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The Haros-Farey sequence at two hundred years. A survey. (English) Zbl 1096.11005

Summary: Let \(\operatorname{Im}_Q\) be the set of representatives for the nonnegative subunitary rational numbers in their lowest terms with denominators at most \(Q\) and arranged in ascending order. This finite sequence of fractions has two remarkable basic properties. The first one asserts that the difference between two consecutives fractions equals the inverse of the product of their denominators. The second called also ’the mediant property’, says that if \(a'/q',\;a''/q''\) and \(a'''/q'''\) are consecutive in \(\operatorname{Im}_Q\) then \(a''/q''=(a'+a''')/(q'+q''')\). These properties are equivalent and they were mentioned without proof for the first time by Haros in 1802 and respectively by Farey in 1816, independently. Thus the proper name \(\operatorname{Im}_Q\) should be “the Haros-Farey sequence” instead of “the Farey sequence” as it is known after Cauchy.
Besides marking the two-hundredth anniversary of the Farey sequence, the main raison d’être of this article is to survey some important properties \(\operatorname{Im}_Q\), most of them discovered recently. We also sketch the impact of these results on different problems of Number Theory or Mathematical Physics. There are many papers (more than five hundred published only in the last fifty years) dealing with \(\operatorname{Im}_Q\), and here we only mention a few of them. Though, starting with the cited articles below, one may easily track most of the remaining ones.

MSC:

11B57 Farey sequences; the sequences \(1^k, 2^k, \dots\)
11-02 Research exposition (monographs, survey articles) pertaining to number theory
11-03 History of number theory
11M26 Nonreal zeros of \(\zeta (s)\) and \(L(s, \chi)\); Riemann and other hypotheses
11N37 Asymptotic results on arithmetic functions
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