Number theory is one of the richest and oldest branches of mathematics. It is notable for its many unsolved but easily stated conjectures. The current chapter touches on issues surrounding prime numbers and their density. In particular, the chapter and book culminate with a proof of the prime number theorem. This highlight of 19th century mathematics was surmised by Legendre and Gauss, attacked by Riemann and Chebyshev, and finally proved by Hadamard and de la Valléé Poussin. These mathematicians created a large part of analytic function theory in the process. In the mid-20th century, Erdös and Selberg succeeded in crafting a proof that avoids analytic functions. Even so, their elementary proof is longer and harder to comprehend than the classical proofs. Our treatment follows the recent trail blazed by Newman  and Zagier  that uses a minimum of analytic function theory. We particularly stress the connections and insight provided by probability.
KeywordsPrime Number Multiplicative Function Arithmetic Function Prime Number Theorem Zipf Distribution
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