# Notation Index

## Number Systems

| Symbol | Meaning |
|---|---|
| $\mathbb{N}$ | natural numbers |
| $\mathbb{Z}$ | integers |
| $\mathbb{Q}$ | rational numbers |
| $\mathbb{R}$ | real numbers |
| $\mathbb{C}$ | complex numbers |
| $\mathbb{F}_p$ | finite field with $p$ elements |
| $\mathbb{Q}_p$ | field of $p$-adic numbers |
| $\mathbb{Z}_p$ | ring of $p$-adic integers |

## Sets and Logic

| Symbol | Meaning |
|---|---|
| $x\in A$ | $x$ belongs to $A$ |
| $x\notin A$ | $x$ does not belong to $A$ |
| $A\subseteq B$ | $A$ is a subset of $B$ |
| $A\cup B$ | union |
| $A\cap B$ | intersection |
| $A\setminus B$ | set difference |
| $\varnothing$ | empty set |
| $A\times B$ | Cartesian product |
| $\forall$ | for all |
| $\exists$ | there exists |
| $\implies$ | implies |
| $\Longleftrightarrow$ | if and only if |

## Divisibility and Congruences

| Symbol | Meaning |
|---|---|
| $a\mid b$ | $a$ divides $b$ |
| $a\nmid b$ | $a$ does not divide $b$ |
| $\gcd(a,b)$ | greatest common divisor |
| $\operatorname{lcm}(a,b)$ | least common multiple |
| $a\equiv b\pmod n$ | $a$ is congruent to $b$ modulo $n$ |
| $\mathbb{Z}/n\mathbb{Z}$ | residue ring modulo $n$ |
| $(\mathbb{Z}/n\mathbb{Z})^\times$ | group of units modulo $n$ |

## Arithmetic Functions

| Symbol | Meaning |
|---|---|
| $\varphi(n)$ | Euler totient function |
| $\mu(n)$ | Möbius function |
| $\tau(n)$ | number of positive divisors of $n$ |
| $\sigma(n)$ | sum of positive divisors of $n$ |
| $\omega(n)$ | number of distinct prime divisors |
| $\Omega(n)$ | number of prime factors counted with multiplicity |
| $f*g$ | Dirichlet convolution |
| $\mathbf{1}(n)$ | constant arithmetic function $1$ |
| $\varepsilon(n)$ | identity for Dirichlet convolution |

## Prime Number Theory

| Symbol | Meaning |
|---|---|
| $p$ | usually a prime number |
| $\pi(x)$ | number of primes at most $x$ |
| $\operatorname{Li}(x)$ | logarithmic integral |
| $\vartheta(x)$ | Chebyshev theta function |
| $\psi(x)$ | Chebyshev psi function |
| $\Lambda(n)$ | von Mangoldt function |

## Algebra

| Symbol | Meaning |
|---|---|
| $G$ | group |
| $e$ | identity element of a group |
| $H\le G$ | $H$ is a subgroup of $G$ |
| $\langle g\rangle$ | subgroup generated by $g$ |
| $|G|$ | order of a finite group |
| $\ker \varphi$ | kernel of a homomorphism |
| $\operatorname{im}\varphi$ | image of a homomorphism |
| $R$ | ring |
| $R^\times$ | group of units of $R$ |
| $I\triangleleft R$ | $I$ is an ideal of $R$ |
| $(a)$ | principal ideal generated by $a$ |
| $R/I$ | quotient ring |

## Field Theory and Algebraic Number Theory

| Symbol | Meaning |
|---|---|
| $K,L$ | fields, often number fields |
| $L/K$ | field extension |
| $[L:K]$ | degree of field extension |
| $\mathcal{O}_K$ | ring of integers of $K$ |
| $N_{K/\mathbb{Q}}(\alpha)$ | norm of $\alpha$ |
| $\operatorname{Tr}_{K/\mathbb{Q}}(\alpha)$ | trace of $\alpha$ |
| $\operatorname{Cl}(K)$ | ideal class group |
| $h_K$ | class number of $K$ |
| $\Delta_K$ | discriminant of $K$ |

## Analysis

| Symbol | Meaning |
|---|---|
| $O(g(x))$ | bounded above by constant multiple of $g(x)$ |
| $o(g(x))$ | negligible compared with $g(x)$ |
| $f(x)\sim g(x)$ | ratio $f(x)/g(x)\to1$ |
| $\sum$ | summation |
| $\prod$ | product |
| $\int$ | integral |
| $\operatorname{Re}(s)$ | real part of $s$ |
| $\operatorname{Im}(s)$ | imaginary part of $s$ |
| $\overline{z}$ | complex conjugate |

## Zeta and $L$-Functions

| Symbol | Meaning |
|---|---|
| $\zeta(s)$ | Riemann zeta function |
| $L(s,\chi)$ | Dirichlet $L$-function |
| $\chi$ | Dirichlet character |
| $\rho$ | usually a nontrivial zero of $\zeta(s)$ |
| $\Gamma(s)$ | gamma function |
| $\xi(s)$ | completed zeta function |

## Geometry and Curves

| Symbol | Meaning |
|---|---|
| $E$ | elliptic curve |
| $E(K)$ | $K$-rational points on $E$ |
| $\#E(\mathbb{F}_q)$ | number of points on $E$ over $\mathbb{F}_q$ |
| $\operatorname{Spec}(R)$ | spectrum of a ring |
| $\mathbb{A}^n$ | affine $n$-space |
| $\mathbb{P}^n$ | projective $n$-space |

## Linear Algebra

| Symbol | Meaning |
|---|---|
| $V,W$ | vector spaces |
| $\dim V$ | dimension of $V$ |
| $\det A$ | determinant of $A$ |
| $\operatorname{rank} A$ | rank of $A$ |
| $\ker T$ | kernel of a linear map |
| $\operatorname{im} T$ | image of a linear map |
| $V^*$ | dual vector space |
| $V\otimes W$ | tensor product |

## Categories

| Symbol | Meaning |
|---|---|
| $\mathcal{C}$ | category |
| $A\to B$ | morphism from $A$ to $B$ |
| $\operatorname{id}_A$ | identity morphism on $A$ |
| $F:\mathcal{C}\to\mathcal{D}$ | functor |
| $\eta:F\Rightarrow G$ | natural transformation |
| $\mathcal{C}^{op}$ | opposite category |
| $\operatorname{Hom}(A,B)$ | morphisms from $A$ to $B$ |

## Common Conventions

The letter $p$ usually denotes a prime. The letter $n$ usually denotes a positive integer. The letter $K$ often denotes a number field. The letter $R$ often denotes a ring. The letter $G$ often denotes a group. The variable $s$ is often complex when zeta functions or Dirichlet series are involved.

When context is clear, the same symbol may carry different meanings in different chapters. For example, $N$ may mean a positive integer, a norm map, or a bound in an asymptotic argument.