In

_{''k''}(''n''): the power functions, defined by Id_{''k''}(''n'') = ''n''^{''k''} for any complex number ''k'' (completely multiplicative). As special cases we have
** Id_{0}(''n'') = 1(''n'') and
** Id_{1}(''n'') = Id(''n'').
* ''ε''(''n''): the function defined by ''ε''(''n'') = 1 if ''n'' = 1 and 0 otherwise, sometimes called ''multiplication unit for _{''C''}(''n''), the _{''C''}(''n'') is multiplicative precisely when the set ''C'' has the following property for any coprime numbers ''a'' and ''b'': the product ''ab'' is in ''C'' if and only if the numbers ''a'' and ''b'' are both themselves in ''C''. This is the case if ''C'' is the set of squares, cubes, or ''k''-th powers, or if ''C'' is the set of _{''k''}(''n''): the _{0}(''n'') = ''d''(''n'') the number of positive _{1}(''n'') = ''σ''(''n''), the sum of all the positive divisors of ''n''.
* ''a''(''n''): the number of non-isomorphic abelian groups of order ''n''.
* ''λ''(''n''): the ^{Ω(''n'')} where Ω(''n'') is the total number of primes (counted with multiplicity) dividing ''n''. (completely multiplicative).
* ''γ''(''n''), defined by ''γ''(''n'') = (−1)^{''ω''(n)}, where the _{2}(''n'') - the number of representations of ''n'' as a sum of squares of two integers, _{2}(1) = 4 ≠ 1. This shows that the function is not multiplicative. However, ''r''_{2}(''n'')/4 is multiplicative.
In the

sequences of values of a multiplicative function

have the keyword "mult". See

^{''a''} ''q''^{''b''} ..., then
''f''(''n'') = ''f''(''p''^{''a''}) ''f''(''q''^{''b''}) ...
This property of multiplicative functions significantly reduces the need for computation, as in the following examples for ''n'' = 144 = 2^{4} · 3^{2}:
Similarly, we have:
$$\backslash varphi(144)\; =\; \backslash varphi(2^4)\; \backslash ,\; \backslash varphi(3^2)\; =\; 8\; \backslash cdot\; 6\; =\; 48$$
In general, if ''f''(''n'') is a multiplicative function and ''a'', ''b'' are any two positive integers, then
Every completely multiplicative function is a

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number theory
Number theory (or arithmetic or higher arithmetic in older usage) is a branch of devoted primarily to the study of the s and . German mathematician (1777–1855) said, "Mathematics is the queen of the sciences—and number theory is the queen ...

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

''f''(''n'') of a positive integer
An integer (from the Latin
Latin (, or , ) is a classical language
A classical language is a language
A language is a structured system of communication
Communication (from Latin ''communicare'', meaning "to share" or "to ...

''n'' with the property that ''f''(1) = 1 and
$$f(ab)\; =\; f(a)f(b)$$ whenever ''a'' and ''b'' are coprime
In number theory, two integer
An integer (from the Latin wikt:integer#Latin, ''integer'' meaning "whole") is colloquially defined as a number that can be written without a Fraction (mathematics), fractional component. For example, 21, 4, 0, ...

.
An arithmetic function ''f''(''n'') is said to be completely multiplicative (or totally multiplicative) if ''f''(1) = 1 and ''f''(''ab'') = ''f''(''a'')''f''(''b'') holds ''for all'' positive integers ''a'' and ''b'', even when they are not coprime.
Examples

Some multiplicative functions are defined to make formulas easier to write: * 1(''n''): the constant function, defined by 1(''n'') = 1 (completely multiplicative) * Id(''n''):identity function
Graph of the identity function on the real numbers
In mathematics, an identity function, also called an identity relation, identity map or identity transformation, is a function (mathematics), function that always returns the same value that was ...

, defined by Id(''n'') = ''n'' (completely multiplicative)
* IdDirichlet convolution
In mathematics, the Dirichlet convolution is a binary operation defined for arithmetic functions; it is important in number theory. It was developed by Peter Gustav Lejeune Dirichlet.
Definition
If f , g : \mathbb\to\mathbb are two arithmetic fun ...

'' or simply the ''unit functionIn number theory, the unit function is a completely multiplicative function
:''Outside number theory, the term multiplicative function is usually used for completely multiplicative functions. This article discusses number theoretic multiplicative f ...

