Distribution of some arithmetic sequences

In this thesis, we first study the correlations of some arithmetic sequences. We prove the existence of the limiting pair correlations of fractions with prime and power denominators, and give the explicit pair correlation density functions. Next, we study the higher level correlations of these fractions, and construct an arithmetic sequence with showing the independence of its different level correlations. We also study the distribution of angles between common tangents of Ford circles, which is a special case of Apollonian circle packing. We provide the limiting distribution functions of these angles in different situations