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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.

Examples

Some multiplicative functions are defined to make formulas easier to write:

• 1(n): the constant function, defined by 1(n) = 1 (completely multiplicative)

• $1_C(n)$ the indicator function of the set $C\subset \mathbb{\mathbb{Z}}$. 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)

• Id_k(n): the power functions, defined by Id_k(n) = n^k 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; the Kronecker delta δ_in; sometimes written as u(n), not to be confused with $\mu$(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.

• $\phi$(n): Euler's totient function $\phi$, counting the positive integers coprime to (but not bigger than) n

• $\mu$(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

• $\sigma$_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). Special cases we have
  • $\sigma$₀(n) = d(n) the number of positive divisors of n,
  • $\sigma$₁(n) = $\sigma$(n), the sum of all the positive divisors of n.

• $a(n)$: the number of non-isomorphic abelian groups of order n.

• $\lambda$(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).

• $\gamma$(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. 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:

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.

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:

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 · 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

$(f*g)(n)=\sum _{d|n}f(d)g\left(\frac{n}{d}\right)$

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 = $ϵ$ (the Möbius inversion formula)
• ($\mu$ * Id_k) * Id_k = $ϵ$ (generalized Möbius inversion)
• $\phi$ * 1 = 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.

Dirichlet series for some multiplicative functions

• $\sum _{n\ge 1}\frac{\mu (n)}{n^s}=\frac{1}{\zeta (s)}$
• $\sum _{n\ge 1}\frac{\phi (n)}{n^s}=\frac{\zeta (s-1)}{\zeta (s)}$
• $\sum _{n\ge 1}\frac{d(n{)}^2}{n^s}=\frac{\zeta (s{)}^4}{\zeta (2s)}$
• $\sum _{n\ge 1}\frac{2^{\omega (n)}}{n^s}=\frac{\zeta (s{)}^2}{\zeta (2s)}$

More examples are shown in the article on Dirichlet series.

See also

• Euler product
• Bell series
• Lambert series

References

• See chapter 2 of Template:Apostol IANT

External links

• Planet Math