Additive function

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:

f(x + y) = f(x) + f(y)

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:

f(ab) = f(a) + f(b).

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.

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 restriction of the logarithmic function to N.

• The multiplicity of a prime factor p in n, that is the largest exponent m for which p^m 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 the OEIS). For example:

a₀(4) = 2 + 2 = 4

a₀(20) = a₀(2² · 5) = 2 + 2+ 5 = 9

a₀(27) = 3 + 3 + 3 = 9

a₀(144) = a₀(2⁴ · 3²) = a₀(2⁴) + a₀(3²) = 8 + 6 = 14

a₀(2,000) = a₀(2⁴ · 5³) = a₀(2⁴) + a₀(5³) = 8 + 15 = 23

a₀(2,003) = 2003

a₀(54,032,858,972,279) = 1240658

a₀(54,032,858,972,302) = 1780417

a₀(20,802,650,704,327,415) = 1240681

• 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 the OEIS). For example;

Ω(1) = 0, since 1 has no prime factors

Ω(4) = 2

Ω(16) = Ω(2·2·2·2) = 4

Ω(20) = Ω(2·2·5) = 3

Ω(27) = Ω(3·3·3) = 3

Ω(144) = Ω(2⁴ · 3²) = Ω(2⁴) + Ω(3²) = 4 + 2 = 6

Ω(2,000) = Ω(2⁴ · 5³) = Ω(2⁴) + Ω(5³) = 4 + 3 = 7

Ω(2,001) = 3

Ω(2,002) = 4

Ω(2,003) = 1

Ω(54,032,858,972,279) = 3

Ω(54,032,858,972,302) = 6

Ω(20,802,650,704,327,415) = 7

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 the OEIS). For example:

ω(4) = 1

ω(16) = ω(2⁴) = 1

ω(20) = ω(2² · 5) = 2

ω(27) = ω(3³) = 1

ω(144) = ω(2⁴ · 3²) = ω(2⁴) + ω(3²) = 1 + 1 = 2

ω(2,000) = ω(2⁴ · 5³) = ω(2⁴) + ω(5³) = 1 + 1 = 2

ω(2,001) = 3

ω(2,002) = 4

ω(2,003) = 1

ω(54,032,858,972,279) = 3

ω(54,032,858,972,302) = 5

ω(20,802,650,704,327,415) = 5

• a₁(n) - the sum of the distinct primes dividing n, sometimes called sopf(n) (sequence A008472 in the OEIS). For example:

a₁(1) = 0

a₁(4) = 2

a₁(20) = 2 + 5 = 7

a₁(27) = 3

a₁(144) = a₁(2⁴ · 3²) = a₁(2⁴) + a₁(3²) = 2 + 3 = 5

a₁(2,000) = a₁(2⁴ · 5³) = a₁(2⁴) + a₁(5³) = 2 + 5 = 7

a₁(2,001) = 55

a₁(2,002) = 33

a₁(2,003) = 2003

a₁(54,032,858,972,279) = 1238665

a₁(54,032,858,972,302) = 1780410

a₁(20,802,650,704,327,415) = 1238677

Multiplicative functions

From any additive function f(n) it is easy to create a related multiplicative function g(n) i.e. with the property that whenever a and b are coprime we have:

g(ab) = g(a) × g(b).

One such example is g(n) = 2^f(n).

See also

• Sigma additivity

Further reading

• Janko Bračič, Kolobar aritmetičnih funkcij (Ring of arithmetical functions), (Obzornik mat, fiz. 49 (2002) 4, pp. 97–108) (MSC (2000) 11A25)