Multiplicative function

In <a href="/facts/Number_theory/zhoa7zrc">number theory</a>, a multiplicative function is an <a href="/facts/Arithmetic_function/hULxwsgm">arithmetic function</a> 
 
 
 
 f
 
 
 {\displaystyle f}
 
 of a positive <a href="/facts/Integer/PUwwrawx">integer</a> 
 
 
 
 n
 
 
 {\displaystyle n}
 
 with the property that 
 
 
 
 f
 (
 1
 )
 =
 1
 
 
 {\displaystyle f(1)=1}
 
 and

f
 (
 a
 b
 )
 =
 f
 (
 a
 )
 f
 (
 b
 )
 
 
 {\displaystyle f(ab)=f(a)f(b)}
 
 whenever 
 
 
 
 a
 
 
 {\displaystyle a}
 
 and 
 
 
 
 b
 
 
 {\displaystyle b}
 
 are <a href="/facts/Coprime/bnCxCXxs">coprime</a>.
An arithmetic function is said to be <a href="/facts/Completely_multiplicative_function/4Sr9ADXQ">completely multiplicative</a> (or totally multiplicative) if 
 
 
 
 f
 (
 1
 )
 =
 1
 
 
 {\displaystyle f(1)=1}
 
 and 
 
 
 
 f
 (
 a
 b
 )
 =
 f
 (
 a
 )
 f
 (
 b
 )
 
 
 {\displaystyle f(ab)=f(a)f(b)}
 
 holds for all positive integers 
 
 
 
 a
 
 
 {\displaystyle a}
 
 and 
 
 
 
 b
 
 
 {\displaystyle b}
 
, even when they are not coprime.

Multiplicative function open-in-new

Multiplicative function