UC Riverside

In this thesis, the reader is provided with a self-contained study of multiplicative characters

modulo composite integers. The results found here are primarily upper bounds on Dirichlet

characters, including those with polynomial arguments. Moreover, the applications of these

character sum bounds will also be discussed; in particular the direct impact on the lower

bound on the size of a subgroup of a finite field containing the image of an interval of

consecutive integers under a polynomial function is discussed in Chapter 4. My main goals

are to extend existing bounds on Dirichlet character sums to characters with composite

moduli and create new generalizations of character sums bounds while also providing the

reader with an understanding of the various applications.