C. Cobeli, A. Zaharescu — The Haros-Farey Sequence at Two Hundred Years
Figure 4.1: The set
❵✴❛❝❜❞❜❢❡ ❣✒❡
❤✰✐ ❥✮❦
❧♥♠ ❦
❧♣♦
q r
r
Figure 4.2: The set
s✉t✇✈②①④③ ⑤
⑥⑧⑦ ❣❍⑨
⑥
.
5.1. Jumping Champions of
❪✡❫
. Finding the maximum or the minimum of a certain
sequence may be both interesting and difficult. Two remarkable examples require to find
⑩■❶
, the length of the longest increasing subsequence of a random permutation of the integers from
☞
to
❷
, and respectively to find
❸ ❶
, the largest eigenvalue of a
❷✫❹❺❷
random GUE matrix. Recently in [BDJ1999] and [TW2000] it was shown that
⑩ ❶
and
❸ ❶
share the same distribution and in [RW2002] are presented many other appearances of the same distribution in limit theorems from widely different areas.
Being confined both in magnitude to the interval
❻ ✓
✍✰☞✲❼
and in size of the denominators to
❻ ☞✎✍
❼
, the definition of
❪❄❫
makes trivial the question on how large is the maximum or the minimum spacing between neighbor fractions. Of course, these are
☞❉❈
and
☞❉❈ ✝
▼❽ ☞
✟
, respectively, attained by the distances of the fraction
✝ ④❽
☞ ✟
❈
to its neighbors. A finer way to analyze the distribution of the gaps is to find the frequency with
which they appear. The most encountered difference is called the jumping champion, or shortly
❾❳❿➁➀
. For the Farey sequence Cobeli, Ford and Zaharescu [CFZ2002] have estimated the size of a set of possible candidates for
❾❳❿➁➀
and studied the arithmetic structure of
❾❳❿➁➀
. For a set
➂ ✠
➃✿➄ ❘
✍ ❬✿❬✿❬
✍ ➄✹➅➇➆
of real numbers ordered increasingly, let
➈ ✝
➂➉✟✉✠ ➃✿➄➋➊
◆ ❘
❽ ➄➋➊➍➌
☞➏➎➑➐➒➎➔➓ ❽
☞ ➆
be the set of gaps between consecutive elements. The elements of
➈ ✝
➂→✟
are arranged in ascending order, keeping in the list all the dif- ferences with their mulitplicities. The champion of
➂
is the element of
➈ ✝
➂→✟
with the highest multiplicity. When more numbers have the same highest multiplicity in
➈ ✝
➂➉✟
,
C. Cobeli, A. Zaharescu — The Haros-Farey Sequence at Two Hundred Years
Figure 4.3: The set
❵✻❛❝❜❝❜✺❡ ❜❢❡
❤❉✐
Figure 4.4: The set
❵➋❣➣❤❝❜❞❜❢❡ ↔❢❡
❣➣❤❉✐
those numbers share the position of champion. Depending on
➂
, finding the
❾❳❿➋➀
of a set may prove a complex task. Certainly this is the case when
➂ ✠➙↕
❶
, the set of primes less than or equal to
❷
. Odlyzko, Rubin- stein and Wolf [ORW1999] give empirical and heuristic evidences based on the
➛
-tuple conjecture that the
❾❳❿➁➀
for primes are the primorials
➜ ✍✣❊
✓ ✍
✜ ☞
✓ ✍
✜ ❊✩☞
✓ ✍
❬✿❬✿❬
, except for a few small values of
❷
, when
❾❳❿➁➀
are
☞✎✍ ✜
or
❨
. Similarities seem to exist between the arithmetical properties of the elements of
➈ ✝
↕ ❶
✟
and
➈ ✝
❪➝❫➞✟
, but in the case of Farey fractions the results are proved unconditionally.
