additive function

Different definitions exist depending on the specific field of application. Traditionally, an additive function 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.

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 coprime. Totally additive is also used in this sense by analogy with totally multiplicative functions.

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

Examples

Arithmetic functions which are completely additive are:

• The restriction of the logarithmic function to N

• a₀(n) - the sum of primes dividing n, sometimes called sopfr(n). We have a₀(20) = a₀(2² · 5) = 2 + 2+ 5 = 9. Some values: (OEIS A001414).

a₀(4) = 4

a₀(27) = 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. It is often called "Big Omega function".This implies Ω(1) = 0 since 1 has no prime factors. Some more values: (OEIS A001222)

Ω(4) = 2

Ω(27) = 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

...

• The function a₁(n) - the sum of the distinct primes dividing n, sometimes called sopf(n), is additive but not completely additive. We have a₁(1) = 0, a₁(20) = 2 + 5 = 7. Some more values: (OEIS A008472)

a₁(4) = 2

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

...

• Another example of an arithmetic function which is additive but not completely additive is ω(n), defined as the total number of different prime factors of n. Some values (compare with Ω(n)) (OEIS A001221)

ω(4) = 1

ω(27) = 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

...

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).

References

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

See also

• Sigma additivity

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