- Outside number theory, the term multiplicative function is usually used for completely multiplicative functions. This article discusses number theoretic multiplicative functions.
- 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.
Some multiplicative functions are defined to make formulas easier to write:
- 1(n): the constant function, defined by 1(n) = 1 (completely multiplicative)
- the indicator function of the set . This is multiplicative if the set C has the property that if a and b are in C, gcd(a, b)=1, then ab is also in C. This is the case if C is the set of squares, cubes, or higher powers, or if C is the set of square-free numbers.
- 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).
- (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; the Kronecker delta δin; sometimes written as u(n), not to be confused with (n) (completely multiplicative).
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.
- (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
- k(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
- 0(n) = d(n) the number of positive divisors of n,
- 1(n) = (n), the sum of all the positive divisors of n.
- (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).
- (n), defined by (n) = (−1)(n), where the additive function (n) is the number of distinct primes dividing n.
- All Dirichlet characters are completely multiplicative functions. For example
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:
- 1 = 12 + 02 = (-1)2 + 02 = 02 + 12 = 02 + (-1)2
and therefore r2(1) = 4 ≠ 1. This shows that the function is not multiplicative. However, r2(n)/4 is multiplicative.
See arithmetic function for some other examples of non-multiplicative functions.
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:
- d(144) = 0(144) = 0(24)0(32) = (10 + 20 + 40 + 80 + 160)(10 + 30 + 90) = 5 · 3 = 15,
- (144) = 1(144) = 1(24)1(32) = (11 + 21 + 41 + 81 + 161)(11 + 31 + 91) = 31 · 13 = 403,
- *(144) = *(24)*(32) = (11 + 161)(11 + 91) = 17 · 10 = 170.
Similarly, we have:
In general, if f(n) is a multiplicative function and a, b are any two positive integers, then
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 . Convolution is commutative, associative, and distributive over addition.
Relations among the multiplicative functions discussed above include:
- * 1 = (the Möbius inversion formula)
- ( Idk) * Idk = (generalized Möbius inversion)
- * 1 = Id
- d = 1 * 1
- = Id * 1 = * d
- k = Idk * 1
- Id = * 1 = *
- Idk = k *
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.
Multiplicative function over Fq[X]
a complex-valued function on A is called multiplicative if , whenever f and g are relatively prime.
Zeta function and Dirichlet series in Fq[X]
Let h be a polynomial arithmetic function (i.e. a function on set of monic polynomials over A). Its corresponding Dirichlet series define to be
The polynomial zeta function is then
Similar to the situation in N, every Dirichlet series of a multiplicative function h has a product representation (Euler product):
Where the product runs over all monic irreducible polynomials P.
Unlike the classical zeta function, is a simple rational function:
In a similar way, If ƒ and g are two polynomial arithmetic functions, one defines ƒ * g, the Dirichlet convolution of ƒ and g, by
where the sum extends over all monic divisors d of m, or equivalently over all pairs (a, b) of monic polynomials whose product is m. The identity still holds.
- See chapter 2 of Template:Apostol IANT