Since
❪➝❫
is symmetric with respect to
☞❉❈ ✜
, we take
➂➠➟➡✠➢❪❄❫✫➤❇❻ ✓
✍✰☞❉❈ ✜
❼
. Then
➥ ➈
✝ ➂→➟➦✟
➥ ✠
✝ ➥
❪ ❫
➥ ❽
☞ ✟
❈ ✜
. For small,
☞✇➎ ➎→➧
, all the elements of
➈ ✝
➂➉➟➦✟
are distinct, so they all share the position of champion. For
✠ ☞
✓
, we have
➂ ❘➩➨
✠④➫ ✓
✍ ☞
☞ ✓
✍ ☞
➧ ✍
☞ ➭
✍ ☞
➯ ✍
☞ ➜
✍ ☞
➲ ✍
✜ ➧
✍ ☞
❨ ✍
✜ ➯
✍ ❊
☞ ✓
✍ ☞
❊ ✍
❊ ➭
✍ ✜
➲ ✍
❊ ➯
✍ ❨
➧ ✍
☞ ✜➦➳
and
➈ ✝
➂ ❘➩➨
✟✕✠ ➫
☞ ☞
✓ ✍
☞ ➧
✓ ✍
☞ ➯
✜ ✍
☞ ➲
➜ ✍
☞ ❨
✜ ✍
☞ ❊
✓ ✍
☞ ❨
➲ ✍
☞ ❊
➜ ✍
☞ ✜
➭ ✍
☞ ➯
✓ ✍
☞ ❊
✓ ✍
☞ ✜
❨ ✍
☞ ❨
✓ ✍
☞ ❊
➲ ✍
☞ ➜
❊ ✍
☞ ☞
➭ ➳
❬
The gap
☞❉❈✎❊ ✓
appears twice in
➈ ✝
➂ ❘➩➨
✟
, so it is the
❾❳❿➁➀
of
➵ ❘➩➨
. In Table 1 are listed some champions with a record number of appearances through different values of
➎ ❨
✓❍✓
, in other words ‘Champs among champions’. One can see that a condition for a number to be a champion is to be highly composite.
C. Cobeli, A. Zaharescu — The Haros-Farey Sequence at Two Hundred Years For any two consecutive elements of
❪ ❫
, say
➸⑧➺ ➻
➺
,
➸⑧➺ ➺
➻ ➺
➺
, the gap between them is
➸⑧➺ ➺
➻ ➺
➺ ❽
➸❢➺ ➻
➺ ✠
❘ ➻
➺ ➻
➺ ➺
, so the basic set to look at is
➼✣➟
, the set of the pairs of consecutive denominators of Farey fractions. This is given by
➼➽➟➾✠➙➚ ✝➶➪
❘ ✍
➪ ❀
✟➹➌ ☞✙➎
➪ ❘
✍ ➪
❀ ➎
✍ ➪
❘ ✚
➪ ❀✗➘
✍➝➴✴➷✿➬ ✝➶➪
❘ ✍
➪ ❀
✟✡✠ ☞✻➮
❬
Table 1: Selected champions with a record number of appearances. 1Champion
➱◗✃➣❐➩❒ ✪✡❮
❒✒❰ ✬
ÏÐ✬ ❒
✭ Ñ
❒ÓÒ➩❒ÕÔ✴Ö ❮✲❮
✃➩ÖÕ×ÐÖ ✭
❐➣✃➩❰
The values of Q
6
❀❁Ø➩Ù Ù
3, 4, 6
12
❀⑧Ú✣Ø❞Ù Û
4–6, 12
30
❀❁Ø➩Ù❳Ø❝Ü Ý
6–12
70
❀❁Ø➩Ü❳Ø❞Ý Ý
11–17
210
❀❁Ø➩Ù❁Ø➩Ü❁Ø❞Ý ❘➩➨
