THE NUMBER OF REPRESENTATIONS OF RATIONALS AS A SUM OF UNIT FRACTIONS

Transcription

1 THE NUMBER OF REPRESENTATIONS OF RATIONALS AS A SUM OF UNIT FRACTIONS T.D. BROWNING AND C. ELSHOLTZ Abstract. For give positive itegers ad, we cosider the frequecy of represetatios of as a su of uit fractios. 1. Itroductio This paper cetres o the questio of represetig fractios as sus of uit fractios. Specifically, for a positive iteger k 2 ad give, N, we would like a better uderstadig of the coutig fuctio f k (, ) = # {(t 1,..., t k ) N k : t 1 t k ad } = 1t tk. We will be aily cocered with upper bouds for f k (, ) which are uifor i k, ad. O observig the trivial upper boud f k (, ) f k (1, ), we will geerally be iterested i bouds for f k (, ) that get sharper as the size of icreases. The easiest case to deal with is the case k = 2, for which we have the followig essetially coplete descriptio. Theore 1. We have ( log ) f 2 (, ) exp (log + o(1)). log log Furtherore, for fixed N, there are ifiitely ay values of for which ( log ) f 2 (, ) exp (log + o(1)). log log Whe k = the equatio appearig i the defiitio of f (, ) has received uch attetio i the cotext of the cojecture 1 of Erdős ad Straus [5]. This predicts that f (4, ) > 0 for ay 2. The cojecture has sice bee geeralised to arbitrary uerators by Schizel [10]. Thus for ay 4 oe expects the existece of N N such that f (, ) > 0 for N. Both of these cojectures are still wide ope ad have geerated a lot of attetio i the literature. A overview of the doai ca be foud i work of the secod author [4]. The followig result provides a upper boud for f (, ) which is uifor i ad. Date: April 24, Matheatics Subject Classificatio. 11D68. 1 The earliest referece i the literature to this cojecture appears to be a paper by Obláth [7], subitted i

2 2 T.D. BROWNING AND C. ELSHOLTZ Theore 2. For ay ε > 0 we have f (, ) ε ( ) 2 ε. It follows fro the theore that f (, ) ε 2 +ε. Nuerical experietatio reveals that f (, ) varies cosiderably as varies but oetheless ought to correspod to a superpositio of divisor fuctios. Ideed we would cojecture that f (, ) ε ε for ay ε > 0. Moreover our uerical ivestigatios lead us to expect that f (, ) as, for fixed. Oce the deoiators are cleared the equatio appearig i f (, ) takes the shape xyz = (xy + xz + yz). This is oe of several affie cubic equatios for which the uber of solutios i positive itegers is expected to grow like the divisor fuctio. I private couicatio with the authors Bria Corey has asked whether the uber of solutios i positive itegers to the equatio = xyz + x + y ca be bouded by O ε ( ε ) for ay ε > 0. Kevi Ford 2 has posed a geeralisatio of this proble, i which oe would like to show that there are O ε ( AB ) ε ) o-trivial positive iteger solutios to the equatio xyz = A(x + y) + B, for give o-zero A, B Z. A further proble of this type has bee posed by Pellig, i which it is asked whether there are O ε ( ε ) solutios to the cubic equatio xyz = (x + y + z), with x, y, z N. For this equatio it is kow that the relevat coutig fuctio grows at ost like O ε ( 1 2 +ε ) but the origial questio is ope. We shall ot say aythig ore about these equatios here. Recordig aythig eaigful for f k (, ) whe k 4 sees to be a harder proble. Noetheless we are able to build Theore 2 ito a iductio arguet which leads us to the followig result. Theore. Let k 4. For ay ε > 0 we have ad for k 5 f 4 (, ) ε ε{( f k (, ) ε (k) ε( k }, 2k 5. The special case f k (1, 1) has received special attetio i the literature. I oe directio Croot [2] has solved a difficult proble of Erdős by showig that ay fiite colourig of the positive itegers allows a oochroatic solutio of the equatio k 1 1 = (1) t i for uspecified k. I a differet directio, for give k N, let K(k) = f k (1, 1) deote the uber of vectors (t 1,..., t k ) N k with t 1 t k, for which (1) 2 First preseted at the DIMACS Meetig i Rutgers i Proble 10745, Solutio i: Aerica Math. Mothly, 108 (2001),

