On some distribution problems in Analytic Number Theory

Show full item record

Title: On some distribution problems in Analytic Number Theory
Author: Homma, Kosuke
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 ) .
URI: http : / /hdl .handle .net /2152 /ETD -UT -2009 -12 -458
Date: 2010-08-26

Citation

On some distribution problems in Analytic Number Theory. Doctoral dissertation, The University of Texas at Austin. Available electronically from http : / /hdl .handle .net /2152 /ETD -UT -2009 -12 -458 .

Files in this item

Files Size Format View

There are no files associated with this item.

This item appears in the following Collection(s)

Show full item record

Search DSpace

Advanced Search

Browse