In number theory, a multiplicative function is an arithmetic function f(n) of a positive integer n with the property that f(1) = 1 and  f(ab) = f(a)f(b) whenever a and b are coprime.
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, defined by Id(n) = n (completely multiplicative)
Idk(n): the power functions, defined by Idk(n) = nk 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 or simply the unit function (completely multiplicative).
Sometimes written as u(n), but not to be confused with μ(n) .
1C(n), the indicator function of the set C ⊂ Z, for certain sets C.
The indicator function 1C(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 numbers.
Other examples of multiplicative functions include many functions of importance in number theory, such as:
gcd(n,k): the greatest common divisor of n and k, as a function of n, where k is a fixed integer.
\varphi(n): Euler's totient function \varphi, counting the positive integers coprime to (but not bigger than) n
μ(n): the Möbius function, the parity (−1 for odd, +1 for even) of the number of prime factors of square-free numbers; 0 if n is not square-free
σk(n): the divisor function, which is the sum of the k-th powers of all the positive divisors of n (where k may be any complex number).
Special cases we have
σ0(n) = d(n) the number of positive divisors of n,
σ1(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, 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 ω(n) is the number of distinct primes dividing n.
τ(n): the Ramanujan tau function.
All Dirichlet characters are completely multiplicative functions.
For example
(n/p), the Legendre symbol, considered as a function of n where p is a fixed prime number.
An example of a non-multiplicative function is the arithmetic function r2(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:
and therefore r2(1) = 4 ≠ 1.
This shows that the function is not multiplicative.
However, r2(n)/4 is multiplicative.
In the On-Line Encyclopedia of Integer Sequences, sequences of values of a multiplicative function have the keyword "mult".
See arithmetic function for some other examples of non-multiplicative 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 = pa qb ..., then  f(n) = f(pa) f(qb) ...
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 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  (f \, * \, g)(n) = \sum_{d|n} f(d) \, g \left( \frac{n}{d} \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; the identity element 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)
(\mu \operatorname{Id}_k) * \operatorname{Id}_k = \varepsilon (generalized Möbius inversion)
\varphi * 1 = \operatorname{Id}
d = 1 * 1
\sigma = \operatorname{Id} * 1 = \varphi * d
\sigma_k = \operatorname{Id}_k * 1
\operatorname{Id} = \varphi * 1 = \sigma * \mu
\operatorname{Id}_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 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{Z}^{+}:  \begin{align} (f \ast g)(ab) & = \sum_{d|ab} f(d) g\left(\frac{ab}{d}\right) \\ &= \sum_{d_1|a} \sum_{d_2|b} f(d_1d_2) g\left(\frac{ab}{d_1d_2}\right) \\ &= \sum_{d_1|a} f(d_1) g\left(\frac{a}{d_1}\right) \times \sum_{d_2|b} f(d_2) g\left(\frac{b}{d_2}\right) \\ &= (f \ast g)(a) \cdot (f \ast g)(b).
\end{align} Dirichlet series for some multiplicative functions
\sum_{n\ge 1} \frac{\mu(n)}{n^s} = \frac{1}{\zeta(s)}
\sum_{n\ge 1} \frac{\varphi(n)}{n^s} = \frac{\zeta(s-1)}{\zeta(s)}
\sum_{n\ge 1} \frac{d(n)^2}{n^s} = \frac{\zeta(s)^4}{\zeta(2s)}
\sum_{n\ge 1} \frac{2^{\omega(n)}}{n^s} = \frac{\zeta(s)^2}{\zeta(2s)}
More examples are shown in the article on Dirichlet series.
Multiplicative function over {{math|''F''<sub>''q''</sub>[''X'']}}
Let , the polynomial ring over the finite field with q elements.
A is a principal ideal domain and therefore A is a unique factorization domain.
A complex-valued function \lambda on A is called multiplicative if \lambda(fg)=\lambda(f)\lambda(g) whenever f and g are relatively prime.
Zeta function and Dirichlet series in {{math|''F''<sub>''q''</sub>[''X'']}}
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_{f\text{ monic}}h(f)|f|^{-s},
where for g\in A, set |g|=q^{\deg(g)} if g\ne 0, and |g|=0 otherwise.
The polynomial zeta function is then \zeta_A(s)=\sum_{f\text{ monic}}|f|^{-s}.
Similar to the situation in , every Dirichlet series of a multiplicative function h has a product representation (Euler product):
D_{h}(s)=\prod_P \left(\sum_{n\mathop =0}^{\infty}h(P^{n})|P|^{-sn}\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_{P}(1-|P|^{-s})^{-1}.
Unlike the classical zeta function, \zeta_A(s) is a simple rational function:
\zeta_A(s)=\sum_f |f|^{-s} = \sum_n\sum_{\deg(f)=n}q^{-sn}=\sum_n(q^{n-sn})=(1-q^{1-s})^{-1}.
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{align} (f*g)(m)  &= \sum_{d \mid m} f(d)g\left(\frac{m}{d}\right) \\ &= \sum_{ab = m}f(a)g(b), \end{align}
where the sum is over all monic divisors d of m, or equivalently over all pairs (a, b) of monic polynomials whose product is m.
The identity D_h D_g = D_{h*g} still holds.
See also
Euler product
Bell series
Lambert series
References
See chapter 2 of
External links
Planet Math
