Axioms for Riemann $\zeta$ function Are there any set of axioms that completely characterize the Riemann zeta function?
 i.e. like Ressayre axioms for the exponential function in an exponential field or functional equations.
 A: Hamburger's Theorem (see Titchmarsh 'Theory of the Riemann Zeta Function' $\S$ 2.13) is in some sense an axiomatic characterization of $\zeta(s)$ among all Dirichlet series by its functional equation. It says:
Let $f(s)=\sum_n a_n n^{-s}$ a Dirichlet series absolutely convergent for $\sigma>1$ such that for some polynomial $P(s)$, $G(s)=P(s)f(s)$ is an integral function of finite order.  Suppose
$$
f(s)\Gamma(s/2)\pi^{-s/2}=g(1-s)\Gamma((1-s)/2)\pi^{-(1-s)/2}
$$
where $g(1-s)=\sum_n b_n n^{1-s}$ is absolutely convergent for $\sigma<-\alpha<0$.  Then
$$
f(s)=C\zeta(s)
$$
for some constant $C$.
Hamburger's theorem was a motivation for Hecke's study of Dirichlet series with functional equations generally, leading to his work on automorphic forms.
A: There are in fact at least two axiomatic characterizations of $\zeta(s).$  One of them is given by Hecke and one of them was given by Hamburger.
Hamburger's Theorem states:  Suppose $D(s)$ is a dirichlet series, convergent in some half plane and whose coefficients have polynomial growth.  If $$$$
1) There exists a polynomial $P(s)$ so that $P(s)D(s)$ is entire and of finite order.
2) $D(s/2)$ is also dirichlet series. That is, the coefficients of $D(s)$ are supported on squares.
3) If $\xi(s)= \pi^{-s}\Gamma(s)D(s)$ then $\xi(1/2-s)=\xi(s).$
Then $D(s)=C\zeta(2s).$
Hecke's version states: Suppose $D(s)$ is a dirichlet series, convergent in some half plane and whose coefficients have polynomial growth.  If $$$$
1) $(s-1/2)D(s)$ is entire and of finite order. 
2) The coefficients of $D(s)$ have arbetrary support.
3) If $\xi(s)= \pi^{-s}\Gamma(s)D(s)$ then $\xi(1/2-s)=\xi(s).$
Then $D(s)=C\zeta(2s).$
A little extra information: These theorems cannot be combined. The so-called "Big Mac Theorem" where$$$$
1) There exists a polynomial $P(s)$ so that $P(s)D(s)$ is entire and of finite order.
2) The coefficients of $D(s)$ have arbetrary support.
3) If $\xi(s)= \pi^{-s}\Gamma(s)D(s)$ then $\xi(1/2-s)=\xi(s).$
produces infinitely many linearly independent dirichlet series!
Sources:
http://www.rowan.edu/open/depts/math/HASSEN/Papers/paper1.pdf
A: I don't know if this is in the spirit you're looking for, but there is the Selberg class -- an attempt at axiomatizing $L$-functions, requiring  a Dirichlet series, functional equation of a certain type, analyticity, and an Euler product (typically) -- and it would be possible to impose extra conditions to isolate the zeta function.  In particular, all degree 1 elements are known to come from Dirichlet L-functions (this was proved by Kaczorowski and Perelli, and then reproved by Soundararajan).  Thus, requiring the degree and the conductor to both be 1 should isolate the zeta function.
A: Perhaps you are looking for something like Hamburger's Theorem? 
It states, essentially, that the only Dirichlet series with a finite number of singularities satisfying the same functional equation as the zeta-function is the zeta-function. You can find the details in Titchmarsh's book. 
Googling I found the following link: http://www.mat.univie.ac.at/~esiprpr/Zetaproc/patterson.pdf
