 Multiplicative function

 Outside number theory, the term multiplicative function is usually used for completely multiplicative functions. This article discusses number theoretic multiplicative functions.
In number theory, a multiplicative function is an arithmetic function f(n) of the positive integer n with the property that f(1) = 1 and whenever a and b are coprime, then
 f(ab) = f(a) f(b).
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.
Contents
Examples
Examples of multiplicative functions include many functions of importance in number theory, such as:
 ϕ(n): Euler's totient function ϕ, counting the positive integers coprime to (but not bigger than) n
 μ(n): the Möbius function, related to the number of prime factors of squarefree numbers
 gcd(n,k): the greatest common divisor of n and k, where k is a fixed integer.
 d(n): the number of positive divisors of n,
 σ(n): the sum of all the positive divisors of n,
 σ_{k}(n): the divisor function, which is the sum of the kth powers of all the positive divisors of n (where k may be any complex number). In special cases we have
 σ_{0}(n) = d(n) and
 σ_{1}(n) = σ(n),
 a(n): the number of nonisomorphic abelian groups of order n.
 1(n): the constant function, defined by 1(n) = 1 (completely multiplicative)
 1_{C}(n) the indicator function of the set C of squares (or cubes, or fourth powers, etc.)
 Id(n): identity function, defined by Id(n) = n (completely multiplicative)
 Id_{k}(n): the power functions, defined by Id_{k}(n) = n^{k} for any natural (or even complex) number k (completely multiplicative). As special cases we have
 Id_{0}(n) = 1(n) and
 Id_{1}(n) = Id(n),
 (n): the function defined by (n) = 1 if n = 1 and 0 otherwise, sometimes called multiplication unit for Dirichlet convolution or simply the unit function; sometimes written as u(n), not to be confused with μ(n) (completely multiplicative).
 (n/p), the Legendre symbol, where p is a fixed prime number (completely multiplicative).
 λ(n): the Liouville function, related to the number of prime factors dividing n (completely multiplicative).
 γ(n), defined by γ(n)=(1)^{ω(n)}, where the additive function ω(n) is the number of distinct primes dividing n.
 All Dirichlet characters are completely multiplicative functions.
An example of a nonmultiplicative function is the arithmetic function r_{2}(n)  the number of representations of n as a sum of squares of two integers, positive, negative, or zero, where in counting the number of ways, reversal of order is allowed. For example: 1 = 1^{2} + 0^{2} = (1)^{2} + 0^{2} = 0^{2} + 1^{2} = 0^{2} + (1)^{2}
and therefore r_{2}(1) = 4 ≠ 1. This shows that the function is not multiplicative. However, r_{2}(n)/4 is multiplicative.
In the OnLine Encyclopedia of Integer Sequences, sequences of values of a multiplicative function have the keyword "mult".
See arithmetic function for some other examples of nonmultiplicative functions.
Properties
A multiplicative function is completely determined by its values at the powers of prime numbers, a consequence of the fundamental theorem of arithmetic. 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 = 2^{4} · 3^{2}:
 d(144) = σ_{0}(144) = σ_{0}(2^{4})σ_{0}(3^{2}) = (1^{0} + 2^{0} + 4^{0} + 8^{0} + 16^{0})(1^{0} + 3^{0} + 9^{0}) = 5 · 3 = 15,
 σ(144) = σ_{1}(144) = σ_{1}(2^{4})σ_{1}(3^{2}) = (1^{1} + 2^{1} + 4^{1} + 8^{1} + 16^{1})(1^{1} + 3^{1} + 9^{1}) = 31 · 13 = 403,
 σ^{*}(144) = σ^{*}(2^{4})σ^{*}(3^{2}) = (1^{1} + 16^{1})(1^{1} + 9^{1}) = 17 · 10 = 170.
Similarly, we have:
 ϕ(144)=ϕ(2^{4})ϕ(3^{2}) = 8 · 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 of monoids 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 of f and g, by
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; the identity element is .
Relations among the multiplicative functions discussed above include:
 μ * 1 = (the Möbius inversion formula)
 (μ * Id_{k}) * Id_{k} = (generalized Möbius inversion)
 ϕ * 1 = Id
 d = 1 * 1
 σ = Id * 1 = ϕ * d
 σ_{k} = Id_{k} * 1
 Id = ϕ * 1 = σ * μ
 Id_{k} = σ_{k} * μ
The Dirichlet convolution can be defined for general arithmetic functions, and yields a ring structure, the Dirichlet ring.
Dirichlet series for some multiplicative functions
More examples are shown in the article on Dirichlet series.
See also
References
 See chapter 2 of Apostol, Tom M. (1976), Introduction to analytic number theory, Undergraduate Texts in Mathematics, New YorkHeidelberg: SpringerVerlag, ISBN 9780387901633, MR0434929
External links
Categories: Multiplicative functions
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