Mathematical Sciences: Analytic Theory of L-functions

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The theory of L-functions is central in the study of number theoretical questions, especially in the realm of analytic number theory. For most applications, the relevant questions usually involve the analytic continuation of the L-functions, their functional equations, estimates for their growth in vertical strips, and knowledge of the distribution of their zeros. This project deals with both very general questions (the classification of Dirichlet series with functional equations and Euler products and their connections with automorphic forms) and particular questions (mean-value theorems for particular L-functions, and the distribution of their zeros) in the theory of L-functions. This research falls into the general mathematical field of Number Theory. Number theory has its historical roots in the study of the whole numbers, addressing such questions as those dealing with the divisibility of one whole number by another. It is among the oldest branches of mathematics and was pursued for many centuries for purely aesthetic reasons. However, within the last half century it has become an indispensable tool in diverse applications in areas such as data transmission and processing, and communication systems.