multiplicative function

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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 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) * Id''k''(''n''): the power functions, defined by Id''k''(''n'') = ''n''''k'' for any complex number ''k'' (completely multiplicative). As special cases we have ** Id0(''n'') = 1(''n'') and ** Id1(''n'') = Id(''n''). * ''ε''(''n''): the function defined by ''ε''(''n'') = 1 if ''n'' = 1 and 0 otherwise, sometimes called ''multiplication unit for
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 ...
'' 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'') . * 1''C''(''n''), the
indicator 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 1''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
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. 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. * $\varphi\left(n\right)$:
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 ...
$\varphi$, 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''''k''(''n''): the
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''0(''n'') = ''d''(''n'') the number of positive
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''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
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)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
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''2(''n'') - the number of representations of ''n'' as a sum of squares of two integers,
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''2(1) = 4 ≠ 1. This shows that the function is not multiplicative. However, ''r''2(''n'')/4 is multiplicative. In the
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 of
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 ...
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''''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 = 24 · 32: Similarly, we have: $\varphi(144) = \varphi(2^4) \, \varphi(3^2) = 8 \cdot 6 = 48$ In general, if ''f''(''n'') is a multiplicative function and ''a'', ''b'' are any two positive integers, then Every completely multiplicative function is a
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 $(f \, * \, g)(n) = \sum_ f(d) \, g \left( \frac \right)$ 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: * $\mu * 1 = \varepsilon$ (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 ...
) * $\left(\mu \operatorname_k\right) * \operatorname_k = \varepsilon$ (generalized Möbius inversion) * $\varphi * 1 = \operatorname$ * $d = 1 * 1$ * $\sigma = \operatorname * 1 = \varphi * d$ * $\sigma_k = \operatorname_k * 1$ * $\operatorname = \varphi * 1 = \sigma * \mu$ * $\operatorname_k = \sigma_k * \mu$ 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 $a,b \in \mathbb^$: $\begin (f \ast g)(ab) & = \sum_ f(d) g\left(\frac\right) \\ &= \sum_ \sum_ f(d_1d_2) g\left(\frac\right) \\ &= \sum_ f(d_1) g\left(\frac\right) \times \sum_ f(d_2) g\left(\frac\right) \\ &= (f \ast g)(a) \cdot (f \ast g)(b). \end$

## Dirichlet series for some multiplicative functions

* $\sum_ \frac = \frac$ * $\sum_ \frac = \frac$ * $\sum_ \frac = \frac$ * $\sum_ \frac = \frac$ More examples are shown in the article on
Dirichlet 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 the
finite 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 $\lambda$ on ''A'' is called multiplicative if $\lambda\left(fg\right)=\lambda\left(f\right)\lambda\left(g\right)$ 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 $D_h(s)=\sum_h(f), f, ^,$ where for $g\in A,$ set $, g, =q^$ if $g\ne 0,$ and $, g, =0$ otherwise. The polynomial zeta function is then $\zeta_A(s)=\sum_, f, ^.$ Similar to the situation in , every Dirichlet series of a multiplicative function ''h'' has a product representation (Euler product): $D_(s)=\prod_P \left(\sum_^h(P^), P, ^\right),$ where the product runs over all monic irreducible polynomials ''P''. For example, the product representation of the zeta function is as for the integers: $\zeta_A(s)=\prod_(1-, P, ^)^.$ Unlike the classical
zeta 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 ...

, $\zeta_A\left(s\right)$ is a simple rational function: $\zeta_A(s)=\sum_f , f, ^ = \sum_n\sum_q^=\sum_n(q^)=(1-q^)^.$ 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 $\begin (f*g)(m) &= \sum_ f(d)g\left(\frac\right) \\ &= \sum_f(a)g(b), \end$ 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.

*
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 IANT