Editing Farey sequence (section)

==Properties==
===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>

===Farey neighbours<!-- This section is linked from [[Farey pair]] -->===
Fractions which are neighbouring terms in any Farey sequence are known as a ''Farey pair'' and have the following properties.

If {{math|{{sfrac|''a''|''b''}}}} and {{math|{{sfrac|''c''|''d''}}}} are neighbours in a Farey sequence, with {{math|{{sfrac|''a''|''b''}} < {{sfrac|''c''|''d''}}}}, then their difference {{math|{{sfrac|''c''|''d''}} &minus; {{sfrac|''a''|''b''}}}} is equal to {{math|{{sfrac|1|''bd''}}}}. Since
<math display=block>\frac{c}{d} - \frac{a}{b} = \frac{bc - ad}{bd},</math>

this is equivalent to saying that 
<math display=block>bc - ad = 1.</math>

Thus {{sfrac|1|3}} and {{sfrac|2|5}} are neighbours in {{math|''F''<sub>5</sub>}}, and their difference is {{sfrac|1|15}}.

The converse is also true. If 
<math display=block>bc - ad = 1</math>

for positive integers {{math|''a'', ''b'', ''c'', ''d''}} with {{math|''a'' < ''b''}} and {{math|''c'' < ''d''}}, then {{math|{{sfrac|''a''|''b''}}}} and {{math|{{sfrac|''c''|''d''}}}} will be neighbours in the Farey sequence of order {{math|max(''b,d'')}}.

If {{math|{{sfrac|''p''|''q''}}}} has neighbours {{math|{{sfrac|''a''|''b''}}}} and {{math|{{sfrac|''c''|''d''}}}} in some Farey sequence, with {{math|{{sfrac|''a''|''b''}} < {{sfrac|''p''|''q''}} < {{sfrac|''c''|''d''}}}}, then {{math|{{sfrac|''p''|''q''}}}} is the [[mediant (mathematics)|mediant]] of {{math|{{sfrac|''a''|''b''}}}} and {{math|{{sfrac|''c''|''d''}}}} &ndash; in other words, 
<math display=block>\frac{p}{q} = \frac{a + c}{b + d}.</math>

This follows easily from the previous property, since if 
<math display=block>\begin{align}
  && bp - aq &= qc - pd = 1, \\[4pt]
  \implies && bp + pd &= qc + aq, \\[4pt]
  \implies && p(b + d) &= q(a + c), \\
  \implies && \frac{p}{q} &= \frac{a+c}{b+d}. 
\end{align}</math>

It follows that if {{math|{{sfrac|''a''|''b''}}}} and {{math|{{sfrac|''c''|''d''}}}} are neighbours in a Farey sequence then the first term that appears between them as the order of the Farey sequence is incremented is 
<math display=block>\frac{a+c}{b+d},</math>

which first appears in the Farey sequence of order {{math|''b'' + ''d''}}.

Thus the first term to appear between {{sfrac|1|3}} and {{sfrac|2|5}} is {{sfrac|3|8}}, which appears in {{math|''F''<sub>8</sub>}}.

The total number of Farey neighbour pairs in {{mvar|F<sub>n</sub>}} is {{math|2{{abs|''F<sub>n</sub>''}} &minus; 3}}.

The ''[[Stern–Brocot tree]]'' is a data structure showing how the sequence is built up from 0 {{pars|{{=}} {{sfrac|0|1}}|150%}} and 1 {{pars|{{=}} {{sfrac|1|1}}|150%}}, by taking successive mediants.

==== Equivalent-area interpretation ====

Every consecutive pair of Farey rationals have an equivalent area of 1.<ref name=Austin2008>{{cite web |last1=Austin |first1=David |date=December 2008 |title=Trees, Teeth, and Time: The mathematics of clock making |website=[[American Mathematical Society]] |location=Rhode Island |language=en |url=http://www.ams.org/publicoutreach/feature-column/fcarc-stern-brocot |access-date=28 September 2020 |url-status=live |archive-url=https://web.archive.org/web/20200204014725/http://www.ams.org/publicoutreach/feature-column/fcarc-stern-brocot |archive-date=4 February 2020}}</ref>  See this by interpreting consecutive rationals 
<math display=block>r_1 = \frac{p}{q} \qquad r_2 = \frac{p'}{q'}</math>
as vectors {{math|(''p'', ''q'')}} in the xy-plane. The area is given by
<math display=block>A \left(\frac{p}{q}, \frac{p'}{q'} \right) = qp' - q'p.</math>
As any added fraction in between two previous consecutive Farey sequence fractions is calculated as the [[Mediant (mathematics)|mediant]] (⊕), then 
<math display=block>\begin{align}
  A(r_1, r_1 \oplus r_2) &= A(r_1, r_1) + A(r_1, r_2) \\
  &= A(r_1, r_2) \\
  &= 1
\end{align}</math> 
(since {{math|1=''r''<sub>1</sub> = {{sfrac|1|0}}}} and {{math|1=''r''<sub>2</sub> = {{sfrac|0|1}}}}, its area must be 1).

