convolution method

Let f, g, and h be multiplicative functionsMathworldPlanetmath such that f=g*h, where * denotes the convolution ( of g and h. The convolution method is a way to nxf(n) by using the fact that f=g*h:


This method for calculating nxf(n) is advantageous when the sums in of g and h are easier to handle.

As an example, the sum nxμ2(n) will be calculated using the convolution method.

Since μ2=μ2*ε=μ2*(μ*1)=(μ2*μ)*1, the functionsMathworldPlanetmath μ2*μ and 1 can be used.

To use the convolution method, a nice way to (μ2*μ)(n) needs to be found. Note that μ2*μ is multiplicative (, so it only needs to be evaluated at prime powers.

Let k. Then

(μ2*μ)(pk)=j=0kμ2(pj)μ(pk-j)=μ(pk)+μ(pk-1)={0if k2-1if k=2.

Since μ2*μ is multiplicative (, then (μ2*μ)(n)={0if n is not a squareμ(n)if n is a square.

The convolution method yields:


Title convolution method
Canonical name ConvolutionMethod
Date of creation 2013-03-22 15:58:24
Last modified on 2013-03-22 15:58:24
Owner Wkbj79 (1863)
Last modified by Wkbj79 (1863)
Numerical id 13
Author Wkbj79 (1863)
Entry type Definition
Classification msc 11N37
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