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词条 Additive function
释义

  1. Completely additive

  2. Examples

  3. Multiplicative functions

  4. See also

  5. References

  6. Further reading

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In number theory, an {{anchor|definition-additive_function-number_theory}}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

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 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 {{OEIS|A001414}}. 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" {{OEIS|A001222}}. 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 {{OEIS|A001221}}. 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) {{OEIS|A008472}}. 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) = 2f(n).

See also

  • Sigma additivity

References

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. [https://www.ncbi.nlm.nih.gov/pmc/articles/PMC1077746/ online]

Further reading

{{refbegin}}
  • Janko Bračič, Kolobar aritmetičnih funkcij (Ring of arithmetical functions), (Obzornik mat, fiz. 49 (2002) 4, pp. 97–108) (MSC (2000) 11A25)
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2 : Arithmetic functions|Additive function
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