elementary results about multiplicative functions and convolution
One of the most important elementary results about multiplicative functions and convolution (http://planetmath.org/DirichletConvolution) is:
Theorem.
If , , and are arithmetic functions with and at least two of them are multiplicative, then all of them are multiplicative.
The above theorem will be proven in two separate parts.
Lemma 1.
If and are multiplicative, then so is .
Proof.
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,
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∎
Lemma 2.
If is an arithmetic function and and are multiplicative functions with , then is multiplicative.
Proof.
Let with . Induction will be used on to establish that .
If , then . Note that . Thus, .
Now let be an arbitrary natural number. The induction hypothesis yields that, if divides , divides , and , then . Thus,
It follows that . ∎
The theorem follows from these two lemmas and the fact that convolution is commutative.
The theorem has an obvious corollary.
Corollary.
If is multiplicative, then so is its convolution inverse.
Proof.
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 |
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Canonical name | ElementaryResultsAboutMultiplicativeFunctionsAndConvolution |
Date of creation | 2013-03-22 16:07:37 |
Last modified on | 2013-03-22 16:07:37 |
Owner | Wkbj79 (1863) |
Last modified by | Wkbj79 (1863) |
Numerical id | 16 |
Author | Wkbj79 (1863) |
Entry type | Theorem |
Classification | msc 11A25 |
Related topic | ConvolutionInversesForArithmeticFunctions |