3 SUMS OF UNIT FRACTIONS holds. Defie the sequece u via u 1 = 1 ad u +1 = u (u + 1). This sequece grows doubly expoetially ad oe has c 0 = li u 2 = Buildig o earlier work of Erdős, Graha ad Straus [6], Sádor [8] has established the upper boud K(k) < c (1+ε)2k 1 0 for ay ε > 0 ad ay k k(ε). Takig = = 1 i Theore we deduce the followig estiate. Corollary. For ay ε > 0 we have K(k) ε k 5 9 2k +ε. For iterediate k this iproves upo Sádor s result. By revisitig Sádor s arguet we achieve the followig sharpeig for large k. Theore 4. Let ε > 0 ad assue that k k(ε). The we have K(k) < c ( ε)2k 1 0. While iterestig i its ow right it traspires that the study of Egyptia fractios has applicatios to various probles i topology. For exaple, Breto ad Hall [1] have established a bijectio betwee solutios (t 1,..., t k ) N k to the equatio k 1 k 1 1 = + t i t i ad hoeoorphis equivalece classes of hoologically trivial coplex surface sigularities whose dual itersectio graph is a star with cetral weight 1 ad weights t i o the ars. I [1, 4] the authors ask for a better uderstadig of the coutig fuctio S(k) for large k, which is defied to be the uber of solutios (t 1,..., t k ) N k to the above equatio with t 1 t k. O observig that S(k) K(k + 1) we observe the followig trivial cosequece of Theore 4. Corollary. Let ε > 0 ad assue that k k(ε). The we have S(k) < c ( ε)2k 0. Ackowledgeet. While workig o this paper the first author was supported by EPSRC grat uber EP/E05262/1. 2. Sus of two uit fractios I this sectio we establish Theore 1. Begiig with the upper boud, Sádor [8, Lea 4] has show that f 2 (, ) f 2 (1, ) = 1 2 (d(2 ) + 1), 1 where d deotes the divisor fuctio. To see this we ote that if = 1 t t 2 the t 2 = t1 t = t, which is a iteger if ad oly if t The coditio t 1 t 2 esures that t 1 2, so that 0 < t 1 ad ideed f 2 (1, ) = 1 2 (d(2 ) + 1). Applyig work of Shiu [9] o the axiu order of ultiplicative fuctios we easily deduce the upper boud i Theore 1.

4 4 T.D. BROWNING AND C. ELSHOLTZ We ow tur to the lower boud for f 2 (, ) for fixed N. It will suffice to exaie g 2 (, ), which is defied as for f 2 (, ), but without the restrictio that t 1 t 2 i each solutio. Ideed we plaily have g 2 (, ) 2f 2 (, ). Let = s q i, where s is odd ad q i deotes the ith prie which is cogruet to 1 od. The we clai that g 2 (, ) s 2. To see this, let x 1 be the product of ay subset of a odd uber i of the s prie factors. Let x 2 be a product of a eve uber j of the reaiig s i prie factors. The x 12 = x 1+x 2 x 1x 2 is a iteger ad we have = x 1 x 12 x 2 x 12 Coutig up the uber of available x 1, x 2 gives the cotributio s i ( )( ) s s i ( ) s S 1 = = 2 s i 1. i j i i odd j eve i odd Likewise we ca istead choose x 1 to cosist of a eve uber i of the s pries, ad x 2 a odd uber j of the reaiig s i pries. This gives the cotributio s i ( )( ) s s i ( ) s S 2 = = 2 s i 1. i j i Thus we deduce that i eve j odd g 2 (, ) S 1 + S 2 = i=0 i eve ( s )2 s i 1 = s i 2, as required. To coplete the proof of the theore we ote that ( s = q i = exp log q i ). By the prie uber theore for arithetic progressios log q i log(i(log i)ϕ()) s log s + s log log s + s log ϕ(). It follows that s log log log +log ϕ() log log log, for fixed. Therefore, there are at least 1 4 s = exp ( (log +o(1)) log log log ) solutios couted by f2 (, ), which thereby copletes the proof of Theore 1.. Sus of three uit fractios I this sectio we establish the upper boud i Theore 2 for f (, ). It will clearly suffice to assue that gcd(, ) = 1. Sice t 1 t 2 t i the defiitio of the coutig fuctio it is clear that < t 1. (2)

