PlanetMath (more info)
 Math for the people, by the people.
Encyclopedia | Requests | Forums | Docs | Wiki | Random | RSS  
Login
create new user
name:
pass:
forget your password?
Main Menu
sections
Encyclopædia
Papers
Books
Expositions

meta
Requests (200)
Orphanage
Unclass'd
Unproven (490)
Corrections (6)
Classification

talkback
Polls
Forums
Feedback
Bug Reports

downloads
Snapshots
PM Book

information
News
Docs
Wiki
ChangeLog
TODO List
Legalese
About
Owner confidence rating: High Entry average rating: No information on entry rating
[parent] formula for the convolution inverse of a completely multiplicative function (Corollary)
Corollary   If $ f$ is a completely multiplicative function, then its convolution inverse is $ f\mu$, where $ \mu$ denotes the Möbius function.
Proof. Recall the Möbius inversion formula $ 1*\mu = \varepsilon$, where $ \varepsilon$ denotes the convolution identity function. Thus, $ f(1*\mu) = f\varepsilon$. Since pointwise multiplication of a completely multiplicative function distributes over convolution, $ (f \cdot 1)*(f\mu)=f\varepsilon$. Note that, for all natural numbers $ n$, $ f(n)1(n)=f(n) \cdot 1=f(n)$ and $ f(n)\varepsilon(n)=\varepsilon(n)$. Thus, $ f*(f \mu)=\varepsilon$. It follows that $ f\mu$ is the convolution inverse of $ f$. $ \qedsymbol$



"formula for the convolution inverse of a completely multiplicative function" is owned by Wkbj79.
(view preamble)

View style:

See Also: criterion for a multiplicative function to be completely multiplicative


This object's parent.

Attachments:
completely multiplicative functions whose convolution inverses are completely multiplicative (Corollary) by Wkbj79
Log in to rate this entry.
(view current ratings)

Cross-references: natural numbers, convolution identity function, Möbius inversion, Möbius function, convolution inverse, completely multiplicative function
There are 2 references to this entry.

This is version 2 of formula for the convolution inverse of a completely multiplicative function, born on 2007-04-14, modified 2007-04-15.
Object id is 9181, canonical name is FormulaForTheConvolutionInverseOfACompletelyMultiplicativeFunction.
Accessed 645 times total.

Classification:
AMS MSC: 11A25 (Number theory :: Elementary number theory :: Arithmetic functions; related numbers; inversion formulas)
Pending Errata and Addenda
None.
Discussion
Style: Expand: Order:
forum policy

No messages.

Interact
post | correct | update request | add example | add (any)