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 Title: Extreme values of L-functions 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 Urbana-Champaign Degree: Ph.D. Genre: Dissertation Subject(s): Extreme values, L-functions 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 Phragmen-Lindel\"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: 2018-06-01 Type: Text URI: http://hdl.handle.net/2142/101751 Rights Information: Copyright 2018 Junxian Li. Date Available in IDEALS: 2018-09-27 Date Deposited: 2018-08
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