Additive function

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In mathematics the term additive function has two different definitions, depending on the specific field of application.

In algebra an additive function (or additive map) is a function that preserves the addition operation:

for any two elements x and y in the domain. For example, any linear map is additive. When the domain is the real numbers, this is Cauchy's functional equation.

In number theory, an additive function is an arithmetic function f(n) of the positive integer n such that whenever a and b are coprime, the function of the product is the sum of the functions:

The remainder of this article discusses number theoretic additive functions, using the second definition. For a specific case of the first definition see additive polynomial. Note also that any homomorphism f between Abelian groups is "additive" by the first definition.

Contents

Completely additive

An additive function f(n) is said to be completely additive if f(ab) = f(a) + f(b) holds for all positive integers a and b, even when they are not co-prime. Totally additive is also used in this sense by analogy with totally multiplicative functions. If f is a completely additive function then f(1) = 0.

Every completely additive function is additive, but not vice versa.

Examples

Example of arithmetic functions which are completely additive are:

  • The multiplicity of a prime factor p in n, that is the largest exponent m for which pm divides n.
  • a0(n) - the sum of primes dividing n counting multiplicity, sometimes called sopfr(n), the potency of n or the integer logarithm of n (sequence A001414 in OEIS). For example:
  • The function Ω(n), defined as the total number of prime factors of n, counting multiple factors multiple times, sometimes called the "Big Omega function" (sequence A001222 in OEIS). For example;

Example of arithmetic functions which are additive but not completely additive are:

  • ω(n), defined as the total number of different prime factors of n (sequence A001221 in OEIS). For example:
  • a1(n) - the sum of the distinct primes dividing n, sometimes called sopf(n) (sequence A008472 in OEIS). For example:

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