5 SUMS OF UNIT FRACTIONS 5 I particular we ust have. We ca get a upper boud for t 2 via the expressio 1 = t 1 t 2 t t 2 Suppose that <. Let = q + r for 0 < r 1. We have t 1 = q + 1 ad it follows that the left had side is at least q + r 1 q (q + 1)(q + r) 2(q + r)(q + r) = 2 2 2, givig t () Suppose ow that >, with. The we have t 1 = 1, whece 1 1 2, whece () holds i this case also. Oce cobied with the uderlyig equatio i f (, ) the iequalities (2) ad () are eough to show that f (, ) 6 2 = Proceedig to the proof of the sharper boud i Theore 2, we ay heceforth assue that t 1, t 2 satisfy t 1 t 2 ad lie i the rages give by (2) ad (), i ay give solutio (t 1, t 2, t ) N couted by f (, ). I what follows let i, j, k deote distict eleets fro the set {1, 2, }. Let with x ij = x ji. The with x 12 = gcd(t 1, t 2, t ), x ij = gcd(t i, t j ) t i x i =, x 12 x ij x ik x 12 t 1 = x 1 x 12 x 1 x 12, t 2 = x 2 x 12 x 2 x 12, t = x x 1 x 2 x 12, (4) gcd(x i x ik, x j x jk ) = 1. (5) Substitutig these values for t 1, t 2, t ito the equatio i the defiitio of f (, ), we obtai x 1 x 2 x x 12 x 1 x 2 x 12 = (x 1 x 2 x 12 + x 1 x x 1 + x 2 x x 2 ). It follows fro (5) that x 1 x 2 x. Sice gcd(, ) = 1, we ay coclude that = x 1 x 2 x h 12 h 1 h 2 h 12, (6) where ( ) ( ) h ij = gcd, x ij, h 12 = gcd, x 12. x 1 x 2 x x 1 x 2 x If we write x ij = h ij y ij ad x 12 = dh 12, the we obtai the siplificatio dy 12 y 1 y 2 = x 1 x 2 h 12 y 12 + x 1 x h 1 y 1 + x 2 x h 2 y 2. (7) Furtherore, we have the additioal copriality relatios gcd(y ij, h ik h jk h 12 ) = gcd(d, h ij ) = 1. Thus (5) ad (7) iply that ay two eleets of the set {x 1, x 2, x, d} ust be coprie.

6 6 T.D. BROWNING AND C. ELSHOLTZ Let D > 0. It will be coveiet to cosider the overall cotributio to f (, ) fro x 1, x 2, x, d, h ij, h 12, y ij such that that d is costraied to lie i a iterval D d < 2D. We will write F (, ; D) for this quatity. It follows fro (2), () ad (4) that y 12 y 1 = x 1x 12 x 12 x 1 x 1 x 12 h 12 h 1 = t 1 x 1 h 12 dh 12 h 1 ad siilarly 6 2 y 12 y 2 x 2 h 12 h 2 h 12 D. We proceed to estiate F (, ; D) i two differet ways. Lea 1. For ay ε > 0 we have F (, ; D) ε 1+ε D. Proof. It follows fro (7) that there exists a iteger r such that y 2 r = x 2 h 12 y 12 + x h 1 y 1. x 1 h 12 h 1 h 12 D, (8) For fixed x 2, x, h 12, h 1, y 12, y 1 the trivial estiate for the divisor fuctio iplies that there are O ε ( ε ) choices for y 2, r. Suig over y 12, y 1 we coclude fro (8) that there are O ε ( 1 D 1 1+ε ) choices for the y ij ad r. A choice of d is fixed by (7). Sice there are O ε ( ε ) possible choices for x 1, x 2, x, h ij, h 12, by (6), so it follows that F (, ; D) ε 1+ε D ε x 1,x 2,x,h ij,h ε D. The stateet of the lea follows o redefiig the choice of ε > 0. Lea 2. For ay ε > 0 we have D ε F (, ; D) ε. Proof. Assue without loss of geerality that y 12 y 1. Fixig y 12, we the estiate the uber of itegers A, B 2 for which 1 2 dy 12 AB = x 1 x 2 h 12 y 12 + x 1 x h 1 A + x 2 x h 2 B. But we ay rewrite this equatio as (dy 12 A x 2 x h 2 )(dy 12 B x 1 x h 1 ) = x 1 x 2 dh 12 y x 1 x 2 x 2 h 1 h 2. For each x 1, x 2, x, d, h ij, h 12, y 12, there are clearly O ε ( ε ) possible values of A, B, by eleetary estiates for the divisor fuctio. Moreover, (8) ad the assuptio y 12 y 1 together iply that y 12 D. Thus we obtai the boud 1 2 +ε D ε F (, ; D) ε (D) 1 ε, x 1,x 2,x,d,h ij o suig over values of d i the rage D d < 2D, ad the O ε ( ε ) possible values of x 1, x 2, x, h ij, h 12 for which (6) holds. The lea follows o redefiig the choice of ε > 0.

