Let us assume we are looking at R3, and a sphere of radius 1. $x^2+y^2+z^2=1$
Now we transform the $xyz$-plane to spherical coordinates. $z,\theta,\phi$. How would the sphere look in the new coordinate system?

$$\tau(n) =\sum_{d \mid n} d^0 = \prod_{k=1}^r (k_i+1)$$
$$\sigma(n)= \sum_{d \mid n} d^1 = \prod_{k=1}^r \frac{p_i^{k_i+1}-1}{p_i-1}$$

$$\tau(n) =\sum_{d \mid n} d^0 = \prod_{k=1}^r \frac{{(p_i)^0}^{k_i+1}-1}{{(p_i)^0}-1}$$
$$\sigma(n)= \sum_{d \mid n} d^1 = \prod_{k=1}^r \frac{p_i^{k_i+1}-1}{p_i-1}$$

$$\eqalign{
& \tau \left( n \right) = {\sum\limits_{d|n} d ^0} = \prod\limits_{i = 1}^r {\frac{{{{\left( {{p_i}^0} \right)}^{{k_i} + 1}} - 1}}{{\left( {{p_i}^0} \right) - 1}}} \cr
& \sigma \left( n \right) = \sum\limits_{d|n} {{d^1}} = \prod\limits_{i = 1}^r {\frac{{{{\left( {{p_i}^1} \right)}^{{k_i} + 1}} - 1}}{{\left( {{p_i}^1} \right) - 1}}} \cr} $$