completely multiplicative functions whose convolution inverses are completely multiplicative
Corollary 1.
The only completely multiplicative function whose convolution inverse is also completely multiplicative is ε, the convolution identity function.
Proof.
Let f be a completely multiplicative function whose convolution inverse is completely multiplicative. By this entry (http://planetmath.org/FormulaForTheConvolutionInverseOfACompletelyMultiplicativeFunction), fμ is the convolution inverse of f, where μ denotes the Möbius function. Thus, fμ is completely multiplicative.
Let p be any prime. Then
(f(p))2=(f(p))2(-1)2=(f(p))2(μ(p))2=(f(p)μ(p))2=f(p2)μ(p2)=f(p2)⋅0=0.
Thus, f(p)=0 for every prime p. Since f is completely multiplicative,
f(n)={1if n=10if n≠1. |
Hence, f=ε. ∎
Title | completely multiplicative functions whose convolution inverses are completely multiplicative |
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Canonical name | CompletelyMultiplicativeFunctionsWhoseConvolutionInversesAreCompletelyMultiplicative |
Date of creation | 2013-03-22 16:55:12 |
Last modified on | 2013-03-22 16:55:12 |
Owner | Wkbj79 (1863) |
Last modified by | Wkbj79 (1863) |
Numerical id | 4 |
Author | Wkbj79 (1863) |
Entry type | Corollary |
Classification | msc 11A25 |