11 at 20:01
• In case it is not clear: the integral kernel is akin to one used by Riemann in one of his proofs for plain-old-zeta. Not the theta series, but $\sum e^{-nx} = 1/(e^x-1)$. The "cone" here is just the positive reals. For other number fields, the cone is a product of positive reals, but the rational structure is more complicated. So, in principle, it is related to the bigger-group Shintani zetas. I think Shintani did a few things about their special values, and I recall Satake wrote a paper about this. – paul garrett Dec 12 '11 at 20:28
• I don't know if this will answer your question but Colmez gave a talk recently in Banff about Shintani's method. The notes (by Matt Greenberg) are available here : temple.birs.ca/~11w5125/colmez.pdf What I remember is that Colmez managed to express special values at negative integers without using the functional equation. – François Brunault Dec 13 '11 at 0:07