7 SUMS OF UNIT FRACTIONS 7 We are ow ready to coplete the proof of the theore. There are O(log ) possible dyadic rages for d, such that d. Theore 2 therefore follows o applyig Lea 1 to deal with the cotributio fro d ( ) 1, ad Lea 2 to hadle d < ( ) Sus of k uit fractios I this sectio we establish Theores ad 4. Begiig with the forer, let (t 1,..., t k ) N k be a poit with t 1 t 2 t k couted by f k (, ). The t 1 t 1 = 1 t 1 = 1 t t k. It is easy to see that f k (, ) = 0 uless k which we ow assue. Furtherore the aalogue of (2) i the precedig sectio is clearly < t 1 k. (9) Our iductio is based o the observatio that f k (, ) t 1 f k 1 (t 1, t 1 ), where the suatio is over t 1 N for which (9) holds. Makig the chage of variables u = t 1 we obtai ( (u + ) ) f k (, ) f k 1 u,. (10) 0<u (k 1) u+ Note that u + k for each u uder cosideratio. Let ε > 0. We begi by establishig the theore i the case k = 4. It follows fro Theore 2 that f 4 (, ) ε ε 0<u u+ ( (u+) ) 2 4 +ε ε u 2 0<u u+ Give θ [0, 1) we ow require the estiate S θ (x) = θ = x1 θ (1 θ)q + O θ(1), x a od q u 2. which is valid uiforly for a Z ad q N. This follows fro cobiig partial suatio with the failiar estiate S 0 (x) = q 1 x + O(1). If θ 1 + δ for soe fixed δ > 0, the S θ (x) δ 1. We ay ow coclude that 4 +ε ( 1 ) f 4 (, ) ε (11) This establishes the theore i the case k = 4. Turig to the case k = 5 we repeat the above aalysis based o (10), but use the iequality i (11) as our boud for f 4 (, ). It follows that f 5 (, ) ε ε 0<u 4 u+ {( 1 2 u + ( 1 2 ) 4 u 2 } ε ε( 2,

8 8 T.D. BROWNING AND C. ELSHOLTZ which thereby establishes the theore whe k = 5. It reais to establish Theore for k 6. We will begi by showig that f k (, ) ε,k ε( 2 2k 5, (12) for k 5, where the iplied costat is allowed to deped o k. This will be achieved by iductio o k, the case k = 5 already havig bee dealt with. Whe k 6 we deduce fro the iductio hypothesis ad (10) that ( f k (, ) ε,k ε 2 (u + ) 2 2k 6 u 2 ε,k ε( 2 2k 5, 0<u (k 1) sice 5 2k 6 5 for k 6. This therefore establishes (12). We ow tur to a boud for f k (, ) which is uifor i k, which we will agai achieve via iductio o k. Let ε > 0. We will take for our iductio hypothesis the estiate f k (, ) ε (k) ε( k θ k 2 2k 5, (1) for a udeteried fuctio θ k. We ay heceforth suppose that 0<u (L 1) u+ k log log(5ε) log 2 + 5, (14) else (1) follows trivially fro (12). Now for ay L k it follows fro (10) that ( (u + ) ) ( (u + ) ) f k (, ) f k 1 u, + f k 1 u,. (L 1)<u (k 1) u+ Oe otes that u + L i the first su ad u + k i the secod. The iductio hypothesis therefore gives f k (, ) ε (k) ε k 5θ k 1 2 k 6{( L 2 ( 2k 5 k 2 } Σ 1 + 2k 5 Σ 2, where ( 1 Σ 1 = u)5 2k 6 1, Σ 2 = We deduce that 0<u (L 1) (L 1)<u (k 1) f k (, ) ε (k) ε( k θk 1 ( 1 u)5 2k 6 ( 1 u)5 2k 6 L 1 5 2k 6. u L 2 2 2k 5{ ( 1 2 L + k L)1 2 (k 5)} 5 5 2k 5. Now (14) esures that (k 5) 1 2 ε. Hece, o takig L = k 2, we coclude that f k (, ) ε k ε( k 4) ε( k θk k 5. Redefiig the choice of ε therefore leads us to the iductio hypothesis (1) with θ k = θ k

