Zeta and L-Functions - Key Properties

Analytic properties of zeta and $L$ -functions reveal deep connections between primes, class numbers, and Galois representations.

TheoremFunctional Equation

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

satisfies the functional equation: $\Lambda_K(s) = \Lambda_K(1-s)$

This symmetry about $s = 1/2$ is fundamental. Here $n = [K : \mathbb{Q}]$ , $r_1$ is the number of real embeddings, $r_2$ is the number of complex conjugate pairs, and $\Delta_K$ is the discriminant.

TheoremResidue at $s = 1$

The Dedekind zeta function has a simple pole at $s = 1$ with residue: $\lim_{s \to 1}(s-1)\zeta_K(s) = \frac{2^{r_1}(2\pi)^{r_2}h_K R_K}{w_K\sqrt{|\Delta_K|}}$

This is the analytic class number formula, connecting:

Analytic data: residue of $\zeta_K(s)$
Algebraic data: class number $h_K$ , regulator $R_K$
Geometric data: discriminant $\Delta_K$ , signature $(r_1, r_2)$
Arithmetic data: number of roots of unity $w_K$

ExampleComputing Class Numbers

For $K = \mathbb{Q}(\sqrt{-5})$ with $\Delta_K = -20, r_1 = 0, r_2 = 1, w_K = 2$ :

Numerical evaluation gives $\lim_{s \to 1}(s-1)\zeta_K(s) \approx 0.962$ . From the formula: $0.962 \approx \frac{(2\pi) h_K \cdot 1}{2\sqrt{20}} \implies h_K \approx 2$

This analytic method confirms the algebraic computation $h_K = 2$ .

TheoremPrime Number Theorem for Number Fields

Let $\pi_K(x)$ count prime ideals of $K$ with norm $\leq x$ . Then: $\pi_K(x) \sim \frac{x}{\log x}$

This generalizes the classical prime number theorem, proven using analytic continuation and non-vanishing of $\zeta_K(s)$ on $\text{Re}(s) = 1$ .

Under GRH: stronger error terms $\pi_K(x) = \text{Li}(x) + O(\sqrt{x}\log x)$ hold, where $\text{Li}(x) = \int_2^x \frac{dt}{\log t}$ .

DefinitionArtin L-Functions

For a Galois representation $\rho: \text{Gal}(\bar{K}/K) \to GL_n(\mathbb{C})$ , the Artin $L$ -function is: $L(s, \rho) = \prod_\mathfrak{p} \det(I - \rho(\text{Frob}_\mathfrak{p})N(\mathfrak{p})^{-s})^{-1}$

Properties:

Product is over unramified primes
For 1-dimensional $\rho$ (abelian case), reduces to Hecke $L$ -functions
Artin conjecture: $L(s, \rho)$ is entire when $\rho$ is irreducible and nontrivial

Artin $L$ -functions encode Galois-theoretic information analytically, central to the Langlands program.

Remark

The Langlands program conjectures that all Artin $L$ -functions arise from automorphic forms, unifying representation theory, number theory, and analysis. Proven cases include:

Abelian case (class field theory)
Solvable case (partial results)
General case remains open, one of the deepest problems in mathematics