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Title:  Extreme values of Lfunctions 
Author(s):  Li, Junxian 
Director of Research:  Zaharescu, Alexandru 
Doctoral Committee Chair(s):  Berndt, Bruce 
Doctoral Committee Member(s):  Ford, Kevin; Hildebrand, A. J. 
Department / Program:  Mathematics 
Discipline:  Mathematics 
Degree Granting Institution:  University of Illinois at UrbanaChampaign 
Degree:  Ph.D. 
Genre:  Dissertation 
Subject(s):  Extreme values, Lfunctions 
Abstract:  The value distribution of the Riemann zeta function $\zeta(s)$ is a classical question. Despite the fact that values of $\zeta(s)$ are approximately Gaussian distributed, $\zeta(s)$ can be very large for infinitely many $s$ as $\Im s\rightarrow\infty$. Exponential sums and random matrix theory have been extensively employed to study the behaviour of extreme values of $\zeta(s)$. This thesis is focused on extreme values of $L$functions using the resonance method together with recent developments on greatest common divisor sums. This thesis consists of five chapters. The first chapter gives some history and recent progress on the extreme values of $L$functions in the critical strip. In Chapter \ref{chap2}, we consider large values of the Dedekind zeta function $\zeta_K(s)$ in the critical strip, where $K$ is an arbitrary number field . We present two different approaches to the problem: one is to use PhragmenLindel\"of principle, and the other is to use the convolution method. This is based on joint work with S. Baluyot and A. Zaharescu. In Chapter \ref{chap3}, we focus on large values of degree $1$ $L$functions in the Selberg class in relation to other $L$functions. It is believed that values of distinct primitive $L$functions behave like independent random variables. For example, if we let $\rho$ denote a zero of a Dirichlet $L$function, we can ask what the behaviour of $\zeta(\rho)$ is. The method is based on the study of simple zeros of $\zeta(s)$, and the results apply to general $L$functions in the Selberg class satisfying appropriate conditions. In Chapter \ref{chap4}, we discuss small values of the derivative of the Dedekind zeta function. It is believed that $\zeta_K'(\rho)$ cannot be zero from the Grand Simplicity Conjecture. We show that $\zeta_K'(\rho)$ can be very small for infinitely many $\rho$. This is a generalization of a result of N. Ng to number fields beyond $\QQ$. The last chapter lists all the references. 
Issue Date:  20180601 
Type:  Text 
URI:  http://hdl.handle.net/2142/101751 
Rights Information:  Copyright 2018 Junxian Li. 
Date Available in IDEALS:  20180927 20200928 
Date Deposited:  201808 
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Dissertations and Theses  Mathematics

Graduate Dissertations and Theses at Illinois
Graduate Theses and Dissertations at Illinois