divisor function

In the parent article there has been proved the formula

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

(1)

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

${\sigma }_z(n)={\displaystyle \prod _{i=1}^k}(1+p_i^z+p_i^{2z}+\mathrm{\dots }+p_i^{m_{i}z})$(2)

usable also for  $z=0$.  For the special case of one primepower the function consists of the singlegeometric sum (http://planetmath.org/GeometricSeries)

$${\sigma }_z(p^m)=\mathrm{\hspace{0.33em}1}+p^z+p^{2z}+\mathrm{\dots }+p^{mz},$$

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 $n$:

Some inequalities

$${\sigma }_m(n)\ge n^{\frac{m}{2}}{\sigma }_0(n)\mathit{\hspace{1em}}\text{for}\mathit{\hspace{1em}}m=0,\mathrm{\hspace{0.17em}1},\mathrm{\hspace{0.17em}2},\mathrm{\dots }$$

$${\sigma }_1(mn)>{\sigma }_1(m)+{\sigma }_1(n)\mathit{\hspace{1em}}\forall m,n\in \mathbb{Z}$$

$${\sigma }_1(n)\le \frac{n+1}{2}{\sigma }_0(n)$$