Dissertation
Moments and zero density of automorphic L-functions for Maass forms and their symmetric squares
University of Iowa
Doctor of Philosophy (PhD), University of Iowa
Spring 2022
DOI: 10.25820/etd.006509
Abstract
The Riemann zeta function, defined as ζ(s) = ?n≥1 1/ns for Re(s) > 1 and endowed with analytic continuation to C and a functional equation, plays an important role in number theory. Two conjectured aspects of the Riemann zeta function are especially important: the Lindel ̈of Hypothesis, which says that ζ(1/2 + it) ≪ tε, and the famous Riemann Hypothesis, which says that all nontrivial zeroes for ζ lie on the critical line Re(s) = 1/2. In this thesis we examine these two aspects for L-functions, which can be thought of as generalizations of the Riemann zeta function.
In the present work we study L-functions associated to Maass forms, which are eigenfunctions of the non-Euclidean Laplacian and can be thought of as generalizations of waveforms which are eigenfunctions of the Euclidean Laplacian. In particular, given f and g belonging to orthonormal bases of Hecke-Maass eigenforms for the space of all Maass cusp forms and all even Maass cusp forms, respectively, we study L(Sym2 f × g) in this thesis.
Our main result consists in nontrivial bounds for double moments of L(1/2, Sym2 f × g) and associated L-functions on the central critical line. On average these bounds improve on the best known convexity or subconvexity bounds. As applications we prove nontrivial average bounds for the number of nontrivial zeroes in the critical strip but not on the central critical line. These two results constitute work towards, or evidence for, the Lindel ̈of Hypothesis and the Riemann Hypothesis, respectively.
In Appendix A a sketch of the similar holomorphic case is also provided.
Details
- Title: Subtitle
- Moments and zero density of automorphic L-functions for Maass forms and their symmetric squares
- Creators
- W. Tyler Reynolds
- Contributors
- Yangbo Ye (Advisor)Sergii Bezuglyi (Committee Member)Miodrag Iovanov (Committee Member)Ryan Kinser (Committee Member)Muthu Krishnamurthy (Committee Member)
- Resource Type
- Dissertation
- Degree Awarded
- Doctor of Philosophy (PhD), University of Iowa
- Degree in
- Mathematics
- Date degree season
- Spring 2022
- Publisher
- University of Iowa
- DOI
- 10.25820/etd.006509
- Number of pages
- vii, 57 pages
- Copyright
- Copyright 2022 W. Tyler Reynolds
- Language
- English
- Description bibliographic
- Includes bibliographical references (pages 55-57).
- Public Abstract (ETD)
The Riemann Zeta function plays an important role in number theory. Two conjectured aspects of the Riemann Zeta function are especially important: the Lindelӧf Hypothesis, which predicts bounds for this function on the central critical line, and the famous Riemann Hypothesis, which predicts that all nontrivial zeroes for this function lie on the central critical line. In this thesis we examine these two aspects for L-functions, which can be thought of as generalizations of the Riemann Zeta function.
In the present work we study L-functions associated to Maass forms, which are eigenfunctions of the non-Euclidean Laplacian and can be thought of as generalizations of waveforms which are eigenfunctions of the Euclidean Laplacian. The particular L-functions we study here are constructed from two Maass forms and their symmetric squares.
Our main result consists in nontrivial bounds for moments of these L-functions on the central critical line. As applications we prove average bounds for the possible number of nontrivial zeroes in the critical strip but not on the central critical line. These two results constitute work towards, or evidence for, the Lindelӧf Hypothesis and the Riemann Hypothesis, respectively.
- Academic Unit
- Mathematics
- Record Identifier
- 9984271453702771
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