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One can describe Analytic Number Theory informally as being the elegant subject where ideas and concepts from real and complex analysis are applied to number-theoretic problems. This book is an overview of some important results in Analytic Number Theory. Topics include Dirichlet L-series, their analytic continuations and functional equations, including relevant supporting material on characters, Gamma functions and the Riemann Zeta-Function. We also examine Dirichlet's Theorem, giving the existence of infinitely many prime numbers congruent to a given "a modulo b" when "a" and "b" are coprime, the Prime Number Theorem for arithmetic progressions and the Poisson Summation Formula. We then discuss how these ideas can be applied to the theory of the so-called Negative Pell Equation, which is an interesting and unlikely application.