divisor function

In the parent article there has been proved the formulaMathworldPlanetmathPlanetmath


giving the sum of all positive divisorsMathworldPlanetmathPlanetmath of an integer n; there the pi’s are the distinct positive prime factorsMathworldPlanetmath of n and mi’s their multiplicitiesMathworldPlanetmath.

It follows that the sum of the z’th powers of those divisors is given by

σz(n)=0<dndz=i=1kpi(mi+1)z-1piz-1. (1)

This complex function of z is called divisor functionMathworldPlanetmath (http://planetmath.org/DivisorFunction).  The equation (1) may be written in the form

σz(n)=i=1k(1+piz+pi2z++pimiz) (2)

usable also for  z=0.  For the special case of one prime power the function consists of the single geometric sum (http://planetmath.org/GeometricSeries)

σz(pm)= 1+pz+p2z++pmz,

which particularly gives m+1 when pz=1, i.e. when z is a multiple of 2iπ/lnp.

A special case of the function (1) is the τ function (http://planetmath.org/TauFunction) of n:


Some inequalities

σm(n)nm2σ0(n)form=0, 1, 2,
Title divisor function
Canonical name DivisorFunction
Date of creation 2013-11-27 18:13:38
Last modified on 2013-11-27 18:13:38
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 8
Author pahio (2872)
Entry type Theorem
Classification msc 11A05
Classification msc 11A25