The concept of prime number dates back at least to Euclid’s Elements circa 300 B.C., wherein the fundamental theorem of arithmetic and the infinitude of primes was proved. The fundamental role of prime numbers is obvious, as they are inextricably bound to the operation of multiplication as its building blocks. Though it is quite easy to show that there must be infinitely many primes, when one begins to ask even basic questions about their finer behaviour such as “approximately how many primes would we expect to find up to a given x?”, it quickly becomes quite non-trivial to come up with satisfactory answers. The earliest such result, the proof of the assertion “there is always a prime between n and 2n”—so- called Bertrand’s Postulate —was obtained as late as the 19th century.
This module will introduce the student to basic techniques involved in investigating the distribution of primes, give a complete demonstration of Bertrand’s Postulate and other assertions of similar strength, and present an overview of how the Prime Number Theorem (an approximate formula for the number of primes up to x) was proved. Along the way the student will be acquainted with the Riemann zeta-function and grasp the relevance of the Riemann Hypothesis—arguably the most important open problem in mathematics—to arithmetic.