Dirichlet hyperbola method
Let , , and be multiplicative functions such that , where denotes the convolution (http://planetmath.org/DirichletConvolution) of and . The Dirichlet hyperbola method (typically shortened to hyperbola method) is a way to by using the fact that :
Note that, since , not both of and can be larger than . The Dirichlet hyperbola method follows from this fact as well as the inclusion-exclusion principle.
This method for calculating is advantageous when the sums in of and are easier to handle and when is relatively small for most .
As an example, the sum will be calculated using the Dirichlet hyperbola method.
Note that . Thus:
|Title||Dirichlet hyperbola method|
|Date of creation||2013-03-22 15:58:27|
|Last modified on||2013-03-22 15:58:27|
|Last modified by||Wkbj79 (1863)|