Let $a, q \in \mathbb{N}$ be coprime, i.e.\ $\gcd(a, q) = 1$. Then there are infinitely many primes in the arithmetic progression $a, a+q, a+2q, a+3q, \ldots$

If $\chi$ is any non-trivial Dirichlet character, then $L(\chi, 1) \neq 0$.

Let $L = \mathbb{Q}(\omega_q)$ and let $\chi_1, \ldots, \chi_{\varphi(q)}$ be the Dirichlet characters of modulus $q$, with $\chi_1$ the trivial character. Then \begin{align*} \zeta_{\mathbb{Q}(\omega_q)}(s) = \prod_{i=1}^{\varphi(q)} L(\chi_i, s), \end{align*} where each $L(\chi_i, s)$ is understood as a modified $L$-function that includes the Euler factors at primes $p \mid q$ (with $\chi_i(p) = 0$, so those factors are $1$, and the convention absorbs any ramified-prime correction).

Let $L = \mathbb{Q}(\omega_q)$ where $\omega_q$ is a primitive $q$th root of unity. Then: 1. $[L:\mathbb{Q}] = \varphi(q)$, where $\varphi(q) = |(\mathbb{Z}/q\mathbb{Z})^\times|$ is Euler's totient. 2. $L/\mathbb{Q}$ is Galois with $\operatorname{Gal}(L/\mathbb{Q}) \cong (\mathbb{Z}/q\mathbb{Z})^\times$, where $r \in (\mathbb{Z}/q\mathbb{Z})^\times$ acts by $\omega_q \mapsto \omega_q^r$. 3. The ring of integers is $\mathcal{O}_L = \mathbb{Z}[\omega_q] = \mathbb{Z}[x]/\Phi_q(x)$, where $\Phi_q(x)$ is the $q$th cyclotomic polynomial. 4. A prime $p$ ramifies in $\mathcal{O}_L$ if and only if $p \mid q$. 5. If $p \nmid q$, then $(p)$ factors as a product of $\varphi(q)/f$ distinct prime ideals, each of norm $p^f$, where $f$ is the order of $p$ in $(\mathbb{Z}/q\mathbb{Z})^\times$.

For any quadratic Dirichlet character $\chi_D$, we have $L(\chi_D, 1) \neq 0$.

If $\chi$ is a non-trivial Dirichlet character, then $L(\chi, s)$ is holomorphic for $\operatorname{Re}(s) > 0$.

For $L = \mathbb{Q}(\sqrt{d})$, the function $\chi_D$ defined by the Legendre symbol and Jacobi symbol data is a Dirichlet character of modulus $D$.

Let $L/\mathbb{Q}$ be a number field of degree $n$. Then: 1. $\zeta_L(s)$ converges to a holomorphic function for $\operatorname{Re}(s) > 1$. 2. **(Analytic class number formula)** $\zeta_L(s)$ extends meromorphically to $\operatorname{Re}(s) > 1 - \frac{1}{n}$, with a simple pole at $s = 1$ with residue \begin{align*} \frac{|\mathrm{Cl}_L|\, 2^{r_1} (2\pi)^{r_2} R_L}{|D_L|^{1/2} |\mu_L|}, \end{align*} where $\mathrm{Cl}_L$ is the ideal class group (Chapter 3), $r_1$ and $r_2$ count the real and complex embeddings respectively (Chapter 2), $R_L$ is the regulator (Chapter 7), $D_L$ is the discriminant (Chapter 2), and $\mu_L$ is the group of roots of unity in $L$. 3. The Euler product holds: \begin{align*} \zeta_L(s) = \prod_{\mathfrak{p} \lhd \mathcal{O}_L,\, \mathfrak{p}\text{ prime}} \left(1 - N(\mathfrak{p})^{-s}\right)^{-1}. \end{align*}…

The Riemann zeta function $\zeta(s)$ satisfies: 1. $\zeta(s)$ is holomorphic for $\operatorname{Re}(s) > 1$. 2. The function $\zeta(s) - \frac{1}{s-1}$ extends to a holomorphic function for $\operatorname{Re}(s) > 0$. In other words, $\zeta(s)$ extends meromorphically to $\operatorname{Re}(s) > 0$ with a simple pole at $s = 1$ with residue $1$. 3. For $\operatorname{Re}(s) > 1$, the **Euler product** holds: \begin{align*} \zeta(s) = \prod_{p\text{ prime}} \left(1 - p^{-s}\right)^{-1}, \end{align*} and the product is absolutely convergent.

Let $\sum_{n \geq 1} a_n n^{-s}$ be a Dirichlet series. If there is a real number $r \in \mathbb{R}$ such that $a_1 + \cdots + a_N = O(N^r)$, then the series converges for all $s$ with $\operatorname{Re}(s) > r$, and defines a holomorphic function there.