===Farey neighbours and continued fractions===
Fractions that appear as neighbours in a Farey sequence have closely related [[continued fraction]] expansions. Every fraction has two continued fraction expansions &mdash; in one the final term is 1; in the other the final term is greater by 1. If {{math|{{sfrac|''p''|''q''}}}}, which first appears in Farey sequence {{mvar|F<sub>q</sub>}}, has the [[simple continued fraction|continued fraction expansion]]s
<math display=block>\begin{align}
  &[0;\ a_1,\ a_2,\ \ldots,\ a_{n-1},\ a_n,\ 1] \\{}
  &[0;\ a_1,\ a_2,\ \ldots,\ a_{n-1},\ a_n + 1]
\end{align}</math>

then the nearest neighbour of {{math|{{sfrac|''p''|''q''}}}} in {{mvar|F{{sub|q}}}} (which will be its neighbour with the larger denominator) has a continued fraction expansion
<math display=block>[0;\ a_1,\ a_2,\ \ldots,\ a_n]</math>

and its other neighbour has a continued fraction expansion 
<math display=block>[0;\ a_1,\ a_2,\ \ldots,\ a_{n-1}]</math>

For example, {{sfrac|3|8}} has the two continued fraction expansions {{math|[0; 2, 1, 1, 1]}} and {{math|[0; 2, 1, 2]}}, and its neighbours in {{math|''F''<sub>8</sub>}} are {{sfrac|2|5}}, which can be expanded as {{math|[0; 2, 1, 1]}}; and {{sfrac|1|3}}, which can be expanded as {{math|[0; 2, 1]}}.

===Farey fractions and the least common multiple===
The [[Least common multiple|lcm]] can be expressed as the products of Farey fractions as
<math display=block> \text{lcm}[1,2,...,N] = e^{\psi(N)} = \frac{1}{2} \left( \prod_{r \in F_N, 0<r \le 1/2} 2 \sin(\pi r) \right)^2 </math>

where {{math|''&psi;''(''N'')}} is the second [[Chebyshev function]].<ref>{{Cite arXiv |eprint = 0907.4384|last1 = Martin|first1 = Greg|title = A product of Gamma function values at fractions with the same denominator|class = math.CA|year = 2009}}</ref><ref>{{Cite arXiv |eprint=0909.1838 |last1=Wehmeier |first1=Stefan |title=The LCM(1,2,...,n) as a product of sine values sampled over the points in Farey sequences |class=math.CA |year=2009}}</ref>

===Farey fractions and the greatest common divisor===
Since the [[Euler's totient function]] is directly connected to the [[Greatest common divisor|gcd]] so is the number of elements in {{mvar|F<sub>n</sub>}},
<math display=block>|F_n| = 1 + \sum_{m=1}^n \varphi(m) = 1+ \sum\limits_{m=1}^{n} \sum\limits_{k=1}^m \gcd(k,m) \cos {2\pi\frac{k}{m}} .</math>

For any 3 Farey fractions {{math|{{sfrac|''a''|''b''}}, {{sfrac|''c''|''d''}}, {{sfrac|''e''|''f''}}}} the following identity between the [[Greatest common divisor|gcd]]'s of the 2x2 [[matrix determinant]]s in absolute value holds:<ref name=TomasGarcia2020>{{cite journal |last=Tomas Garcia |first=Rogelio |url=http://rtomas.web.cern.ch/rtomas/NNTDM-26-3-005-007.pdf|title=Equalities between greatest common divisors involving three coprime pairs| journal=Notes on Number Theory and Discrete Mathematics |date=August 2020 |volume=26 |issue=3|pages=5–7 |doi= 10.7546/nntdm.2020.26.3.5-7 |s2cid=225280271 |doi-access=free  <!-- |access-date=20 January 2022 --> }}</ref>
<math display=block>
  \gcd\left(\begin{Vmatrix} a & c\\b & d \end{Vmatrix}, \begin{Vmatrix} a & e\\b & f \end{Vmatrix} \right)
  = \gcd\left(\begin{Vmatrix} a & c\\b & d \end{Vmatrix}, \begin{Vmatrix} c & e\\d & f \end{Vmatrix} \right)
  = \gcd\left(\begin{Vmatrix} a & e\\b & f \end{Vmatrix}, \begin{Vmatrix} c & e\\d & f \end{Vmatrix} \right)
</math>

