Dissertation
NON-VANISHING OF THE DERIVATIVE OF L-FUNCTIONS AT THE CENTRAL POINT
Doctor of Philosophy (PhD), Washington State University
01/2020
Handle:
https://hdl.handle.net/2376/111226
Abstract
L-functions are complex analytic functions attached to various objects of interest in number theory. The distribution of the zeros of an L-function is often of particular interest, due to their relationship with the objects associated to L-functions as well as connections to deep questions about the theory of L-functions in general. Here we consider derivatives of L-functions associated to classical modular forms of large weight.
In particular, we derive a lower bound for the proportion of such derivatives which are non-vanishing at the point s=1/2, which is the central point of a functional equation satisfied by the L-functions. This is accomplished by computing the first and second moments of the L-function derivatives. We then attach objects known as mollifiers to these moments, and attempt to choose coefficients within the mollifiers to optimize the ratio between the square of the first mollified moment and the second mollified moment. By Cauchy's inequality, this ratio gives a lower bound for the proportion of L-function derivatives which are non-vanishing. By our choice of mollifier, we can deduce that the proportion of L-function derivatives associated to modular forms of large weight which do not vanish at s=1/2 is at least .398.
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Details
- Title
- NON-VANISHING OF THE DERIVATIVE OF L-FUNCTIONS AT THE CENTRAL POINT
- Creators
- Matthew Dolan Jobrack
- Contributors
- Sheng-Chi Liu (Advisor)Charles Moore (Committee Member)Michael Tsatsomeros (Committee Member)
- Awarding Institution
- Washington State University
- Academic Unit
- Mathematics and Statistics, Department of
- Theses and Dissertations
- Doctor of Philosophy (PhD), Washington State University
- Number of pages
- 69
- Identifiers
- 99900581612801842
- Language
- English
- Resource Type
- Dissertation