Farey series

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

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

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

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

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

References

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

Comments

Of course, in \ref{*} $\phi$ denotes the Euler function.