Additive function

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For the algebraic meaning, see Additive map.

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:[1]

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

Completely additive[edit]

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[edit]

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:
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 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(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 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[edit]

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) = 2f(n).

See also[edit]

References[edit]

  1. ^ Erdös, P., and M. Kac. On the Gaussian Law of Errors in the Theory of Additive Functions. Proc Natl Acad Sci USA. 1939 April; 25(4): 206–207. online

Further reading[edit]

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