<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>

===Applications===
Farey sequences are very useful to find rational approximations of irrational numbers.<ref>{{cite web |url=https://nrich.maths.org/6596 |title=Farey Approximation |website=NRICH.maths.org |access-date=18 November 2018 |archive-url=https://web.archive.org/web/20181119092100/https://nrich.maths.org/6596 |archive-date=19 November 2018 |url-status=dead}}</ref> For example, the construction by Eliahou<ref>{{cite journal |last1=Eliahou |first1=Shalom |title=The 3x+1 problem: new lower bounds on nontrivial cycle lengths |journal=Discrete Mathematics |date=August 1993 |volume=118 |issue=1–3 |pages=45–56 |doi=10.1016/0012-365X(93)90052-U|doi-access=free }}</ref> of a lower bound on the length of non-trivial cycles in the [[Collatz conjecture#Cycles|3''x''+1 process]] uses Farey sequences to calculate a continued fraction expansion of the number {{math|log<sub>2</sub>(3)}}.

In physical systems with resonance phenomena, Farey sequences provide a very elegant and efficient method to compute resonance locations in 1D<ref>{{cite journal |last1=Zhenhua Li |first1=A. |last2=Harter |first2=W.G. |arxiv=1308.4470 |title=Quantum Revivals of Morse Oscillators and Farey–Ford Geometry |journal=Chem. Phys. Lett. |year=2015 |volume=633 |pages=208–213 |doi=10.1016/j.cplett.2015.05.035|bibcode=2015CPL...633..208L |s2cid=66213897 }}</ref> and 2D.<ref>{{cite journal |last1=Tomas |first1=R. |year=2014 |doi=10.1103/PhysRevSTAB.17.014001|title=From Farey sequences to resonance diagrams |journal=Physical Review Special Topics - Accelerators and Beams |volume=17 |issue=1 |page=014001|bibcode=2014PhRvS..17a4001T |doi-access=free |url=http://cds.cern.ch/record/2135825/files/10.1103_PhysRevSTAB.17.014001.pdf }}</ref>

Farey sequences are prominent in studies of [[any-angle path planning]] on square-celled grids, for example in characterizing their computational complexity<ref>{{cite journal |last1=Harabor |first1=Daniel Damir |last2=Grastien |first2=Alban |last3=Öz |first3=Dindar |last4=Aksakalli |first4=Vural |title=Optimal Any-Angle Pathfinding In Practice |journal=Journal of Artificial Intelligence Research |date=26 May 2016 |volume=56 |pages=89–118 |doi=10.1613/jair.5007|doi-access=free }}</ref> or optimality.<ref>{{cite journal |last1=Hew |first1=Patrick Chisan |title=The Length of Shortest Vertex Paths in Binary Occupancy Grids Compared to Shortest ''r''-Constrained Ones |journal=Journal of Artificial Intelligence Research |date=19 August 2017 |volume=59 |pages=543–563 |doi=10.1613/jair.5442|doi-access=free }}</ref> The connection can be considered in terms of {{mvar|r}}-constrained paths, namely paths made up of line segments that each traverse at most {{mvar|r}} rows and at most {{mvar|r}}  columns of cells. Let {{mvar|Q}} be the set of vectors {{math|(''q'', ''p'')}} such that <math>1 \leq q \leq r</math>, <math>0 \leq p \leq q</math>, and {{mvar|p}}, {{mvar|q}} are coprime. Let {{mvar|Q*}} be the result of reflecting {{mvar|Q}} in the line {{math|1=''y'' = ''x''}}. Let <math>S = \{ (\pm x, \pm y) : (x, y) \in Q \cup Q* \}</math>. Then any {{mvar|r}}-constrained path can be described as a sequence of vectors from {{mvar|S}}. There is a bijection between {{mvar|Q}} and the Farey sequence of order {{mvar|r}} given by {{math|(''q'', ''p'')}} mapping to <math>\tfrac{p}{q}</math>.

