DISJOINTNESS OF THE MÖBIUS TRANSFORMATION AND MÖBIUS FUNCTION

Overview & Synopsis

We study the distribution of the sequence of elements of the discrete dynamical system generated by the Möbius transformation x → (ax + b)/(cx + d) over a finite field of p elements at the moments of time that correspond to prime numbers. 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 Möbius function. We also obtain an equidistribution result for such trajectories at prime moments of time.