M?bius Disjointness For Skew Products

One of the central problems in analytic number theory is to investigate the distribution of prime numbers,which is equivalent to the estimate of the mean value of the Mobius function ?(n).This inspires us to investigate the behaviour of the Mobius function farther.A lot of results imply that ?(n)is linearly disjointed with a "determined"sequence.In 2010s,Sarnak summarizes this phenomenon into the Mobius disjointness conjecture,which claims that(?)1/N(?)?(n)?(n)=0 for every sampling sequence ?(n)in a zero-entropy topological dynamical system.In 2015,Liu and Sarnak study the Mobius disjointness for the skew product on the 2-torus T2,which is a building block of distal flows,one of important examples of zero-entropy systems.In this thesis,we proceed with this progression to prove the Mobius disjointness for more skew products.We first consider the generalization of skew products on the torus.We consider the skew product on the product space of the unit circle and the simplest non-commutative nilmanifold,the Heisenberg nilmanifold.Specifically,let G be the group consisting of all 3x3 real upper triangular matrices with all diagonal entries being 1.Let ? be the subgroup of G consisting of all the integral matices in G.Then the quotient space ?\G is called the Heisenberg nilmanifold.We prove that if ?,? is smooth periodic functions of period 1 and the constant Fourier coefficient of ? vanishes,then the skew product system(T x ?\G,T)given by#12 satisfies the Mobius disjointness conjecture.The number theoretical results we use in the proof include the estimate of exponential sum jointed with the Mobius function and the work on the averaged form of Chowla's conjecture by Matomaki-Radziwill-Tao.For the latter one,we cite indirectly via the measure complexity theory introduced by Huang-Wang-Ye.We next consider the Mobius disjointness conjecture in short intervals.We prove that under some assumptions,the Anzai skew product on the 2-torus T2 is linearly disjointed with ?(n)in short intervals of the size N? with 5/8<?<1.In particu-lar,our result covers Furstenberg's irregular flow,hence gives an example of irregular system satisfying the Mobius disjointness conjecture in short intervals.In the proof of this result,we mainly use the Fourier analysis and the estimate of exponential sums over primes by Zhan.