Course description: Given a positive integer Q, the sequence obtained by arranging in increasing order all positive rational numbers with denominator (in reduced form) at most Q, is called the sequence of Farey fractions of order Q. Farey fractions arise in many problems in number theory, and in other areas of mathematics. Examples include the "circle method" approach to the Goldbach and twin primes conjectures; the approximation of irrational numbers by rationals; and the behavior of trajectories of billiard balls.
Despite their elementary definition, the finer properties of Farey sequences are quite mysterious and, to a large extent, unexplored. For example, one of the most famous problems in number theory, the Riemann Hypothesis, is equivalent to a certain property of Farey sequences.
References: Basic properties of Farey sequences can be found in most texts on number theory; a good source is the classic text "An introduction to the theory of numbers" by G.H. Hardy and E. Wright. A more specialized reference, which might form the starting point for research projects, is R.R. Hall, "A note on Farey series", J. London Math. Soc. 2 (1970), 139-148.
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