17, 21–28, 30
390
❀❁Ø❝Ù❳Ø➩Ü❁Ø ❘
Ù Þ
30–32, 39–41
420
❀ Ú
Ø➩Ù❁Ø❞Ü❁Ø❝Ý ❘
❀
28–32, 35–41
546
❀❁Ø❝Ù❳Ø❝Ý◗Ø ❘
Ù ❘➩➨
41–50
840
❀✒ß◗Ø➩Ù❁Ø❞Ü❁Ø❝Ý ❘✒❘
41, 47–50, 55–60
1260
❀ Ú
Ø➩Ù Ú
Ø➩Ü❁Ø❝Ý ❘
Ý
47–50, 55, 59–70
2310
❀❳Ø➩Ù❁Ø❝Ü❁Ø❝Ý◗Ø ❘✒❘
❀⑧Ý
70–96
6930
❀❁Ø➩Ù Ú
Ø➩Ü❳Ø❝Ý◗Ø ❘✒❘
Û⑧Ù
126–168
8190
❀❁Ø➩Ù⑧Ú◗Ø➩Ü❳Ø❝Ý◗Ø ❘
Ù Ù✒à
130–153, 167–180
10010
❀◗Ø❝Ü❁Ø❝Ý❁Ø ❘✒❘
Ø ❘
Ù Û
➨
167–206
18018
❀❳Ø➩Ù Ú
Ø❝Ý◗Ø ❘✒❘
Ø ❘
Ù Ý
➨
201–270
30030
❀❁Ø❞Ù❁Ø➩Ü❁Ø❝Ý◗Ø ❘✒❘
Ø ❘
Ù à⑧Ý
231–233, 269–352
39270
❀❁Ø❞Ù❁Ø➩Ü❁Ø❝Ý◗Ø ❘✒❘
Ø ❘
Ý Ü✒Þ
269–272, 349–400
We denote by
á ✝
➈ ✍
✟
the number of gaps of length
☞❉❈ ➈
in
➂➉➟
. Then
á ✝
➈ ✍
✟
is the multiplicity of
☞❉❈ ➈
in
➈ ✝
➂➉➟➦✟
, and this can be written as:
á ✝
➈ ✍
✟✕✠ãâ â
â ➚
✝➶➪ ❘
✍ ➪
❀ ✟✞ä❱➼➽➟✂➌
➪ ❘
➪ ❀
✠å➈ ✍
➪ ❘
✘ ➪
❀ ➮
â â
â ✠ãâ
â â
➚ ➪
➥ ➈æ➌
➴✴➷✿➬ ✝➶➪
✍✻ç ➻
✟➦✠ ☞✎✍✥ç
➻ ✘
➪ ➎
✍✗ç ➻
✚ ➪
➘ ➮
â â
â ❬
This gives a partition of
➈ ✝
➂➉➟➦✟
, therefore
â â
➈ ✝
➂→➟➦✟ â
â ✠
✼ ç❑è
❘ á
✝ ➈
✍ ✟
❬
Then any champion verifies:
➓ ✝
✟✕✠åé✧ê✦ë ç
á ✝
➈ ✍
✟ ❬
The size of
❾ì❿➁➀
is given by
➓ ✝
✟✡✠îíÓë✩ï➇ðñ✜❄ò ❋
➴ ✜
ò ❋
➴ ò
❋ ➴
ò ❋
➴ ✚✉ó➙ð
ò ❋
➴ ✝
ò ❋
➴ ò
❋ ➴
✟ ❀✹ô✤ô
❬
C. Cobeli, A. Zaharescu — The Haros-Farey Sequence at Two Hundred Years Next we look at the set of champions strictly speaking inverses of champions:
õ÷ö ê❍é■ïñø
✝ ✟➦✠➙➚✻➈æ➌■á
✝ ➈
✍ ✟❑✠
➓ ✝
✟ ➮
❬
The characteristic of
õùö ê❍é■ï✣ø
✝ ✟
is that its elements have close to the maximum possi- ble number of prime factors for integers of their size. Thus, if
➈úä õùö
ê❍é■ïñø ✝
✟
then
ûü✝ ➈❯✟✡✠ý✜
ò ❋
➴ ò
❋ ➴
ò ❋
➴ ✚✉ó➙ð
ò ❋
➴ ✝
