The Riemann zeta function is defined for $\text{Re}(s) > 1$ by: $\zeta(s) = \sum_{n=1}^\infty \frac{1}{n^s} = \prod_p \frac{1}{1 - p^{-s}}$

The product is over all primes $p$ (Euler product). This fundamental function:

• Converges absolutely for $\text{Re}(s) > 1$
• Has analytic continuation to $\mathbb{C} \setminus \{1\}$
• Has a simple pole at $s = 1$ with residue 1
• Satisfies functional equation relating $\zeta(s)$ and $\zeta(1-s)$

For a number field $K$, the Dedekind zeta function is: $\zeta_K(s) = \sum_{\mathfrak{a}} \frac{1}{N(\mathfrak{a})^s} = \prod_\mathfrak{p} \frac{1}{1 - N(\mathfrak{p})^{-s}}$

where the sum is over nonzero ideals of $\mathcal{O}_K$ and product is over prime ideals. This generalizes the Riemann zeta function:

• For $K = \mathbb{Q}$: $\zeta_{\mathbb{Q}}(s) = \zeta(s)$
• Converges for $\text{Re}(s) > 1$
• Has analytic continuation and functional equation
• Pole at $s = 1$ encodes class number and regulator

For a Dirichlet character $\chi: (\mathbb{Z}/n\mathbb{Z})^* \to \mathbb{C}^*$, the Dirichlet $L$-function is: $L(s, \chi) = \sum_{n=1}^\infty \frac{\chi(n)}{n^s} = \prod_p \frac{1}{1 - \chi(p)p^{-s}}$

These generalize $\zeta(s)$ (taking $\chi = 1$, the trivial character). Properties:

• Analytic continuation to $\mathbb{C}$ (entire if $\chi \neq 1$)
• Functional equation relating $L(s, \chi)$ and $L(1-s, \bar{\chi})$
• Non-vanishing at $s = 1$ when $\chi$ is non-principal