'' (completely multiplicative). Sometimes written as ''u''(''n''), but not to be confused with ''μ''(''n'') .
* 1indicator function
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...

of the set ''C'' ⊂ Z, for certain sets ''C''. The indicator function 1square-free {{no footnotes, date=December 2015
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained ...

numbers.
Other examples of multiplicative functions include many functions of importance in number theory, such as:
* gcd(''n'',''k''): the 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. For two integers ''x'', ''y'', the greatest common divisor of ''x'' and ''y'' is ...

of ''n'' and ''k'', as a function of ''n'', where ''k'' is a fixed integer.
* $\backslash varphi(n)$: Euler's totient function
In number theory
Number theory (or arithmetic or higher arithmetic in older usage) is a branch of devoted primarily to the study of the s and . German mathematician (1777–1855) said, "Mathematics is the queen of the sciences—and numbe ...

$\backslash varphi$, counting the positive integers coprime
In number theory, two integer
An integer (from the Latin wikt:integer#Latin, ''integer'' meaning "whole") is colloquially defined as a number that can be written without a Fraction (mathematics), fractional component. For example, 21, 4, 0, ...

to (but not bigger than) ''n''
* ''μ''(''n''): the Möbius function
The Möbius function is an important multiplicative function
:''Outside number theory, the term multiplicative function is usually used for completely multiplicative functions. This article discusses number theoretic multiplicative functions.'' ...

, the parity (−1 for odd, +1 for even) of the number of prime factors of square-free {{no footnotes, date=December 2015
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained ...

numbers; 0 if ''n'' is not square-free
* ''σ''divisor function
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities ...

, which is the sum of the ''k''-th powers of all the positive divisors of ''n'' (where ''k'' may be any complex number
In mathematics, a complex number is an element of a number system that contains the real numbers and a specific element denoted , called the imaginary unit, and satisfying the equation . Moreover, every complex number can be expressed in the for ...

). Special cases we have
** ''σ''divisor
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...

s of ''n'',
** ''σ''Liouville function The Liouville Lambda function, denoted by λ(''n'') and named after Joseph Liouville, is an important arithmetic function.
Its value is +1 if ''n'' is the product of an even number of prime numbers, and −1 if it is the product of an odd number of ...

, ''λ''(''n'') = (−1)additive function
In number theory
Number theory (or arithmetic or higher arithmetic in older usage) is a branch of devoted primarily to the study of the s and . German mathematician (1777–1855) said, "Mathematics is the queen of the sciences—and number ...

''ω''(''n'') is the number of distinct primes dividing ''n''.
* ''τ''(''n''): the Ramanujan tau function The Ramanujan tau function, studied by , is the function \tau : \mathbb \rarr\mathbb defined by the following identity:
:\sum_\tau(n)q^n=q\prod_\left(1-q^n\right)^ = \eta(z)^=\Delta(z),
where with and is the Dedekind eta function and the functio ...

.
* All Dirichlet character In analytic number theory 300px, Riemann zeta function ''ζ''(''s'') in the complex plane. The color of a point ''s'' encodes the value of ''ζ''(''s''): colors close to black denote values close to zero, while hue encodes the value's Argument (com ...

s are completely multiplicative functions. For example
** (''n''/''p''), the Legendre symbol
In number theory
Number theory (or arithmetic or higher arithmetic in older usage) is a branch of devoted primarily to the study of the s and . German mathematician (1777–1855) said, "Mathematics is the queen of the sciences—and number t ...

, considered as a function of ''n'' where ''p'' is a fixed prime number
A prime number (or a prime) is a natural number greater than 1 that is not a Product (mathematics), product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime ...

.
An example of a non-multiplicative function is the arithmetic function ''r''positive
Positive is a property of Positivity (disambiguation), positivity and may refer to:
Mathematics and science
* Converging lens or positive lens, in optics
* Plus sign, the sign "+" used to indicate a positive number
* Positive (electricity), a po ...

, negative, or zero
0 (zero) is a number
A number is a mathematical object
A mathematical object is an abstract concept arising in mathematics.
In the usual language of mathematics, an ''object'' is anything that has been (or could be) formally defined, and ...

