Answer
$\phi \,\,which\,\,is\,defined\,as\,follows:\\
For\,each\,integer\,\,\,\,
n \geq 1, \\\phi(n)\,is\,the\,number\,of\,positive\,integers\,less\,than\,or\,equal\,\,
to\,n\,\,\\that\,have\,no\,common\,factors\,with\,\,n\,\,except\,\,\pm 1. \\
notice\,that\,p^n\,factors\,are\,\\p,2p,3p,4p,....,p^{n-1}(p)(they\,are\,p^{n-1}integers)\\
because\,\,there\,are\,p^n\,positive\,integers\,less\,than\,\,\\
or\,\,equal\,to\,p^n\,\,and\,there\,are\,p^{n-1}integers \,\\that\,have\,common\,factors \,with\,p^n\,\\
therefore\\
there\,are\,p^n-p^{n-1}\,positive\,integers\,less\,than\\
or\,equal\,to\,p^n\,that\,have\,no\,common\,factors\,with\,p^n\,except\,\pm 1
so\,\,\,\\
\phi(p^n)=p^n-p^{n-1}.
$
Work Step by Step
$\phi \,\,which\,\,is\,defined\,as\,follows:\\
For\,each\,integer\,\,\,\,
n \geq 1, \\\phi(n)\,is\,the\,number\,of\,positive\,integers\,less\,than\,or\,equal\,\,
to\,n\,\,\\that\,have\,no\,common\,factors\,with\,\,n\,\,except\,\,\pm 1. \\
notice\,that\,p^n\,factors\,are\\\,p,2p,3p,4p,....,p^{n-1}(p)(they\,are\,p^{n-1}integers)\\
because\,\,there\,are\,p^n\,positive\,integers\,less\,than\,\,\\
or\,\,equal\,to\,p^n\,\,and\,there\,are\,p^{n-1}integers \,\\that\,have\,common\,factors \,with\,p^n\,\\
therefore\\
there\,are\,p^n-p^{n-1}\,positive\,integers\,less\,than\\
or\,equal\,to\,p^n\,that\,have\,no\,common\,factors\,with\,p^n\,except\,\pm 1
so\,\,\,\\
\phi(p^n)=p^n-p^{n-1}.
$
