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 .