, where in counting the number of ways, reversal of order is allowed. For example:
and therefore ''r''On-Line Encyclopedia of Integer Sequences
The On-Line Encyclopedia of Integer Sequences (OEIS) is an online database of integer sequences. It was created and maintained by Neil Sloane while a researcher at AT&T Labs. He transferred the intellectual property
Intellectual property (I ...

sequences of values of a multiplicative function

have the keyword "mult". See

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

for some other examples of non-multiplicative functions.
Properties

A multiplicative function is completely determined by its values at the powers ofprime number
A prime number (or a prime) is a natural number greater than 1 that is not a Product (mathematics), product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime ...

s, a consequence of the 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
An integer (from the Latin wikt:integer#Latin, ''integer'' meaning "wh ...

. Thus, if ''n'' is a product of powers of distinct primes, say ''n'' = ''p''homomorphism
In algebra
Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical analysis, analysis. I ...

of monoid
In abstract algebra
In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathemati ...

s and is completely determined by its restriction to the prime numbers.
Convolution

If ''f'' and ''g'' are two multiplicative functions, one defines a new multiplicative function $f\; *\; g$, the ''Dirichlet convolution
In mathematics, the Dirichlet convolution is a binary operation defined for arithmetic functions; it is important in number theory. It was developed by Peter Gustav Lejeune Dirichlet.
Definition
If f , g : \mathbb\to\mathbb are two arithmetic fun ...

'' of ''f'' and ''g'', by
$$(f\; \backslash ,\; *\; \backslash ,\; g)(n)\; =\; \backslash sum\_\; f(d)\; \backslash ,\; g\; \backslash left(\; \backslash frac\; \backslash right)$$
where the sum extends over all positive divisors ''d'' of ''n''.
With this operation, the set of all multiplicative functions turns into an abelian group
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...

; the identity element
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...

is ''ε''. Convolution is commutative, associative, and distributive over addition.
Relations among the multiplicative functions discussed above include:
* $\backslash mu\; *\; 1\; =\; \backslash varepsilon$ (the Möbius inversion formula
In mathematics, the classic Möbius inversion formula is a relation between pairs of arithmetic functions, each defined from the other by sums over divisors. It was introduced into number theory in 1832 by August Ferdinand Möbius.
A large general ...

)
* $(\backslash mu\; \backslash operatorname\_k)\; *\; \backslash operatorname\_k\; =\; \backslash varepsilon$ (generalized Möbius inversion)
* $\backslash varphi\; *\; 1\; =\; \backslash operatorname$
* $d\; =\; 1\; *\; 1$
* $\backslash sigma\; =\; \backslash operatorname\; *\; 1\; =\; \backslash varphi\; *\; d$
* $\backslash sigma\_k\; =\; \backslash operatorname\_k\; *\; 1$
* $\backslash operatorname\; =\; \backslash varphi\; *\; 1\; =\; \backslash sigma\; *\; \backslash mu$
* $\backslash operatorname\_k\; =\; \backslash sigma\_k\; *\; \backslash mu$
The Dirichlet convolution can be defined for general arithmetic functions, and yields a ring structure, the Dirichlet ring.
The Dirichlet convolution
In mathematics, the Dirichlet convolution is a binary operation defined for arithmetic functions; it is important in number theory. It was developed by Peter Gustav Lejeune Dirichlet.
Definition
If f , g : \mathbb\to\mathbb are two arithmetic fun ...

of two multiplicative functions is again multiplicative. A proof of this fact is given by the following expansion for relatively prime $a,b\; \backslash in\; \backslash mathbb^$:
$$\backslash begin\; (f\; \backslash ast\; g)(ab)\; \&\; =\; \backslash sum\_\; f(d)\; g\backslash left(\backslash frac\backslash right)\; \backslash \backslash \; \&=\; \backslash sum\_\; \backslash sum\_\; f(d\_1d\_2)\; g\backslash left(\backslash frac\backslash right)\; \backslash \backslash \; \&=\; \backslash sum\_\; f(d\_1)\; g\backslash left(\backslash frac\backslash right)\; \backslash times\; \backslash sum\_\; f(d\_2)\; g\backslash left(\backslash frac\backslash right)\; \backslash \backslash \; \&=\; (f\; \backslash ast\; g)(a)\; \backslash cdot\; (f\; \backslash ast\; g)(b).\; \backslash end$$
Dirichlet series for some multiplicative functions

* $\backslash sum\_\; \backslash frac\; =\; \backslash frac$ * $\backslash sum\_\; \backslash frac\; =\; \backslash frac$ * $\backslash sum\_\; \backslash frac\; =\; \backslash frac$ * $\backslash sum\_\; \backslash frac\; =\; \backslash frac$ More examples are shown in the article onDirichlet series
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...

