The Dedekind zeta function $\zeta_K(s)$, initially defined for $\text{Re}(s) > 1$, has:

1. Analytic continuation to a meromorphic function on all $\mathbb{C}$
2. A simple pole at $s = 1$ with residue given by the class number formula
3. A functional equation relating values at $s$ and $1-s$

The completed zeta function $\Lambda_K(s) = |\Delta_K|^{s/2}\pi^{-ns/2}\prod_{i=1}^{r_1}\Gamma(s/2)\prod_{j=1}^{r_2}\Gamma(s)\zeta_K(s)$ satisfies: $\Lambda_K(s) = \Lambda_K(1-s)$

This symmetry about the critical line $\text{Re}(s) = 1/2$ is fundamental to understanding zeros and the Riemann hypothesis.

Let $L/K$ be Galois with group $G$. For a conjugacy class $C \subseteq G$, define: $\delta_C = \lim_{x \to \infty}\frac{\#\{\mathfrak{p} : N(\mathfrak{p}) \leq x, \text{Frob}_\mathfrak{p} \in C\}}{\#\{\mathfrak{p} : N(\mathfrak{p}) \leq x\}}$

Then $\delta_C = |C|/|G|$.

Proof method: Uses analytic properties of Artin $L$-functions. The non-vanishing $L(1, \rho) \neq 0$ for nontrivial irreducible $\rho$ implies equidistribution via Tauberian theorems.

This generalizes Dirichlet's theorem and determines splitting statistics in arbitrary Galois extensions.

For coprime $a, n$, let $\pi(x; a, n)$ count primes $p \leq x$ with $p \equiv a \pmod{n}$. Then: $\pi(x; a, n) \sim \frac{1}{\varphi(n)}\frac{x}{\log x}$

Proof: Requires showing $L(1, \chi) \neq 0$ for all Dirichlet characters $\chi$ modulo $n$. This uses:

1. Functional equation and analytic continuation
2. Product formula relating $L(s, \chi)$ to primes
3. Tauberian theorem (Wiener-Ikehara) relating series and asymptotics

The non-vanishing at $s = 1$ is the key analytic input, following from the pole structure of $\zeta(s)$.

Conjecture: For every Dedekind zeta function $\zeta_K(s)$ and Dirichlet $L$-function $L(s, \chi)$, all nontrivial zeros lie on the critical line $\text{Re}(s) = 1/2$.

Consequences if true:

• Prime number theorems with optimal error $O(\sqrt{x}\log x)$
• Class number bounds: Effective Minkowski bounds, Bach's bound $h_K = O(|\Delta_K|^{1/2+\epsilon})$
• Deterministic primality testing: Miller's test becomes deterministic under GRH
• Cryptographic security: Many algorithms' correctness depends on GRH

GRH remains one of the most important open problems, with vast implications across number theory and computer science.

The Stark conjectures predict that derivatives of $L$-functions at special points are related to logarithms of algebraic units.

For an abelian extension $L/K$ and character $\chi$: $L'(0, \chi) = -\log|\epsilon|$

for some unit $\epsilon \in L^*$ (the Stark unit).

These conjectures, when proven, provide explicit constructions of units and abelian extensions, generalizing Kronecker's Jugendtraum beyond imaginary quadratic fields.