# divisor function

In the parent article there has been proved the formula

 $\sigma_{1}(n)\;=\;\sum_{0

giving the sum of all positive divisors of an integer $n$; there the $p_{i}$’s are the distinct positive prime factors of $n$ and $m_{i}$’s their multiplicities.

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

 $\displaystyle\sigma_{z}(n)\;=\;\sum_{0 (1)

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

 $\displaystyle\sigma_{z}(n)\;=\;\prod_{i=1}^{k}(1+p_{i}^{z}+p_{i}^{2z}+\ldots+p% _{i}^{m_{i}z})$ (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)

 $\sigma_{z}(p^{m})\;=\;1+p^{z}+p^{2z}+\ldots+p^{mz},$

which particularly gives $m\!+\!1$ when $p^{z}=1$, i.e. when $z$ is a multiple of $2i\pi/\ln{p}$.

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

 $\sigma_{0}(n)\;=\;\sum_{0

Some inequalities

 $\sigma_{m}(n)\,\geq\;n^{\frac{m}{2}}\sigma_{0}(n)\quad\mbox{for}\quad m=0,\,1,% \,2,\,\ldots$
 $\sigma_{1}(mn)>\sigma_{1}(m)+\sigma_{1}(n)\quad\forall\,m,n\in\mathbb{Z}$
 $\sigma_{1}(n)\,\leq\;\frac{n\!+\!1}{2}\sigma_{0}(n)$
 $n\!+\!\sqrt{n}\,<\sigma_{1}(n)<\frac{6}{\pi^{2}}\!\cdot\!n\sqrt{n}$
Title divisor function DivisorFunction 2013-11-27 18:13:38 2013-11-27 18:13:38 pahio (2872) pahio (2872) 8 pahio (2872) Theorem msc 11A05 msc 11A25