.
Multiplicative function over

Let , the polynomial ring over thefinite field
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

with ''q'' elements. ''A'' is a principal ideal domain
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...

and therefore ''A'' is a unique factorization domain
In mathematics, a unique factorization domain (UFD) (also sometimes called a factorial ring following the terminology of Nicolas Bourbaki, Bourbaki) is a Ring (mathematics), ring in which a statement analogous to the fundamental theorem of arithme ...

.
A complex-valued function $\backslash lambda$ on ''A'' is called multiplicative if $\backslash lambda(fg)=\backslash lambda(f)\backslash lambda(g)$ whenever ''f'' and ''g'' are relatively prime
In number theory, two integer
An integer (from the Latin wikt:integer#Latin, ''integer'' meaning "whole") is colloquially defined as a number that can be written without a Fraction (mathematics), fractional component. For example, 21, 4, 0, ...

.
Zeta function and Dirichlet series in

Let ''h'' be a polynomial arithmetic function (i.e. a function on set of monic polynomials over ''A''). Its corresponding Dirichlet series is defined to be $$D\_h(s)=\backslash sum\_h(f),\; f,\; ^,$$ where for $g\backslash in\; A,$ set $,\; g,\; =q^$ if $g\backslash ne\; 0,$ and $,\; g,\; =0$ otherwise. The polynomial zeta function is then $$\backslash zeta\_A(s)=\backslash sum\_,\; f,\; ^.$$ Similar to the situation in , every Dirichlet series of a multiplicative function ''h'' has a product representation (Euler product): $$D\_(s)=\backslash prod\_P\; \backslash left(\backslash sum\_^h(P^),\; P,\; ^\backslash right),$$ where the product runs over all monic irreducible polynomials ''P''. For example, the product representation of the zeta function is as for the integers: $$\backslash zeta\_A(s)=\backslash prod\_(1-,\; P,\; ^)^.$$ Unlike the classicalzeta function
In mathematics, a zeta function is (usually) a function (mathematics), function analogous to the original example, the Riemann zeta function
: \zeta(s) = \sum_^\infty \frac 1 .
Zeta functions include:
* Airy zeta function, related to the zeros of ...

, $\backslash zeta\_A(s)$ is a simple rational function:
$$\backslash zeta\_A(s)=\backslash sum\_f\; ,\; f,\; ^\; =\; \backslash sum\_n\backslash sum\_q^=\backslash sum\_n(q^)=(1-q^)^.$$
In a similar way, If ''f'' and ''g'' are two polynomial arithmetic functions, one defines ''f'' * ''g'', the ''Dirichlet convolution'' of ''f'' and ''g'', by
$$\backslash begin\; (f*g)(m)\; \&=\; \backslash sum\_\; f(d)g\backslash left(\backslash frac\backslash right)\; \backslash \backslash \; \&=\; \backslash sum\_f(a)g(b),\; \backslash end$$
where the sum is over all monic divisor
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...

s ''d'' of ''m'', or equivalently over all pairs (''a'', ''b'') of monic polynomials whose product is ''m''. The identity $D\_h\; D\_g\; =\; D\_$ still holds.
See also

*Euler product In number theory, an Euler product is an expansion of a Dirichlet series into an infinite product indexed by prime numbers. The original such product was given for Proof of the Euler product formula for the Riemann zeta function, the sum of all posi ...

* Bell series In mathematics, the Bell series is a formal power series used to study properties of arithmetical functions. Bell series were introduced and developed by Eric Temple Bell.
Given an arithmetic function f and a Prime number, prime p, define the forma ...

* Lambert series
In mathematics, a Lambert series, named for Johann Heinrich Lambert, is a Series (mathematics), series taking the form
:S(q)=\sum_^\infty a_n \frac .
It can be resumed Formal series, formally by expanding the denominator:
:S(q)=\sum_^\infty a_ ...

References

* See chapter 2 of {{Apostol IANTExternal links

Planet Math