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
$\ker \varphi$	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

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.