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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