===Ford circles===
[[File:Comparison_Ford_circles_Farey_diagram.svg|thumb|250px|link={{filepath:Comparison Ford circles Farey diagram.svg}}|Comparison of Ford circles and a Farey diagram with circular arcs for ''n'' from 1 to 9. Each arc intersects its corresponding circles at right angles. In [[Media:Comparison Ford circles Farey diagram.svg|the SVG image]], hover over a circle or curve to highlight it and its terms.]]

There is a connection between Farey sequence and [[Ford circle]]s.

For every fraction {{math|{{sfrac|''p''|''q''}}}} (in its lowest terms) there is a Ford circle {{math|''C''[''p''/''q'']}}, which is the circle with radius <math>\tfrac{1}{2q^2}</math> and centre at <math>\bigl(\tfrac{p}{q}, \tfrac{1}{2q^2}\bigr).</math> Two Ford circles for different fractions are either [[Disjoint sets|disjoint]] or they are [[tangent]] to one another—two Ford circles never intersect. If {{math|0 < {{sfrac|''p''|''q''}} < 1}} then the Ford circles that are tangent to {{math|''C''[''p''/''q'']}} are precisely the Ford circles for fractions that are neighbours of {{math|{{sfrac|''p''|''q''}}}} in some Farey sequence.

Thus {{math|''C''[2/5]}} is tangent to {{math|''C''[1/2]}}, {{math|''C''[1/3]}}, {{math|''C''[3/7]}}, {{math|''C''[3/8]}}, etc.

Ford circles appear also in the [[Apollonian gasket]] {{math|(0,0,1,1)}}. The picture below illustrates this together with Farey resonance lines.<ref name=Tomas2020>{{cite arXiv |last=Tomas |first=Rogelio |eprint=2006.10661|title=Imperfections and corrections |class= physics.acc-ph|year=2020 <!-- |access-date=4 July 2020 --> }}</ref>

[[File:Apolloinan gasket Farey.png|650px|thumb|center|[[Apollonian gasket]] {{math|(0,0,1,1)}} and the Farey resonance diagram.]]

===Riemann hypothesis===
Farey sequences are used in two equivalent formulations of the [[Riemann hypothesis]]. Suppose the terms of {{mvar|F{{sub|n}}}} are <math>\{a_{k,n} : k = 0, 1, \ldots, m_n\}.</math> Define <math>d_{k,n} = a_{k,n} - \tfrac{k}{m_n},</math> in other words <math>d_{k,n}</math> is the difference between the {{mvar|k}}th term of the {{mvar|n}}th Farey sequence, and the {{mvar|k}}th member of a set of the same number of points, distributed evenly on the unit interval. In 1924 [[Jérôme Franel]]<ref>{{cite journal |author-link=Jérôme Franel |first=Jérôme |last=Franel |url=http://www.digizeitschriften.de/dms/resolveppn/?PID=GDZPPN00250653X |title=Les suites de Farey et le problème des nombres premiers |journal=Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen |series=Mathematisch-Physikalische Klasse |year=1924 |pages=198–201 |language=fr}}</ref> proved that the statement

<math display=block>\sum_{k=1}^{m_n} d_{k,n}^2 = O (n^r) \quad \forall r > -1</math>

is equivalent to the Riemann hypothesis, and then [[Edmund Landau]]<ref>{{cite journal |author-link=Edmund Landau |first=Edmund |last=Landau |url=http://www.digizeitschriften.de/dms/resolveppn/?PID=GDZPPN002506548 |title=Bemerkungen zu der vorstehenden Abhandlung von Herrn Franel |journal=Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen |series=Mathematisch-Physikalische Klasse |year=1924 |pages=202–206 |language=de}}</ref> remarked (just after Franel's paper) that the statement
<math display=block>\sum_{k=1}^{m_n} |d_{k,n}| = O (n^r) \quad \forall r > \frac{1}{2}</math>

is also equivalent to the Riemann hypothesis.

=== Other sums involving Farey fractions ===
The sum of all Farey fractions of order {{mvar|n}} is half the number of elements:
<math display=block>\sum_{r\in F_n} r = \frac{1}{2} |F_n| .</math>

The sum of the denominators in the Farey sequence is twice the sum of the numerators and relates to Euler's totient function:

