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

Completely additive

An additive function f(n) is said to be completely additive 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

Examples of arithmetic functions which are completely additive are:

• The restriction of the logarithmic function

• 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 . For example:

  1. a0(4) = 2 + 2 = 4

  2. a0(20) = a0(22 · 5) = 2 + 2 + 5 = 9

  3. a0(27) = 3 + 3 + 3 = 9

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

  5. a0(2000) = a0(24 · 53) = a0(24) + a0(53) = 8 + 15 = 23

  6. a0(2003) = 2003

  7. a0(54,032,858,972,279) = 1240658

  8. a0(54,032,858,972,302) = 1780417

  9. 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" . For example;

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

  2. Ω(4) = 2

  3. Ω(16) = Ω(2·2·2·2) = 4

  4. Ω(20) = Ω(2·2·5) = 3

  5. Ω(27) = Ω(3·3·3) = 3

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

  7. Ω(2000) = Ω(24 · 53) = Ω(24) + Ω(53) = 4 + 3 = 7

  8. Ω(2001) = 3

  9. Ω(2002) = 4

  10. Ω(2003) = 1

  11. Ω(54,032,858,972,279) = Ω(11 ⋅ 19932 ⋅ 1236661) = 4

  12. Ω(54,032,858,972,302) = Ω(2 ⋅ 72 ⋅ 149 ⋅ 2081 ⋅ 1778171) = 6

  13. Ω(20,802,650,704,327,415) = Ω(5 ⋅ 7 ⋅ 112 ⋅ 19932 ⋅ 1236661) = 7.

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

• ω(n), defined as the total number of distinct prime factors of n . For example:

  1. ω(4) = 1

  2. ω(16) = ω(24) = 1

  3. ω(20) = ω(22 · 5) = 2

  4. ω(27) = ω(33) = 1

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

  6. ω(2000) = ω(24 · 53) = ω(24) + ω(53) = 1 + 1 = 2

  7. ω(2001) = 3

  8. ω(2002) = 4

  9. ω(2003) = 1

  10. ω(54,032,858,972,279) = 3

  11. ω(54,032,858,972,302) = 5

  12. ω(20,802,650,704,327,415) = 5

  a1(n) – the sum of the distinct primes dividing n, sometimes called sopf(n) . For example:

  1. a1(1) = 0

  2. a1(4) = 2

  3. a1(20) = 2 + 5 = 7

  4. a1(27) = 3

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

  6. a1(2000) = a1(24 · 53) = a1(24) + a1(53) = 2 + 5 = 7

  7. a1(2001) = 55

  8. a1(2002) = 33

  9. a1(2003) = 2003

  10. a1(54,032,858,972,279) = 1238665

  11. a1(54,032,858,972,302) = 1780410

  12. a1(20,802,650,704,327,415) = 1238677

Multiplicative functions

it is possible to create a related is a nonzero real constant.

Summatory functions

such that for all natural numbers

is the Gaussian distribution function

Examples of this result related to the prime omega function

See also

• Sigma additivity

• Prime omega function

• Multiplicative function

• Arithmetic function

References

Further reading

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

• Iwaniec and Kowalski, Analytic number theory, AMS (2004).