Multiplicative function

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

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

• ϕ(n): Euler's totient function ϕ, counting the positive integers coprime to (but not bigger than) n
• μ(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,
• σ(n): the sum of all the positive divisors of n,
• σ_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
  • σ₀(n) = d(n) and
  • σ₁(n) = σ(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 otherwise, sometimes called multiplication unit for Dirichlet convolution or simply the unit function; sometimes written as u(n), not to be confused with μ(n) (completely multiplicative).
• (n/p), the Legendre symbol, where p is a fixed prime number (completely multiplicative).
• λ(n): the Liouville function, related to the number of prime factors 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.
• 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.

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) = σ₀(144) = σ₀(2⁴)σ₀(3²) = (1⁰ + 2⁰ + 4⁰ + 8⁰ + 16⁰)(1⁰ + 3⁰ + 9⁰) = 5 · 3 = 15,

σ(144) = σ₁(144) = σ₁(2⁴)σ₁(3²) = (1¹ + 2¹ + 4¹ + 8¹ + 16¹)(1¹ + 3¹ + 9¹) = 31 · 13 = 403,

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

Similarly, we have:

ϕ(144)=ϕ(2⁴)ϕ(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 $\epsilon$.

Relations among the multiplicative functions discussed above include:

• μ * 1 = $\epsilon$ (the Möbius inversion formula)
• (μ * Id_k) * Id_k = $\epsilon$ (generalized Möbius inversion)
• ϕ * 1 = Id
• d = 1 * 1
• σ = Id * 1 = ϕ * d
• σ_k = Id_k * 1
• Id = ϕ * 1 = σ * μ
• Id_k = σ_k * μ

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

Dirichlet series for some multiplicative functions

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

More examples are shown in the article on Dirichlet series.

See also

• Euler product
• Bell series
• Lambert series

References

• See chapter 2 of Apostol, Tom M. (1976), Introduction to analytic number theory, Undergraduate Texts in Mathematics, New York-Heidelberg: Springer-Verlag, ISBN 978-0-387-90163-3, MR0434929 

External links

• Planet Math