Disjointness of the M?bius Transformation and M?bius Function

摘要

We study the distribution of the sequence of elements of the discrete dynamical system generated by the M?bius transformation $$x mapsto (ax + b)/(cx + d)$$ x ? ( a x + b ) / ( c x + d ) over a finite field of p elements. Motivated by a recent conjecture of P.?Sarnak, we obtain nontrivial estimates of exponential sums with such sequences that imply that trajectories of this dynamical system are disjoined with the M?bius function.