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A Study of L-Functions: at the Edge of the Critical Strip and Within

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A Study of L-Functions: at the Edge of the Critical Strip and Within A STUDY OF L-FUNCTIONS: AT THE EDGE OF THE CRITICAL STRIP AND WITHIN. ALLYSA LUMLEY A DISSERTATION SUBMITTED TO THE FACULTY OF GRADUATE STUDIES IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY GRADUATE PROGRAM IN MATHEMATICS AND STATISTICS YORK UNIVERSITY TORONTO, ONTARIO AUGUST 2019 c Allysa Lumley, 2019 Abstract In analytic number theory, and increasingly in other surprising places, L-functions arise naturally when describing algebraic and geometric phenomena. For example, when attempting to prove the Prime Number Theorem the values of L-functions on the 1-line played a crucial role. In this thesis we discuss the theory of L-functions in two different settings. In the classical context we provide results which give estimates for the size of a general L-function on the right edge of the critical strip Re(s) = 1 and provide a bound for the number of zeros for the classical Riemann zeta function inside the critical strip commonly referred to as a zero density estimate. In the second setting we study L-functions over the polynomial ring A = Fq[T ], where Fq is a finite field. As A and Z have similar structure, A is a natural candidate for analyzing classical number theoretic questions. Additionally, the truth of the Riemann Hypothesis (RH) in A yields deeper unconditional results currently unattainable over Z. We will focus on the distribution of values of specific L-functions in two different places: 1 On the right edge of the critical strip, that is Re(s) = 1 and values where 2 < Re(s) < 1. ii Acknowledgements In what follows I will convey my sincere thanks to the many people who helped me on my doctorate journey. For those I do not mention, it is not that I have forgotten your efforts and time, I just ran out of steam. Before thanking the people in my life, the research in this thesis was supported by the Ontario Graduate Scholarship for two years of my degree. These funds made it much easier to survive in Toronto. Firstly, I would like to thank sincerely my supervisor Dr. Youness Lamzouri for his guidance and invaluable support. Throughout my degree he has provided me with new opportunities to share my work and supported my efforts to build up the various communities I am a part of. To my co-authors Dr. Nathan Ng and Dr. Habiba Kadiri, I would not be here without you. May there be many more papers and summer visits to come. I am extremely grateful to all the members of my examining committee: My external, Dr. Andrew Granville, for his very thorough reading of my thesis and his enthusiasm regarding the results therein. The chair of the defence, Dr. Neal Madras, who gave very interesting comments in the thesis and was a very kind and supportive person throughout my time at York. The internal external, Dr. Michael Haslam, for the very good suggestion to consider differential equations as a source of inspiration. Finally, Dr. Patrick Ingram, who was on my committee from the beginning, and often explained random mathematics to me. He also supported my efforts in establishing an active student chapter of the Association of Women in Mathematics (AWM), encouraging me and my team (mentioned below) to run programs for young women in math at the Fields Institute. Various members of the faculty and staff in the Department of Mathematics and Statistics at York University deserve recognition. The current chair, Dr. Paul Szeptycki, one of my first professors at York, has been a iii major source of strength throughout my degree. I especially appreciate his embrace of all the crazy ideas I tried to implement over the years, ranging from the establishment of a graduate seminar to running extra tutorials for undergrads to running AWM related events. The current Graduate Program Director (GPD), Dr. Alexey Kuznetsov, who moonlighted as a supervisory committee member, therapist and supporter all while giving a master class in sarcasm, I guess. To Dr. Ada Chan and Dr. Amenda Chow, I have worked with them on many different projects. We first worked together via the pilot year of the Math Kangaroo training sessions which was an excellent opportunity for expanding my pedagogical skills. These women have been strong advocates for improving math education at various levels and their advice and counsel served to expand my many efforts at community building within the department and beyond. Dr. Paul Skoufranis with whom I had many conversations where the topic ranged from politics to pedagogy. Dr. Mike Zabrocki, the former GPD for his candor and very