Thesis Defense

Discrete Moments and Linear Combinations of L-functions

Scott Kirila

Tuesday, July 31st, 2018
1:00 PM - 2:00 PM
Hylan 1106A

This thesis is divided into two main parts. First, assuming the Riemann hypothesis (RH), we establish an upper bound for the \(2k\)-th discrete moment of the derivative of the Riemann zeta-function at nontrivial zeros, where \(k\) is a positive real number. Our upper bound agrees with conjectures of Gonek and Hejhal and of Hughes, Keating, and O’Connell, and it sharpens a result of Milinovich.

In the second part of the thesis, we investigate how often two different linear combinations of Dirichlet \(L\)-functions can have common zeros in certain regions. With certain mild conditions on the linear combinations, we show that a positive proportion of zeros of one linear combination are not zeros of the other in the following regions: the half-plane \(\sigma > 1\), and closed vertical strips contained in the right half of the critical strip. We also show, assuming RH and other hypotheses, that a positive proportion of zeros of the Riemann zeta-function on the critical line are not zeros of a linear combination of two Dirichlet \(L\)-functions.

Event contact: hazel dot mcknight at rochester dot edu