Multiplicative function

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 (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.

Outside number theory, the term multiplicative is usually used for functions with the property f(ab) = f(a) f(b) for all arguments a and b. This article discusses number theoretic multiplicative functions.

Examples

Examples of multiplicative functions include many functions of importance in number theory, such as:

• $\phi$(n): the Euler $\phi$ function, counting the positive integers coprime to (but not bigger than) n
• $\mu$(n): the Möbius function, related to the number of prime factors of square-free 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,
• $\sigma$(n): the sum of all the positive divisors of n,
• $\sigma$_k(n): the sum of the k-th powers of all the positive divisors of n (where k may be any complex number). In special cases we have
  • $\sigma$₀(n) = d(n) and
  • $\sigma$₁(n) = $\sigma$(n),
• 1(n): the constant function, defined by 1(n) = 1 (completely multiplicative)
• 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
  • Id₀(n) = 1(n) and
  • Id₁(n) = Id(n),
• $\epsilon$(n): the function defined by $\epsilon$(n) = 1 if n = 1 and = 0 if n > 1, sometimes called multiplication unit for Dirichlet convolution (completely multiplicative).
• (n/p), the Legendre symbol, where p is a fixed prime number (completely multiplicative).
• $\lambda$(n): the Liouville function, related to the number of prime factors dividing n (completely multiplicative).
• $\gamma$(n), defined by $\gamma$(n)=(-1)^$\omega$(n), where the additive function $\omega$(n) is the number of distinct primes dividing n.
• All Dirichlet characters are completely multiplicative functions.

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.

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 = 2⁴ · 3²:

d(144) = $\sigma$₀(144) = $\sigma$₀(2⁴)$\sigma$₀(3²) = (1⁰ + 2⁰ + 4⁰ + 8⁰ + 16⁰)(1⁰ + 3⁰ + 9⁰) = 5 · 3 = 15,

$\sigma$(144) = $\sigma$₁(144) = $\sigma$₁(2⁴)$\sigma$₁(3²) = (1¹ + 2¹ + 4¹ + 8¹ + 16¹)(1¹ + 3¹ + 9¹) = 31 · 13 = 403,

$\sigma$^*(144) = $\sigma$^*(2⁴)$\sigma$^*(3²) = (1¹ + 16¹)(1¹ + 9¹) = 17 · 10 = 170.

Similarly, we have:

$\phi$(144)=$\phi$(2⁴)$\phi$(3²) = 8 &middot 6 = 48

In general, if f(n) is a multiplicative function and a, b are any two positive integers, then

f(a) · f(b) = f(gcd(a,b)) · f(lcm(a,b)).

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) = ∑_d|n f(d)g(n/d)

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 $\epsilon$.

Relations among the multiplicative functions discussed above include:

• $\epsilon$ = $\mu$ * 1 (the Möbius inversion formula)
• $\phi$ = $\mu$ * Id
• d = 1 * 1
• $\sigma$ = Id * 1 = $\phi$ * d
• $\sigma$_k = Id_k * 1
• Id = $\phi$ * 1 = $\sigma$ * $\mu$
• Id_k = $\sigma$_k * $\mu$

The Dirichlet convolution can be defined for general arithmetic functions, and yields a ring structure, the Dirichlet ring.

See also: Euler product.