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“Dirichlet’s Theorem and the Rise of Analytic Number Theory”

In 1837, Peter Dirichlet proved the following theorem: If a and d are relatively prime integers, then the arithmetic progression a, a+d, a+2d, . . . contains infinitely many prime numbers. His proof ushered in a revolution in number theory because it relied in a critical way on complex analysis. The use of analytic methods to solve problems in number theory was a tremendous innovation at the time. We will consider some of the details of Dirichlet’s proof, focusing on understanding why there is a deep connection between these seemingly unrelated branches of mathematics. Throughout, we will be more concerned with the ideas underlying the proof than we will with presenting an excessive amount of technical detail.