Logo
Unionpedia
Communication
Get it on Google Play
New! Download Unionpedia on your Android™ device!
Free
Faster access than browser!
 

Multiplicative function

Index 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). [1]

42 relations: Abelian group, Additive function, Arithmetic function, Bell series, Completely multiplicative function, Complex number, Coprime integers, Dirichlet character, Dirichlet convolution, Dirichlet series, Divisor, Divisor function, Euler product, Euler's totient function, Finite field, Fundamental theorem of arithmetic, Greatest common divisor, Homomorphism, Identity element, Identity function, Indicator function, Integer, Lambert series, Least common multiple, Legendre symbol, Liouville function, List of zeta functions, Möbius function, Möbius inversion formula, Monoid, Negative number, Number theory, On-Line Encyclopedia of Integer Sequences, Prime number, Principal ideal domain, Ramanujan tau function, Sign (mathematics), Square-free element, Square-free integer, Unique factorization domain, Unit function, 0.

Abelian group

In abstract algebra, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written.

New!!: Multiplicative function and Abelian group · See more »

Additive function

In number theory, an additive function is an arithmetic function f(n) of the positive integer n such that whenever a and b are coprime, the function of the product is the sum of the functions:Erdös, P., and M. Kac.

New!!: Multiplicative function and Additive function · See more »

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.

New!!: Multiplicative function and Arithmetic function · See more »

Bell series

In mathematics, the Bell series is a formal power series used to study properties of arithmetical functions.

New!!: Multiplicative function and Bell series · See more »

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.

New!!: Multiplicative function and Completely multiplicative function · See more »

Complex number

A complex number is a number that can be expressed in the form, where and are real numbers, and is a solution of the equation.

New!!: Multiplicative function and Complex number · See more »

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.

New!!: Multiplicative function and Coprime integers · See more »

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.

New!!: Multiplicative function and Dirichlet character · See more »

Dirichlet convolution

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

New!!: Multiplicative function and Dirichlet convolution · See more »

Dirichlet series

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

New!!: Multiplicative function and Dirichlet series · See more »

Divisor

In mathematics, a divisor of an integer n, also called a factor of n, is an integer m that may be multiplied by some integer to produce n. In this case, one also says that n is a multiple of m. An integer n is divisible by another integer m if m is a divisor of n; this implies dividing n by m leaves no remainder.

New!!: Multiplicative function and Divisor · See more »

Divisor function

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

New!!: Multiplicative function and Divisor function · See more »

Euler product

In number theory, an Euler product is an expansion of a Dirichlet series into an infinite product indexed by prime numbers.

New!!: Multiplicative function and Euler product · See more »

Euler's totient function

In number theory, Euler's totient function counts the positive integers up to a given integer that are relatively prime to.

New!!: Multiplicative function and Euler's totient function · See more »

Finite field

In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field that contains a finite number of elements.

New!!: Multiplicative function and Finite field · See more »

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.

New!!: Multiplicative function and Fundamental theorem of arithmetic · See more »

Greatest common divisor

In mathematics, the greatest common divisor (gcd) of two or more integers, which are not all zero, is the largest positive integer that divides each of the integers.

New!!: Multiplicative function and Greatest common divisor · See more »

Homomorphism

In algebra, a homomorphism is a structure-preserving map between two algebraic structures of the same type (such as two groups, two rings, or two vector spaces).

New!!: Multiplicative function and Homomorphism · See more »

Identity element

In mathematics, an identity element or neutral element is a special type of element of a set with respect to a binary operation on that set, which leaves other elements unchanged when combined with them.

New!!: Multiplicative function and Identity element · See more »

Identity function

Graph of the identity function on the real numbers In mathematics, an identity function, also called an identity relation or identity map or identity transformation, is a function that always returns the same value that was used as its argument.

New!!: Multiplicative function and Identity function · See more »

Indicator function

In mathematics, an indicator function or a characteristic function is a function defined on a set X that indicates membership of an element in a subset A of X, having the value 1 for all elements of A and the value 0 for all elements of X not in A. It is usually denoted by a symbol 1 or I, sometimes in boldface or blackboard boldface, with a subscript specifying the subset.

