Multiplicative function Totally Explained

Everything about Multiplicative Function totally explained

ť 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 ť 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're not coprime.

Examples

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

$phi$ (n): Euler's totient function $phi$ , 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 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). In special cases we have
- $sigma$ ₀(n) = d(n) and
- $sigma$ ₁(n) = $sigma$ (n),
$a(n)$ : the number of non-isomorphic 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
- 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 or simply the unit function; sometimes written as u(n), not to be confused with $mu$ (n) (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:
ť 1 = 1² + 0² = (-1)² + 0² = 0² + 1² = 0² + (-1)²

and therefore r₂(1) = 4 ≠ 1. This shows that the function isn't 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 = 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: ť 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, ť $sigma$ ^*(144) = $sigma$ ^*(2⁴) $sigma$ ^*(3²) = (1¹ + 16¹)(1¹ + 9¹) = 17 · 10 = 170.

Similarly, we have:
ť $phi$ (144)= $phi$ (2⁴) $phi$ (3²) = 8 · 6 = 48

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

(f , * , g)(n) = sum_.

Further Information

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