There are several formulations and consequences of the Riemann Hypothesis over Function Fields (RH, from now on). I am interested in the logical implications between those, and in proofs\references for those implications (which are surely known to the experts).

Personally, I would be most satisfied with proofs which are explicit and which only use the knowledge that was available to Weil at the time of his proof of the RH.

I will state below the various formulations\consequences. Let $\mathbb{F}_q$ be the finite field with $q$ elements.

- Curves: Let $C/\overline{\mathbb{F}_q}$ be a smooth, projective algebraic curve defined over $\mathbb{F}_q$. Let $\zeta_C(s)$ be the zeta function of $C$, defined as
$$\zeta_C(s) = \exp(\sum_{n \ge 1} \frac{N_m}{m}q^{-ms}),$$
where $N_m$ is the number of points of $C$ defined over the degree $m$ extension $\mathbb{F}_{q^m}$ of $\mathbb{F}_q$.
**RH**: All the zeros of $\zeta_C(s)$ lie on the line $\Re(s)=\frac{1}{2}$. - Field Extensions: Let $K/\mathbb{F}_q$ be a function field with constant field $\mathbb{F}_q$. Let $\zeta_K(s)$ be the zeta function of $K$, defined as follows:
$$\zeta_K(s) = \sum_{A \ge 0} (NA)^{-s},$$
where the summation is over all effective divisors $A$ of $K$, and $NA=q^{\deg A}$.
**RH**implies: All the zeros of $\zeta_K(s)$ lie on the line $\Re(s)=\frac{1}{2}$. - Rings of Integers (Dedekind zeta functions): Let $K/\mathbb{F}_q(T)$ be a field extension of finite degree. Let $O_K$ be the integral closure of $\mathbb{F}_q[T]$ in $K$. Let $\zeta_{O_K}(s)$ be the zeta function of $O_K$, defined as follows:
$$\zeta_{O_K}(s) = \sum_{I \ge 0} (NI)^{-s},$$
where the summation is over all ideals $I$ of $O_K$, and $NI=|O_K/I|$.
**RH**implies: All the zeros of $\zeta_{O_K}(s)$ lie on the line $\Re(s)=\frac{1}{2}$. - Characters (L-functions): Let $\chi:\mathbb{F}_q[T] \to \mathbb{C}$ be a Dirichlet character. Let $$L(s,\chi) = \sum_{f \in \mathbb{F}_q[T], f \text{ monic}} \chi(f) |f|^{-s}$$
be its L-function, where $|f|=|\mathbb{F}_q[T]/f|$.
**RH**implies: All the zeros of $L(s,\chi)$ lie on the line $\Re(s) = \frac{1}{2}$.

I believe the implications are $1 \leftrightarrow 2 \leftrightarrow 3 \implies 4$, although I cannot show this rigorously. Are these the only implications, and are they correct?

Below are some hand-waving arguments that partially explain the implications. They are not proofs, and I am *not* satisfied with them.

- I am most comfortable with Variant 2, mostly because there is an elementary proof for it, due to Stepanov-Bombieri, found in the appendix "Number Theory in Function Fields" by Michael Rosen.
- Morally, Variant 1 and Variant 2 are equivalent, since one can associate a function field to any curve, and vice versa.
- The only difference between Variant 2 and Variant 3 seems to be the contribution of the prime at infinity, which only contributes a pole ($s=1$).
- Variant 3 implies Variant 4, as follows: Note $L(s,\chi)$ is a polynomial. Construct an abelian extension $K$ of $\mathbb{F}_q(T)$ whose set $S$ of "associated Dirichlet characters" contains $\chi$. In that case, $\zeta_{O_K}(s) = \prod_{\chi' \in S} L(s,\chi')$. $RH$ for $\zeta_{O_K}(s)$ implies $RH$ for $L(s,\chi)$. The hard part, which I am not sure how to do, is the construction of the abelian extension - Is there a simple procedure for producing such an extension?

(Cross-posted from MSE. Got no answer there in the duration of 8 months.)