Multiplicative function


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.

Contents

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,
  • σk(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)
  • 1C(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)
  • Idk(n): the power functions, defined by Idk(n) = nk 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 = 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) = 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 * Idk) * Idk = \epsilon (generalized Möbius inversion)
  • ϕ * 1 = Id
  • d = 1 * 1
  • σ = Id * 1 = ϕ * d
  • σk = Idk * 1
  • Id = ϕ * 1 = σ * μ
  • Idk = σk * 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

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


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