# Mathematicial notation

## Sets

: the integers (a unique factorization domain).

: the natural numbers. Unfortunately, this notation is ambiguous -- some authors use it for the positive integers, some for the nonnegative integers.

: Also an ambiguous notation, use for the positive primes or the positive integers.

: the reals (a field).

: the complex numbers (an algebraically closed and complete field).

: the -adic numbers (a complete field); also and are used sometimes.

: the residues (a ring; a field for prime).

When is one of the sets from above, then denotes the numbers (when defined), analogous for . The meaning of will depend on : for most cases it denotes the invertible elements, but for it means the nonzero integers (note that these definitions coincide in most cases). A zero in the index, like in , tells us that is also included.

## Definitions

For a set , denotes the number of elements of .

divides (both integers) is written as , or sometimes as .
Then for , or is their **greatest common divisor**, the greatest with and ( is defined as ) and or denotes their least common multiple, the smallest non-negative integer such that and
.
When , one often says that are called "coprime".

For to be **squarefree** means that there is no integer with . Equivalently, this means that no prime factor occurs more than once in the decomposition.

**Factorial** of :

**Binomial Coefficients**:

For two functions the **Dirichlet convolution** is defined as .
A (weak) **multiplicative function** is one such that for all with .

Some special types of such functions:

**Euler's totient function**: .

**Möbius' function**: .

**Sum of powers of divisors**: ; often is used for , the number of divisors, and simply for .

For any it denotes the **number of representations of as sum of squares**.

Let be coprime integers. Then , the "**order of **" is the smallest with .

For and , the **-adic valuation ** can be defined as the multiplicity of in the factorisation of , and can be extended for by .
Additionally often is used.

For any function we define as the (upper) finite difference of . Then we set and then iteratively for all integers .

**Legendre symbol**: for and odd we define

Then the **Jacobi symbol** for and odd (prime factorization of ) is defined as:

**Hilbert symbol**: let and . Then
is the "Hilbert symbol of in respect to " (nontrivial means here that not all numbers are ).

When , then we can define a **counting function** .
One special case of a counting function is the one that belongs to the primes , which is often called .
With counting functions, some types of densities can be defined:

**Lower asymptotic density**:

**Upper asymptotic density**:

**Asymptotic density** (does not always exist):

**Shnirelman's density**:

**Dirichlet's density**(does not always exist):

and are equal iff the asymptotic density exists and all three are equal then and equal to Dirichlet's density.

Often, **density** is meant **in relation to some other set** (often the primes). Then we need with counting functions and simply change into and into :

**Lower asymptotic density**:

**Upper asymptotic density**:

**Asymptotic density** (does not always exist):

**Shnirelman's density**:

**Dirichlet's density**(does not always exist):

Again the same relations as above hold.