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MATH is an introduction to Analytic Number Theory, a foundational subject in mathematics which dates back to the s and is still a major research area today. The subject generally uses tools and techniques which are analytic in nature to solve problems primarily related to integers.
Asymptotic and summation results and methods are of great significance in Analytic Number Theory. Two primary course objectives are to state and prove two major theorems: Dirichlet's Theorem on Primes in Arithmetic Progressions, and the Prime Number Theorem.
In particular, we will study Selberg's "elementary" proof of the Prime Number Theorem, as well as an analytic proof. Rings: basic properties of rings and examples including polynomial rings, matrix rings, and number rings ; subrings, ideals and ring homomorphisms; divisibility in integral domains; greatest common divisors; Euclidean rings and unique factorisation; applications to number theory; principal ideal domains.
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