Weil Conjectures Analog for Multivariate Zeta Functions

Question

We know that the Riemann zeta function can be generalized to multivariate zeta functions.

Is there a multivariate analog of the Weil conjectures?

Answer 1

The Weil conjectures have to do with local zeta-function over finite fields.

The Riemann zeta function and the multiple zeta functions are defined over $\mathbb{Q}$

So Weil conjectures are simply the wrong type of conjecture to make about this kind of L-functions. On the other hand, the Weil conjectures were modeled after conjectures about the Riemman zeta, so probably the question that makes sense is: "what conjectures about the Riemann zeta function also hold for multiple zeta functions?"

Take the Riemann hypothesis for example. Is there an analogous conjecture in the multiple case?

The answer is no. The zeros of multiple zeta functions for $d>1$ are expected to behave quite differently from those of the $d=1$ case. In fact, we have the following result.

Theorem. [Nakamura & Pankowski] The multiple zeta function $\zeta(s_1,...,s_d)$ has zeros in $1/2<\Re(s_1),...,\Re(s_d)<1$.

For another example, even the location of trivial zeros is an open problem, as far as I know.

The functional equation (another "Weil conjecture") is also an open problem.