Jumping Champions of survey3.ps pages21 38

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