convolution method
Let , , and be multiplicative functions such that , where denotes the convolution (http://planetmath.org/MultiplicativeFunction) of and . The convolution method is a way to by using the fact that :
This method for calculating is advantageous when the sums in of and are easier to handle.
As an example, the sum will be calculated using the convolution method.
Since , the functions and can be used.
To use the convolution method, a nice way to needs to be found. Note that is multiplicative (http://planetmath.org/MultiplicativeFunction), so it only needs to be evaluated at prime powers.
Let . Then
Since is multiplicative (http://planetmath.org/MultiplicativeFunction), then
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 |
Related topic | DirichletHyperbolaMethod |
Related topic | MoebiusFunction |
Related topic | DisplaystyleYOmeganOleftFracxlogXy12YRightFor1LeY2 |
Related topic | DisplaystyleXlog2xOleftsum_nLeX2OmeganRight |