Editing Multiplicative function (section)

== Properties ==
A multiplicative function is completely determined by its values at the powers of [[prime number]]s, a consequence of the [[fundamental theorem of arithmetic]]. Thus, if ''n'' is a product of powers of distinct primes, say ''n'' = ''p''<sup>''a''</sup> ''q''<sup>''b''</sup> ..., then 
''f''(''n'') = ''f''(''p''<sup>''a''</sup>) ''f''(''q''<sup>''b''</sup>) ...

This property of multiplicative functions significantly reduces the need for computation, as in the following examples for ''n'' = 144 = 2<sup>4</sup> · 3<sup>2</sup>:
<math display="block">d(144) = \sigma_0(144) = \sigma_0(2^4) \, \sigma_0(3^2) = (1^0 + 2^0 + 4^0 + 8^0 + 16^0)(1^0 + 3^0 + 9^0) = 5 \cdot 3 = 15</math>
<math display="block">\sigma(144) = \sigma_1(144) = \sigma_1(2^4) \, \sigma_1(3^2) = (1^1 + 2^1 + 4^1 + 8^1 + 16^1)(1^1 + 3^1 + 9^1) = 31 \cdot 13 = 403</math>
<math display="block">\sigma^*(144) = \sigma^*(2^4) \, \sigma^*(3^2) = (1^1 + 16^1)(1^1 + 9^1) = 17 \cdot 10 = 170</math>

Similarly, we have:
<math display="block">\varphi(144) = \varphi(2^4) \, \varphi(3^2) = 8 \cdot 6 = 48</math>

In general, if ''f''(''n'') is a multiplicative function and ''a'', ''b'' are any two positive integers, then
{{block indent|em=1.2|text=''f''(''a'') · ''f''(''b'') = ''f''([[greatest common divisor|gcd]](''a'',''b'')) · ''f''([[least common multiple|lcm]](''a'',''b'')).}}

Every completely multiplicative function is a [[homomorphism]] of [[monoid]]s and is completely determined by its restriction to the prime numbers.