Multiplicative Function - Examples

Examples

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

  • 1(n): the constant function, defined by 1(n) = 1 (completely multiplicative)
  • the indicator function of the set . This is multiplicative if the set C has the property that if a and b are in C, gcd(a, b)=1, than 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)
  • Idk(n): the power functions, defined by Idk(n) = nk for any complex number k (completely multiplicative). As special cases we have
    • Id0(n) = 1(n) and
    • Id1(n) = Id(n).
  • (n): the function defined by (n) = 1 if n = 1 and 0 otherwise, sometimes called multiplication unit for Dirichlet convolution or simply the unit function; the Kronecker delta δin; sometimes written as u(n), not to be confused with (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.
  • (n): Euler's totient function, counting the positive integers coprime to (but not bigger than) n
  • (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
  • 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
    • 0(n) = d(n) the number of positive divisors of n,
    • 1(n) = (n), the sum of all the positive divisors of n.
  • : the number of non-isomorphic abelian groups of order n.
  • (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).
  • (n), defined by (n) = (−1)(n), where the additive function (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.

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