completely multiplicative functions whose convolution inverses are completely multiplicative
The only completely multiplicative function whose convolution inverse is also completely multiplicative is , the convolution identity function.
Let be a completely multiplicative function whose convolution inverse is completely multiplicative. By this entry (http://planetmath.org/FormulaForTheConvolutionInverseOfACompletelyMultiplicativeFunction), is the convolution inverse of , where denotes the Möbius function. Thus, is completely multiplicative.
Let be any prime. Then
Thus, for every prime . Since is completely multiplicative,
Hence, . ∎
|Title||completely multiplicative functions whose convolution inverses are completely multiplicative|
|Date of creation||2013-03-22 16:55:12|
|Last modified on||2013-03-22 16:55:12|
|Last modified by||Wkbj79 (1863)|