Moments and zeros of L-functions
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Date
2021
Authors
Shen, Quanli
University of Lethbridge. Faculty of Arts and Science
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Publisher
Lethbridge, Alta. : University of Lethbridge, Dept. of Mathematics and Computer Science
Abstract
We study moments and zeros of L-functions in this thesis. In Chapter 2, by following closely Soundararajan-Young's method, we prove an asymptotic for the fourth moment of quadratic Dirichlet L-functions under the generalized Riemann hypothesis. Unconditionally, we are able to give a sharp lower bound that agrees with Keating-Snaith's conjecture. In Chapter 3, we use a recursive method that was pioneered by Heath-Brown and developed by Young to give an asymptotic with an error O(X1/2+E) for the smoothed first moment of quadratic twists of modular L-functions. The result is analogous to Sono's work on the second moment of quadratic Dirichlet L-functions. It improves previous results of Iwaniec and Soundararajan-Radziwill. In Chapter 4, we obtain an explicit result for the number of zeros, in a box, of Dedekind zeta functions, which improves a result of Trudgian. Our argument is based on previous works of Bennett-Martin-O'Bryant-Rechnitzer, Kadiri-Ng and Trudgian.
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Keywords
moments of L-functions , zeros of L-functions , Number theory , L-functions , Dissertations, Academic