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 |
| $\ker \varphi$ |
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.