On the distribution of central values of Hecke L-functions

Questions regarding the behavior of the Riemann zeta function on the critical line 1 2 + it can be naturally interpreted as questions regarding the family of L-functions over Q associated to the archimedian characters ψ(k) = k−it at the center point 1 2 . There are many families of characters besides those strictly of archimedean-type, especially as one expands their scope to proper finite extensions of Q. Consideration of these Hecke characters leads immediately to analogous questions concerning their associated L-functions. Using tools from p-adic analysis which are analogues of traditional archimedean techniques, we prove the q-aspect analogue of Heath-Brown’s result on the twelfth power moment of the Riemann zeta function for Dirichlet L-functions to odd prime power moduli. In particular, our results rely on the p-adic method of stationary phase for sums of products and complement Nunes’ bound for smooth square-free moduli. We additionally prove the frequency-aspect analogue of Soundararajan’s result on extreme values of the Riemann zeta function for Hecke L-functions to angular characters over imaginary quadratic number fields. This result relies on the resonance method, which is applied for the first time to this family of L-functions, where the classification and extraction of diagonal terms depends on the geometry of the associated field’s complex embedding. To my friends and family I would like to extend the most sincere and deserved thanks to all of those who have supported me on this long academic journey. To my wife, Jessie, and daughter, Rowan. You were willing to sacrifice your stability so that I could travel the east coast on my tour of higher education over the past ten years. This fact is not lost on me and I know it has been unbearable. Thank you for sticking with me and providing an unbreakable support system. To my grandmother, Iris. Thank you for believing in my potential and convincing me to do something with it. You propped me up and assumed I had a sense of responsibility when, at times, I had absolutely none to speak of. This was the most critical component to my success. To my undergraduate research advisor and friend, Lenny. Thank you for believing in a loser. You were the first person to show me what mathematics is all about and spent a ridiculous amount of time and energy preparing me for graduate school. This is appreciated to the highest possible degree. To my father, Frank. Grinding through years of mathematical research was only possible because of the work ethic you tried so hard to teach. I know that sometimes it seems like I haven’t learned anything at all, but I learned this much at least: just get the damn job done. To my long-time friends, Chad and Rob. Thank you for fostering my love for science. I’m not sure I would have my sense of curiosity and analytic world-view without either of you. Let’s see each other again soon. To the rest of my immediate family and close friends. Your love and support has provided me the opportunity to focus on my work in uncountably-many ways. I know I can be distant at times, so I’ll work on that. Thank you all so much. To my advisor and friend, Djordje. Your healthy mix of patience and expectation, combined with your insightful and concept-driven instruction was the exact proper fit for my learning style. Thank you for shaping my perspective of mathematics in all the right ways. To my readers and committee. Thank you for your time, diligence, and support over the past four years. You have shaped my experience at Bryn Mawr in too many positive ways to list. I’ll remember it fondly. To my office-mate and friend, Isaac. You made things enjoyable during the times that no reasonable person could describe as fun. You made sure I didn’t break when I doubted what I was doing. And when all else failed, we grabbed a beer. My only hope is that I helped you a fraction of the amount that you helped me. To the rest of my Bryn Mawr family. The support at the College is unmatched. I never felt like an outsider and everybody holds stock in the success of others. I truly appreciate the opportunity to be a part of this community. Best wishes.