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# Completely multiplicative function

In number theory, functions of positive integers which respect products are important and are called completely multiplicative functions or totally multiplicative functions. 

## Arithmetic function

In number theory, an arithmetic, arithmetical, or number-theoretic function is for most authors any function f(n) whose domain is the positive integers and whose range is a subset of the complex numbers.

## Coprime integers

In number theory, two integers and are said to be relatively prime, mutually prime, or coprime (also written co-prime) if the only positive integer (factor) that divides both of them is 1.

## Dirichlet character

In number theory, Dirichlet characters are certain arithmetic functions which arise from completely multiplicative characters on the units of \mathbb Z / k \mathbb Z. Dirichlet characters are used to define Dirichlet ''L''-functions, which are meromorphic functions with a variety of interesting analytic properties.

## Dirichlet convolution

In mathematics, the Dirichlet convolution is a binary operation defined for arithmetic functions; it is important in number theory.

## Dirichlet L-function

In mathematics, a Dirichlet L-series is a function of the form Here χ is a Dirichlet character and s a complex variable with real part greater than 1.

## Dirichlet series

In mathematics, a Dirichlet series is any series of the form where s is complex, and a_n is a complex sequence.

## Divisor function

In mathematics, and specifically in number theory, a divisor function is an arithmetic function related to the divisors of an integer.

## Domain of a function

In mathematics, and more specifically in naive set theory, the domain of definition (or simply the domain) of a function is the set of "input" or argument values for which the function is defined.

## Fundamental theorem of arithmetic

In number theory, the fundamental theorem of arithmetic, also called the unique factorization theorem or the unique-prime-factorization theorem, states that every integer greater than 1 either is a prime number itself or can be represented as the product of prime numbers and that, moreover, this representation is unique, up to (except for) the order of the factors.

In mathematics, the Hadamard product (also known as the Schur product or the entrywise product) is a binary operation that takes two matrices of the same dimensions, and produces another matrix where each element i,j is the product of elements i,j of the original two matrices.

## Jacobi symbol

Jacobi symbol for various k (along top) and n (along left side).

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## Liouville function

The Liouville function, denoted by λ(n) and named after Joseph Liouville, is an important function in number theory.

## Möbius function

The classical Möbius function is an important multiplicative function in number theory and combinatorics.

## Monomial

In mathematics, a monomial is, roughly speaking, a polynomial which has only one term.

## Multiplicative function

In number theory, a multiplicative function is an arithmetic function f(n) of a positive integer n with the property that f(1).

## Natural number

In mathematics, the natural numbers are those used for counting (as in "there are six coins on the table") and ordering (as in "this is the third largest city in the country").

## Number theory

Number theory, or in older usage arithmetic, is a branch of pure mathematics devoted primarily to the study of the integers.

## References

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