9 SUMS OF UNIT FRACTIONS 9 It is ow easy to deduce that θ k < 4, which copletes the proof of Theore. We ow tur to the proof of Theore 4, for which we will odify the arguet i [8]. Recall the defiitio of the sequece u fro the itroductio ad let c 0 = li u 2. Sice u 2 is ootoically icreasig we have u < c 2 0. Suppose that 1 = k 1 t i, with t 1 t k. The Curtiss [] has show that 1 1 1, t i u +1 for 1 k 1. It follows that t j (k j + 1)u j for each j sice otherwise k 1 j 1 1 k 1 1 = = + < k j + 1 = 1, t i t i t i u j (k j + 1)u j i=j which is a cotradictio. Let ε > 0 ad let L be chose to be the least positive iteger for which 2 5 L < ε 2. The uber of tuples (t 1,..., t k L ) with t j (k j + 1)u j is therefore k L (k j + 1)u j (k L)! j=1 k L j=1 c 2j 0 < k!c 2k L+1 0. For a give (t 1,..., t k L )-tuple, it reais to estiate the uber of vectors (t k L+1,..., t k ) that coplete the su k 1 t i = 1. We write 1 1 t 1 1 t k L =, where t 1 t k L < k!c 2k L+1 0. Applyig Theore we deduce that the uber of available (t k L+1,..., t k ) is at ost f L (, ) ε ( k!c 2 k L+1 0 2L 5 +ε ε (k! 2L 5 +ε c 5 2k 4 +ε 0, for ay ε > 0. Cobiig our two estiates we ay ow coclude that K(k) ε (k!) e L c 2k L+1 0 c 5 2k 4 +ε 0 ε (k!) e L c ( 5 +ε) 2k 4 0, where e L = L 5 + ε. This therefore cocludes the proof of Theore 4 o redefiig the choice of ε. Refereces [1] L. Breto ad R. Hill, O the Diophatie equatio 1 = P 1/ i + 1/ Q i ad a class of hoologically trivial coplex surface sigularities. Pacific J. Math. 1 (1988), [2] E.S. Croot III, O a colorig cojecture about uit fractios. Aals of Math. 157 (200), [] D.R. Curtiss, O Kellogg s Diophatie Proble, Aerica Math. Mothly 29 (1922), [4] C. Elsholtz, Sus of k uit fractios. Tras. Aer. Math. Soc. 5 (2001), = a egyelet egész száú egoldásairól (O a Diophatie b equatio). Mat. Lapok 1 (1950), [6] P. Erdős ad R.L. Graha, Old ad ew probles ad results i cobiatorial uber theory. Moographies de L Eseigeet Mathéatique 28, Geeva, [5] P. Erdős, Az 1 x x x [7] M.R. Obláth, Sur l équatio diophatiee 4 = 1 x x x. Mathesis 59 (1950), [8] C. Sádor, O the uber of solutios of the Diophatie equatio P 1 x i = 1. Period. Math. Hug. 47 (200), [9] P. Shiu, The axiu order of ultiplicative fuctios. Quart. J. Math. 1 (1980),

10 10 T.D. BROWNING AND C. ELSHOLTZ [10] W. Sierpiński, Sur les décopositios de obres ratioels e fractios priaires. Mathesis 65 (1956), School of Matheatics, Uiversity of Bristol, Bristol BS8 1TW, UK E-ail address: t.d.browig@bristol.ac.uk Istitut für Matheatik A, Techische Uiversität Graz, Steyrergasse 0, A-8010 Graz, Austria E-ail address: elsholtz@ath.tugraz.at