elementary results about multiplicative functions and convolution
One of the most important elementary results about multiplicative functions and convolution (http://planetmath.org/DirichletConvolution) is:
The above theorem will be proven in two separate parts.
If and are multiplicative, then so is .
Note that since and are multiplicative.
Let with . Then any divisor of can be uniquely factored as , where divides and divides . When a divisor of is factored in this manner, the fact that implies that and . Thus,
If is an arithmetic function and and are multiplicative functions with , then is multiplicative.
Let with . Induction will be used on to establish that .
If , then . Note that . Thus, .
It follows that . ∎
The theorem has an obvious corollary.
If is multiplicative, then so is its convolution inverse.
Let be multiplicative. Since , has a convolution inverse . (See convolution inverses for arithmetic functions for more details.) Since , where denotes the convolution identity function, and both and are multiplicative, the theorem yields that is multiplicative. ∎
|Title||elementary results about multiplicative functions and convolution|
|Date of creation||2013-03-22 16:07:37|
|Last modified on||2013-03-22 16:07:37|
|Last modified by||Wkbj79 (1863)|