Dirichlet hyperbola method

Let f, g, and h be multiplicative functions such that f=g*h, where * denotes the convolution (http://planetmath.org/DirichletConvolution) of g and h. The Dirichlet hyperbola method (typically shortened to hyperbola method) is a way to nxf(n) by using the fact that f=g*h:


Note that, since ab=nx, not both of a and b can be larger than x. The Dirichlet hyperbola method follows from this fact as well as the inclusion-exclusion principleMathworldPlanetmath.

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

As an example, the sum nxτ(n) will be calculated using the Dirichlet hyperbola method.

Note that τ=1*1. Thus:


Title Dirichlet hyperbola method
Canonical name DirichletHyperbolaMethod
Date of creation 2013-03-22 15:58:27
Last modified on 2013-03-22 15:58:27
Owner Wkbj79 (1863)
Last modified by Wkbj79 (1863)
Numerical id 14
Author Wkbj79 (1863)
Entry type Definition
Classification msc 11N37
Synonym hyperbola method
Related topic ConvolutionMethod
Related topic TauFunction
Related topic EulersConstant