Arithmetic Function - Multiplicative and Additive Functions

Multiplicative and Additive Functions

An arithmetic function a is

  • completely additive if a(mn) = a(m) + a(n) for all natural numbers m and n;
  • completely multiplicative if a(mn) = a(m)a(n) for all natural numbers m and n;

Two whole numbers m and n are called coprime if their greatest common divisor is 1; i.e., if there is no prime number that divides both of them.

Then an arithmetic function a is

  • additive if a(mn) = a(m) + a(n) for all coprime natural numbers m and n;
  • multiplicative if a(mn) = a(m)a(n) for all coprime natural numbers m and n.

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Famous quotes containing the word functions:

    “Empirical science is apt to cloud the sight, and, by the very knowledge of functions and processes, to bereave the student of the manly contemplation of the whole.”
    —Ralph Waldo Emerson (1803–1882)