New!!: Multiplicative function and Indicator function · See more »

Integer

An integer (from the Latin ''integer'' meaning "whole")Integer 's first literal meaning in Latin is "untouched", from in ("not") plus tangere ("to touch").

New!!: Multiplicative function and Integer · See more »

Lambert series

In mathematics, a Lambert series, named for Johann Heinrich Lambert, is a series taking the form It can be resummed formally by expanding the denominator: where the coefficients of the new series are given by the Dirichlet convolution of an with the constant function 1(n).

New!!: Multiplicative function and Lambert series · See more »

Least common multiple

In arithmetic and number theory, the least common multiple, lowest common multiple, or smallest common multiple of two integers a and b, usually denoted by LCM(a, b), is the smallest positive integer that is divisible by both a and b. Since division of integers by zero is undefined, this definition has meaning only if a and b are both different from zero.

New!!: Multiplicative function and Least common multiple · See more »

Legendre symbol

No description.

New!!: Multiplicative function and Legendre symbol · See more »

Liouville function

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

New!!: Multiplicative function and Liouville function · See more »

List of zeta functions

In mathematics, a zeta function is (usually) a function analogous to the original example: the Riemann zeta function Zeta functions include.

New!!: Multiplicative function and List of zeta functions · See more »

Möbius function

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

New!!: Multiplicative function and Möbius function · See more »

Möbius inversion formula

In mathematics, the classic Möbius inversion formula was introduced into number theory during the 19th century by August Ferdinand Möbius.

New!!: Multiplicative function and Möbius inversion formula · See more »

Monoid

In abstract algebra, a branch of mathematics, a monoid is an algebraic structure with a single associative binary operation and an identity element.

New!!: Multiplicative function and Monoid · See more »

Negative number

In mathematics, a negative number is a real number that is less than zero.

New!!: Multiplicative function and Negative number · See more »

Number theory

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

New!!: Multiplicative function and Number theory · See more »

On-Line Encyclopedia of Integer Sequences

The On-Line Encyclopedia of Integer Sequences (OEIS), also cited simply as Sloane's, is an online database of integer sequences.

New!!: Multiplicative function and On-Line Encyclopedia of Integer Sequences · See more »

Prime number

A prime number (or a prime) is a natural number greater than 1 that cannot be formed by multiplying two smaller natural numbers.

New!!: Multiplicative function and Prime number · See more »

Principal ideal domain

In abstract algebra, a principal ideal domain, or PID, is an integral domain in which every ideal is principal, i.e., can be generated by a single element.

New!!: Multiplicative function and Principal ideal domain · See more »

Ramanujan tau function

The Ramanujan tau function, studied by, is the function \tau:\mathbb\to\mathbb defined by the following identity: where q.

New!!: Multiplicative function and Ramanujan tau function · See more »

Sign (mathematics)

In mathematics, the concept of sign originates from the property of every non-zero real number of being positive or negative.

New!!: Multiplicative function and Sign (mathematics) · See more »

Square-free element

In mathematics, a square-free element is an element r of a unique factorization domain R that is not divisible by a non-trivial square.

New!!: Multiplicative function and Square-free element · See more »

Square-free integer

In mathematics, a square-free integer is an integer which is divisible by no perfect square other than 1.

New!!: Multiplicative function and Square-free integer · See more »

Unique factorization domain

In mathematics, a unique factorization domain (UFD) is an integral domain (a non-zero commutative ring in which the product of non-zero elements is non-zero) in which every non-zero non-unit element can be written as a product of prime elements (or irreducible elements), uniquely up to order and units, analogous to the fundamental theorem of arithmetic for the integers.

New!!: Multiplicative function and Unique factorization domain · See more »

Unit function

In number theory, the unit function is a completely multiplicative function on the positive integers defined as: It is called the unit function because it is the identity element for Dirichlet convolution.

New!!: Multiplicative function and Unit function · See more »

0

0 (zero) is both a number and the numerical digit used to represent that number in numerals.

New!!: Multiplicative function and 0 · See more »

Redirects here:

Multiplicative functions, Multiplicative series, Multiplicativeness, Multiplicativity, Non-multiplicative function.

References

[1] https://en.wikipedia.org/wiki/Multiplicative_function

OutgoingIncoming
Hey! We are on Facebook now! »