PhD Oral Exam - Martin Cech, Mathematics

When studying for a doctoral degree (PhD), candidates submit a thesis that provides a critical review of the current state of knowledge of the thesis subject as well as the student’s own contributions to the subject. The distinguishing criterion of doctoral graduate research is a significant and original contribution to knowledge.

Once accepted, the candidate presents the thesis orally. This oral exam is open to the public.

Abstract

We consider several classical problems in analytic number theory from the point of view of multiple Dirichlet series.

In Chapter 1, we review the required background and give an overview of the thesis.

In Chapter 2, we introduce multiple Dirichlet series and the relevant ideas used in later chapters.

In Chapter 3, we study the double character sum S(X,Y) = ∑_(m≤X,m odd) ∑_(n≤Y,n odd) (m/n)

The asymptotic formula for this sum was obtained by Conrey, Farmer and Soundararajan. We recover this formula using our approach with an improved error term.

In Chapter 4, we use multiple Dirichlet series to give a conditional proof of the ratios conjecture for the family of real Dirichlet L-functions in some region of the shifts. As an application of our result, we compute the one-level density in our family for test functions whose Fourier transform is supported in $(-2,2)$, including all lower-order terms.

In Chapter 5, we elaborate on an ongoing work of the author with Siegfred Baluyot. We compare two methods to come up with conjectural asymptotic formulas for moments in the family of real Dirichlet L-functions -- the recipe developed by Conrey, Farmer, Keating, Rubinstein and Snaith, and the multiple Dirichlet series approach introduced by Diaconu, Goldfeld and Hoffstein. We consider shifted moments and show that the two methods are essentially equivalent, in that they give rise to the same terms which come from the functional equation of the individual L-functions.