<math display=block>\sum_{a/b \in F_n} b = 2 \sum_{a/b \in F_n} a = 1 + \sum_{i=1}^{n} i\varphi(i) , </math>

which was conjectured by Harold L. Aaron in 1962 and demonstrated by Jean A. Blake in 1966.<ref>{{Cite journal|first1=Jean A.|last1=Blake|doi=10.2307/2313922 |title=Some Characteristic Properties of the Farey Series|journal=The American Mathematical Monthly| volume=73|issue=1|year=1966|pages=50–52 |jstor=2313922 }}</ref> A one line proof of the Harold L. Aaron conjecture is as follows.
The sum of the numerators is 
<math display=block>1 + \sum_{2 \le b \le n} \ \sum_{(a,b)=1} a = 1 + \sum_{2 \le b \le n} b\frac{\varphi(b)}{2}.</math> 
The sum of denominators is
<math display=block>2 + \sum_{2 \le b \le n} \ \sum_{(a,b)=1} b  = 2 + \sum_{2 \le b \le n} b\varphi(b).</math>
The quotient of the first sum by the second sum is {{sfrac|1|2}}.

Let {{mvar|b<sub>j</sub>}} be the ordered denominators of {{mvar|F<sub>n</sub>}}, then:<ref>{{Cite journal |jstor=10.4169/000298910X475005 |doi=10.4169/000298910X475005 |title=Farey Sums and Dedekind Sums |journal=The American Mathematical Monthly |volume=117 |issue=1 |pages=72–78 |year=2010 |last1=Kurt Girstmair |last2=Girstmair |first2=Kurt|s2cid=31933470 }}</ref>

<math display=block>\sum_{j=0}^{|F_n|-1} \frac{b_j}{b_{j+1}} = \frac{3|F_n|-4}{2}  </math>
and
<math display=block>\sum_{j=0}^{|F_n|-1} \frac{1}{b_{j+1}b_{j}} = 1.</math>

Let <math>\tfrac{a_j}{b_j}</math> the {{mvar|j}}th Farey fraction in {{mvar|F<sub>n</sub>}}, then
<math display=block>
  \sum_{j=1}^{|F_n|-1} (a_{j-1}b_{j+1} - a_{j+1}b_{j-1}) 
  = \sum_{j=1}^{|F_n|-1} \begin{Vmatrix} 
    a_{j-1} & a_{j+1} \\
    b_{j-1} & b_{j+1} 
    \end{Vmatrix} = 3(|F_n|-1) - 2n - 1, </math>

which is demonstrated in.<ref>{{cite journal|first1=R. R. |last1=Hall |first2= P. |last2=Shiu |title= The Index of a Farey Sequence|journal= Michigan Math. J. | volume=51 | year =2003| doi=10.1307/mmj/1049832901|number=1 |pages=209–223|doi-access=free }}
</ref> Also according to this reference the term inside the sum can be expressed in many different ways:
<math display=block> a_{j-1} b_{j+1} - a_{j+1} b_{j-1} = \frac{b_{j-1}+b_{j+1}}{b_{j}} = \frac{a_{j-1}+a_{j+1}}{a_{j}} = \left\lfloor\frac{n+ b_{j-1}}{b_{j}} \right\rfloor, </math>

obtaining thus many different sums over the Farey elements with same result. Using the symmetry around 1/2 the former sum can be limited to half of the sequence as

<math display=block>\sum_{j=1}^{\left\lfloor \frac{|F_n|}{2} \right\rfloor} (a_{j-1} b_{j+1} - a_{j+1} b_{j-1}) = \frac{3(|F_n|-1)}{2} - n - \left\lceil \frac{n}{2} \right\rceil , </math>

The [[Mertens function]] can be expressed as a sum over Farey fractions as
<math display=block>M(n)= -1+ \sum_{a\in \mathcal{F}_n} e^{2\pi i a}</math>
where <math> \mathcal{F}_n</math> is the Farey sequence of order {{mvar|n}}.

This formula is used in the proof of the [[#Riemann hypothesis|Franel–Landau theorem]].<ref>{{cite book|last1=Edwards |first1=Harold M. |author1-link=Harold Edwards (mathematician) |editor1-last=Smith|editor1-first=Paul A.|editor1-link=Paul Althaus Smith|editor2-last=Ellenberg|editor2-first=Samuel|editor2-link=Samuel Eilenberg|year=1974 |title=Riemann's Zeta Function |chapter=12.2 Miscellany. The Riemann Hypothesis and Farey Series |publisher=[[Academic Press]] |series=Pure and Applied Mathematics|pages=263{{ndash}}267 |location=New York |language=en |isbn=978-0-08-087373-2 |oclc=316553016 |chapter-url=https://archive.org/details/riemannszetafunc00edwa_0/page/262 |chapter-url-access=registration |access-date=30 September 2020}}</ref>