ò ❋
➴ ò
❋ ➴
✟ ❀
ô ✍
where
ûü✝ ❷þ✟
is the number of distinct prime factors of
❷
. As a consequence, most of the prime factors of a champion are small. Indeed, one can show that the largest prime
factor of any
➈úä õùö
ê❍é■ïñø ✝
✟
, is
ÿ ✝
ò ❋
➴ ✟
Ù
. Various other results related to the champions of
❪✡❫
are proved in [CFZ2002]. Specifi- cally, it is studied
✝ ✟
, the number of distinct gaps participants in the competition for
❾❳❿➋➀
and
✁✄✂ ✝
✟
, the number of gaps with multiplicity
❚ ➛
. For
✝ ✟
it is shown that
❀ ✝
ò ❋
➴ ✟✆☎
íÓë✩ï ✝
❽✟✞ ❘✡✠
ò ❋
➴ ò
❋ ➴
ò ❋
➴ ò
❋ ➴
ò ❋
➴ ☞☛
ÿ✌ ✝
✟➦ÿ ❀
✝ ò
❋ ➴
✟✆☎✎✍ ☞
✏ ò
❋ ➴
ò ❋
➴ ✍
where
✑ã✠ ☞
❽ ❘
◆✓✒ ❒✕✔
✒ ❒✕✔
❀ ✒
❒✕✔ ❀
and
✞ ➘
✓
is constant. Depending on
➛
, different bounds for
✁☞✂ ✝
✟
are also proved. An example is one given in a closed form:
✁✄✂ ✝
✟➦✠ ❀
✝ ò
❋ ➴
✟✆☎ ◆✗✖✙✘
❘✛✚ ✍
which is valid when
➛ ✠✢✜
✖✤✣✦✥ ✧✩★✡✥
✧✩★ ❫
✥ ✧✩★✡✥
✧✩★✡✥ ✧✩★
❫✫✪
.
5.2. Moments of the index of Farey fractions. Here we present some asymptotic for- mulae concerning the distribution of the index of Farey fractions of order
as
✂✁☎✄
. First let us have a closer look at the definition of the index. With the notations introduced
in Section 3.4, one sees that
✬ ✝✮✭
✍✕✯ ✟✗✠
✝ ✯◗✍
❂ ✯
❽ ✭
✟
for any
✝✮✭ ✍✕✯
✟ ä✢➼
✽
. Starting with
❂ ❘
✝✮✭ ✍✕✯
✟✕✠✱✰ ❘
◆✓✲ ✳✵✴
, for
✶ ❚
✜
one has recursively
❂✸✷ ✝✮✭
✍✕✯ ✟✕✠
✝ ❂✸✷
❙❳❘✺✹ ✬✗✟
✝✮✭ ✍✕✯
✟ ❬
Also, for
✶ ❚
☞ ✻
✷ ◆
❘ ✝✮✭
✍✕✯ ✟➦✠
❂✸✷ ✝✮✭
✍✕✯ ✟
✻ ✷
✝✮✭ ✍✕✯
✟ ❽
✻ ✷
❙❳❘ ✝✮✭
✍✕✯ ✟
✍
with
✻ ➨
✝✮✭ ✍✕✯
✟➦✠ ✭
,
✻ ❘
✝✮✭ ✍✕✯
✟✕✠ ✯
.
