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The lower bound calculation is easy .

But for the upper bound , observe that each $$z \in K$$ lies in some cycle of length m(z) and we these cycles by $$C_1 , C_2 …..,C_q$$ . Further , we denote the length of the cycle by $$m_j$$ , so , if $$z \in C_j$$ then , $$m(z)= m_j$$ $$\sum_{j=1}^{q} \sum_{z \in C_j} [\mu(N,z) – \mu(m_j ,z)]$$ we can confine our attention xparatly .

Now , $$\mu(N,z) = \mu(m_j -, z)$$ whenever $$z \in C_j \rightarrow$$ rationally indifferent .

So , nonzero contribution comes from rationally different cycles , $$C_j$$ .

#### Theorem:

1. If m|n , then $$R^n$$ has no fixed at $$\zeta_j$$ .
1. If m|n but $$m_q \not | n$$ , then $$R^n$$ has our fixed point at $$\zeta$$ .
2. If $$m_q |n$$ then $$R^n$$ has fixed point .