#### Abstract

The thesis is in two parts. The first part is the paper “The Distribution of k-free integers” that my advisor, Dr. Roger Baker, and I submitted in February 2009. The reader will note that I have inserted additional commentary and explanations which appear in smaller text. Dr. Baker and I improved the asymptotic formula for the number of k-free integers less than x by taking advantage of exponential sum techniques developed since the 1980's. Both of us made substantial contributions to the paper. I discovered the exponent in the error term for the cases k=3,4, and worked the case k=3 completely. Dr. Baker corrected my work for k=4 and proved the result for k=5. He then generalized our work into the paper as it now stands. We also discussed and both contributed to parts of section 3 on bounds for exponential sums. The second part represents my own work guided by my advisor. I study the zeros of derivatives of Dirichlet L-functions. The first theorem gives an analog for a result of Speiser on the zeros of ζ'(s). He proved that RH is equivalent to the hypothesis that ζ'(s) has no zeros with real part strictly between 0 and ½. The last two theorems discuss zero-free regions to the left and right for L^{(k)}(s,χ).

#### Degree

MS

#### College and Department

Physical and Mathematical Sciences; Mathematics

#### Rights

http://lib.byu.edu/about/copyright/

#### BYU ScholarsArchive Citation

Powell, Kevin James, "Topics in Analytic Number Theory" (2009). *All Theses and Dissertations*. 2084.

http://scholarsarchive.byu.edu/etd/2084

#### Date Submitted

2009-03-31

#### Document Type

Thesis

#### Handle

http://hdl.lib.byu.edu/1877/etd2871

#### Keywords

Derivative, Dirichlet L-function, k-free integer, exponential sum, Heath-Brown Decomposition, Zeros, Zero-free region, left, right, Generalized Riemann Hypothesis, GRH, r-free integers, Equivalence, Type I sum, Type II sum, Asymptotic formula