Abstract:

This dissertation consists of three parts . In the first part we consider the equidistribution of roots of quadratic congruences . The roots of quadratic congruences are known to be equidistributed . However ,we establish a bound for the discrepancy of this sequence using a spectral method involvingautomorphic forms , especially Kuznetsov's formula , together with an Erdős Turán inequality . Then we discuss the implications of our discrepancy estimate for the reducibility problem of arctangents of integers . In the second and third part of this dissertation we consider some aspects of Farey fractions . The set of Farey fractions of order at most [mathematical formula] is , of course , a classical object in Analytic Number Theory . Our interest here is in certain sumsets of Farey fractions . Also , in this dissertation we study Farey fractions by working in the quotient group Q /Z , which is the modern point of view . We first derive an identity which involves the structure of Farey fractions in the group ring of Q /Z . Then we use these identities to estimate the asymptotic magnitude of the size of the sumset [mathematical formula] . Our method uses results about divisors in short intervals due to K . Ford . We also prove a new form of the Erdős Turán inequality in which the usual complex exponential functions are replaced by a special family of functions which are orthogonal in L² (R /Z ) . 