C. Cobeli, A. Zaharescu — The Haros-Farey Sequence at Two Hundred Years On the other hand, noticing that for any
➪✤✼ ✍
➪✽✼ ✼
✍ ➪✽✼
✼ ✼
, consecutive denominators of fractions from
❪➝❫
we have
✬ ✖
➻ ➺
➟ ✍
➻ ➺
➺ ➟
★ ✠
✖ ➻
➺ ➺
➟ ✍
➻ ➺
➺ ➺
➟ ★
, we get
❂ ❘
✖ ➻✮✾
✶➋✵ ➟
✍ ➻✿✾
➟ ★
✠ ✰
➟ ◆
➻✿✾ ✶➋✵
➻✿✾ ✴
, and then, for any
➛✛ä❁❀
it follows that,
❂ ✂
◆ ❘
✖ ➻✮✾
✶➋✵ ➟
✍ ➻✿✾
➟ ★
✠ ❂
❘✺✹ ✬
✂ ✖
➻✿✾ ✶➁✵
➟ ✍
➻✮✾ ➟
★ ✠
❂ ❘
✖ ➻✮✾
✱❃❂ ✶➋✵
➟ ✍
➻✮✾ ✱❃❂
➟ ★
✠✱✰ ➟
◆ ➻✿✾
✱❄❂ ✶➁✵
➻✿✾ ✱❄❂
✴ ❬
Finally, one should notice that the index of a Farey fraction
➄
is the integer that reduces the fraction that gives
➄
as the mediant of its neighbor fractions in
❪➦❫
. Indeed, the index satisfy the equalities:
❅ì✝ ➄
✷ ✟➦✠❇❆
✚ ➪
✷ ❙❳❘
➪ ✷
❈ ✠
❂ ❘
ð ➪
✷ ❙❳❘
✍ ➪
✷ ô
✠ ➪
✷ ◆
❘ ✚
➪ ✷
❙❳❘ ➪
✷ ✠❊❉
✷ ◆
❘ ✚
❉ ✷
❙❳❘ ❉
✷ ❬
The index has interesting properties. Hall and Shiu [HS2001] discovered some very re- markable exact formulae involving the index of Farey fractions, and they also proved a
number of asymptotic results. Boca, Gologan and Zaharescu [BGZ2002] found asymp- totic formulae for the moments of the index. They proved that
â â
â â
â ✼
❋❍●✡■ ❫
❅ì✝ ➄❳✟✆❏
❽ ✜✤❑
❏▼▲ ✝
✟ â
â â
â â
ÿ ❏❖◆P
◗
P❘ ò
❋ ➴
✍
for
❙ ✘
☞✎✍ ò
❋ ➴
❀ ✍
for
❙②✠ ☞✎✍
❏ ò
❋ ➴
✍
for
☞✗✘ ❙
✘ ✜
❬
5.1 Here
❑ ❏
is a constant given by
❑ ❏
✠❯❚❱❚ ❲
❆ ☞
✚❨❳ ❩
❈ ❏❭❬
❳ ❬
❩ ✠❫❪
✼ ✽✿✾
❘ ❂
❏❵❴ ●
í✰ê ✝
➼ ✽
✟✕ÿ ❪
✼ ✽✿✾
❘ ❂
❏ ❙
Ù ✘
✄ ❬
The moment of order
✜
was calculated in [HS2001] with an error term of size
ó ✝
ò ❋
➴ ❀
✟
. Similar formulae for moments of order
❙✫ä ✝
✓ ✍❢❊✴❈
✜✴✟
were also established for the Farey fractions from a subinterval of
❻ ✓
✍✰☞✲❼
. In [HS2001] and [BGZ2002] it is also investigated the twisted sum
❛✗❜ ✸
❝ ✝
✟✕✠ ✼
❋✆✾❞●✸■ ❫❢❡❵❣
➨ ✸
❝✐❤ ❅
➟ ✝
➄ ✷
✟ ❅
➟ ✝
➄ ✷
◆ ❜
✟ ✍
where
✓ ➎
❩ ➎❇☞
. Boca, Gologan and Zaharescu [BGZ2002] proved that for any integer
á ❚
☞
and
✓ ➎
❩ ➎❇☞
, we have
❛✗❜ ✸
❝ ✝
✟➦✠ ❩✆❥
✝ á✣✟
▲ ✝
✟þ✚✉ó ❜
✸ ❦
✖ Ù♠❧Õ❀
◆ ❦
★ ✍
where
❥ ✝
á✣✟ ✠
✜❱♥❃♥ ❲
❂ ❘
✝ ❳
✍ ❩
✟ ❂
❜ ◆
❘ ✝
❳ ✍
❩ ✟
❬ ❳
❬ ❩
are rational positive constants of size
ÿ ☞
✚❏ò ❋
➴ á
. In the case
❩ ✠
☞
, they get a better error term of size
ó ✝
ò ❋
➴ ❀
✟
.
C. Cobeli, A. Zaharescu — The Haros-Farey Sequence at Two Hundred Years
5.3. The distribution of lattice points visible from the origin. Let