View source for Farey series

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''of order $n$''

The increasing sequence of non-negative irreducible fractions not exceeding 1 with denominators not exceeding $n$. For example, the Farey series of order 5 is the sequence

$$\frac01,\frac15,\frac14,\frac13,\frac25,\frac12,\frac35,\frac23,\frac34,\frac45,\frac11.$$

The following assertions hold.

1) If $a/b$ and $a'/b'$ are two consecutive terms of the Farey series of order $n$, then

$$ba'-ab'=1.$$

2) If $a/b$, $a'/b'$, $a''/b''$ are three consecutive terms of the Farey series of order $n$, then the [[mediant]]

$$\frac{a''}{b''}=\frac{a+a'}{b+b'}.$$

3) The number of terms in the Farey series of order $n$ is equal to

$$1+\sum_{x=1}^n\phi(x).\label{*}\tag{*}$$

Farey series were investigated by J. Farey (1816).

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.A. Bukhshtab,   "Number theory" , Moscow  (1966)  (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  R.R. Hall,   "A note on Farey series"  ''J. London Math. Soc.'' , '''2'''  (1970)  pp. 139–148</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  G.H. Hardy,   E.M. Wright,   "An introduction to the theory of numbers" , Oxford Univ. Press  (1979)</TD></TR></table>



====Comments====
Of course, in \eqref{*} $\phi$ denotes the [[Euler function|Euler function]].