Editing Farey sequence (section)

===Sequence length and index of a fraction===
The Farey sequence of order {{mvar|n}} contains all of the members of the Farey sequences of lower orders. In particular {{mvar|F<sub>n</sub>}} contains all of the members of {{math|''F''<sub>''n''&minus;1</sub>}} and also contains an additional fraction for each number that is less than {{mvar|n}} and [[coprime]] to {{mvar|n}}. Thus {{math|''F''<sub>6</sub>}} consists of {{math|''F''<sub>5</sub>}} together with the fractions {{sfrac|1|6}} and {{sfrac|5|6}}.

The middle term of a Farey sequence {{mvar|F<sub>n</sub>}} is always {{sfrac|1|2}}, 
for {{math|''n'' > 1}}. From this, we can relate the lengths of {{mvar|F<sub>n</sub>}} and {{math|''F''<sub>''n''&minus;1</sub>}} using [[Euler's totient function]] {{math|''&phi;''(''n'')}}:

<math display=block>|F_n| = |F_{n-1}| + \varphi(n).</math>

Using the fact that {{math|1={{abs|''F''<sub>1</sub>}} = 2}}, we can derive an expression for the length of {{mvar|F<sub>n</sub>}}:<ref>{{Cite OEIS|A005728}}</ref>

<math display=block>|F_n| = 1 + \sum_{m=1}^n \varphi(m) = 1 + \Phi(n),</math>
where {{math|&Phi;(''n'')}} is the [[Totient summatory function|summatory totient]].

We also have :
<math display=block>|F_n| = \frac{1}{2}\left(3+\sum_{d=1}^n \mu(d) \left\lfloor \tfrac{n}{d} \right\rfloor^2 \right),</math>
and by a [[Möbius inversion formula]] : 
<math display=block>|F_n| = \frac{1}{2} (n+3)n - \sum_{d=2}^n|F_{\lfloor n/d \rfloor}|,</math>
where {{math|''&mu;''(''d'')}} is the number-theoretic [[Möbius function]], and  <math>\lfloor n/d \rfloor </math> is the [[Floor and ceiling functions|floor function]].

The asymptotic behaviour of {{math|{{abs|''F<sub>n</sub>''}}}} is :
<math display=block>|F_n| \sim \frac {3n^2}{\pi^2}.</math>

The number of Farey fractions with denominators equal to {{mvar|k}} in {{mvar|F<sub>n</sub>}} is given by {{math|''&phi;''(''k'')}} when {{math|''k'' &le; ''n''}} and zero otherwise. Concerning the numerators one can define the function <math>\mathcal{N}_n(h)</math> that returns the number of Farey fractions with numerators equal to {{mvar|h}} in {{mvar|F<sub>n</sub>}}. This function has some interesting properties as<ref name=Tomas2024>{{cite journal |last=Tomas Garcia |first=Rogelio |url=https://math.colgate.edu/~integers/y63/y63.pdf|title=Farey Fractions with Equal Numerators and the Rank of Unit Fractions
 | journal=Integers | date=July 2024 |volume=24   <!-- |access-date=13 July 2024 --> |doi=10.5281/zenodo.12685697 |arxiv=2404.08283 }}</ref>  
:<math>\mathcal{N}_n(1)=n</math>,
:<math>\mathcal{N}_n(p^m)=\left\lceil(n-p^m) \left(1- 1/p \right)\right\rceil</math> for any prime number <math>p</math>,
:<math>\mathcal{N}_{n+mh}(h)=\mathcal{N}_{n}(h) + m\varphi(h)</math>  for any integer  {{math|''m'' &ge; 0}},
:<math>\mathcal{N}_{n}(4h)=\mathcal{N}_{n}(2h) - \varphi(2h).</math>
In particular, the property in the third line above implies <math>\mathcal{N}_{mh}(h)=(m-1)\varphi(h)</math> and, further, <math>\mathcal{N}_{2h}(h)=\varphi(h).</math> The latter means that, for Farey sequences of even order {{mvar|n}},  the number of fractions with numerators equal to {{math|{{sfrac|n|2}}}} is the same as the number of fractions with denominators equal to {{math|{{sfrac|n|2}}}}, that is <math>\mathcal{N}_{n}(n/2) = \varphi(n/2)</math>.

The index <math>I_n(a_{k,n}) = k</math> of a fraction <math>a_{k,n}</math> in the Farey sequence <math>F_n=\{a_{k,n} : k = 0, 1, \ldots, m_n\}</math> is simply the position that <math>a_{k,n}</math>  occupies in the sequence. This is of special relevance as it is used in an alternative formulation of the [[Riemann hypothesis]], see [[#Riemann hypothesis|below]]. Various useful properties follow:
<math display=block>\begin{align}
  I_n(0/1) &= 0, \\[6pt]
  I_n(1/n) &= 1, \\[2pt]
  I_n(1/2) &= \frac{|F_n|-1}{2}, \\[2pt]
  I_n(1/1) &= |F_n|-1 , \\[2pt]
  I_n(h/k) &= |F_n|-1 - I_n\left(\frac{k-h}{k}\right).
\end{align}</math>

The index of {{math|{{sfrac|1|''k''}}}} where {{math|{{sfrac|n|''i''+1}} < ''k'' &le; {{sfrac|''n''|''i''}}}} and {{mvar|n}} is the [[least common multiple]] of the first {{mvar|i}} numbers, {{math|1=''n'' = lcm([2, ''i''])}}, is given by:<ref name=Tomas2018>{{cite journal |last=Tomas |first=Rogelio |url=https://cs.uwaterloo.ca/journals/JIS/VOL25/Tomas/tomas5.pdf|title=Partial Franel sums | journal=Journal of Integer Sequences | date=January 2022 |volume=25 |issue=1  <!-- |access-date=16 January 2022 --> }}</ref>
<math display=block>I_n(1/k) = 1 + n \sum_{j=1}^{i} \frac{\varphi(j)}{j} - k\Phi(i).</math>

A similar expression was used as an approximation of <math>I_n(x)</math> for low values of <math>x</math> in the classical paper by F. Dress.<ref name=Dress1999>{{cite journal |last=Dress |first=F. |url=http://archive.numdam.org/article/JTNB_1999__11_2_345_0.pdf|title=Discrépance des suites de Farey| journal= J. Théorie des Nr. Bordx.| date=1999 |volume=11   <!-- |access-date=11 January 2025 --> }}</ref> A general expression for <math>I_n(h/k)</math> for any Farey fraction <math>h/k</math> is given in.<ref name=Tomas2025>{{cite journal |last=Tomas Garcia |first=Rogelio |url=https://www.mdpi.com/2227-7390/13/1/140|title=New Analytical Formulas for the Rank of Farey Fractions and Estimates of the Local Discrepancy| journal=Mathematics| date=2025 |volume=13 |issue=1 |page=140 |doi=10.3390/math13010140   <!-- |access-date=11 January 2025 --> |doi-access=free }}</ref>