important advice to live in downtown Toronto rather than on campus. Dr. Nantel Bergeron who gave me have a new appreciation for algebra and a new perspective on my own contributions to mathematics. Of course I must thank Steven Chen, the resident computer expert, for his efforts to support all of my endeavours. The projector in Ross N638 is truly a beast to be reckoned with. Finally, the administrative staff: Primrose Miranda (who knew everything), Ann-Marie Carless (who knew the things Primrose did not), Anne Marie Ridley (who helped me with random things and loved to dish on clothes), Susan Rainey (who was kind enough to listen to my worries and talk with me when time allowed) and Karishma Karim (who somehow remembers everyone and how to help them). For my friends in the department, there are more of you than I can name and all of you managed to really make me laugh, cry and think. I will be mentioning the extreme kindness and support of a few of you, in no particular order, let's begin. A huge thank you to Yohana Solomon, the president of our little AWM chapter. It's really hard for me to describe your impact on my time at York. We did so many things together some of them crazy (running workshops at Fields) and some of them mundane (ordering way too much Starbucks and fighting over who would pay). I will always appreciate your cheerleading, kindness and the stress relieving nail appointments. Heaps of gratitude to Neda Aminnejad, one of the kindest and strongest people I have ever met and a true inspiration for me. Thank you for your patience, your kind words and for touring the campus with me, I will miss our walks. May you keep smiling and inspiring others to be a better version of themselves. I can't thank Arash Islami enough for his constant friendship and honest conversations. You were my first iv friend at York and I hope we can always talk to one another about math and life. A big thank you to Robert Jordan for expanding my pedagogy, including me in many cool projects and for generally supporting me. I appreciate our coffee chats and working together, it was truly a great experience. I am grateful for the very calm presence of Sergio Garcia and Najla Muhee. My first study group, you got me through a series of dark spots when we were studying for comprehensives. Sergio I can't thank you enough for your kindness and your sense of humour, sometimes all we need is a well-timed joke to make things better. Najla, you have the patience of a saint and I am so lucky to have seen you grow as a mathematician. I am incredibly thankful for the friendship and support of Nathan Gold, without him \Left to the reader" would only be half as great as it is. I deeply appreciate talking to you about mathematics and your ability to keep me updated on all the lit new words, although I think \lit" might not be in favour anymore. Looking forward to writing a paper with you some day. My sincere thanks to Marco Tosato, who was always there to listen and learn. Your optimism and enthusiasm for life will always inspire me to try new things. Thank you also for sharing your music with me. May we stay friends for life. I appreciate the great kindness of Vishal Siewnarine, someone who always knew when to say a kind word and when to play a prank. You are a true friend and I wish you all the best. Many heartfelt thanks to Masoud Ataei who supported the graduate seminar sincerely. I appreciate your interest in my research and in the larger world of number theory, may we have many more discussions about math in the future. Finally I want to mention the following people for their general kindness and support. Ben Fraser (the best baker and fellow Raptors fan), Georgios Katsimpas (many discussions about basketball and other random topics), Daniel Calder´on(a genuinely kind person who happens to be amazing at delivering math talks), Richard Le (the most positive person on Earth), Elena Aruffo (whose passion will never be forgotten), Kaveh Arabpour (who is hilarious and made me laugh way too much), Kelvin Chan (a great addition to the department and successor of the grad seminar), Sam Dupuis (for modelling exactly how not to sleep and his genuine curiosity in all things number theory), and the CUPE 3903 strike team (for showing strength in the face of adversity). v For my family, especially my mom and dad, there are no words that can express what you have done for me. For my sister Brenna, without you I would not know as many incredible lame puns. For my brother Ryan, you have grown so much since I moved away and I am proud of how strong and kind you are. For my nephew Aidan who makes me smile every day with his sweetness and heartfelt messages.
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