(2)-CONSTRUCTING SOME FUNCTION ZEROS (ii)
(2.5) POWERS OF A PRIME * ODD SQUAREFREE:
$latex \displaystyle f(N)=f(p_{1}^{k}*M)=f(p_{1}^{k}*p_2*...*p_n)=0\;\rightarrow (k+1)|p_{1}^{k-1}* \phi\left(\frac{N}{p_{1}^{k-1}}\right) $, then the zeros can fall into two cases:
(2.5.1) Case: $latex \displaystyle (k+1)| \phi\left(\frac{N}{p_{1}^{k-1}}\right) $
Then if $latex \displaystyle d_i| \phi\left(\frac{N}{p_{1}^{k-1}}\right) $, $latex \displaystyle f(p_{1}^{d_{i}-1}*M)=0$, and we can built $latex \displaystyle \sigma_{0} \left(\frac{N}{p_{1}^{k-1}}\right) $, zeros,
one for every divisor of the product applied to every distinct prime factor
$latex \displaystyle \prod_{i=1}^{i=n}{ (p_{i}-1)}$ of $latex \displaystyle N$
Note that, in the particular case:
$latex \displaystyle (k+1)=(p_{i}-1) \rightarrow{k=p_{i}-2}\rightarrow{f(p_{1}^{p_{i}-2}*M)=0}$.
And $latex \displaystyle f(p_{1}^{p_{1}-1}*M)=Mod(p_{1}^{p_{1}-2}*\phi\left(\frac{N}{p_{1}^{k-1}}\right),p_{1})=0$.
(2.5.2) Case: $latex \displaystyle (k+1)|p_{1}^{k-1}$
$latex \displaystyle f(p_{1}^{p_{1}^{n}-1})=0$
