On the number of pairs of positive integers x,y H such that x 2 +y 2 +1, x 2 +y 2 +2 are square-free

Transcription

1 arxiv: v [math.nt] 5 Jan 09 On th numbr of pairs of positiv intgrs x,y H such that x +y +, x +y + ar squar-fr S. I. Dimitrov Abstract In th prsnt papr w show that thr xist infinitly many conscutiv squarfr numbrs of th form x +y +, x +y +. W also giv an asymptotic formula for th numbr of pairs of positiv intgrs x,y H such that x +y +, x +y + ar squar-fr. Kywords: Squar-fr numbrs, Asymptotic formula, Gauss sums. 00 Math. Subjct Classification: L05 N5 N37 Notations Lt H b a sufficintly larg positiv numbr. By ε w dnot an arbitrary small positiv numbr, not th sam in all apparancs. Th lttrs d,h,k,l,q,r,x,y with or without subscript will dnot positiv intgrs. By th lttrs D,D, H 0 and t w dnot ral numbrs and by m,n intgrs. As usual µn is Möbius function and τn dnots th numbr of positiv divisors of n. Furthr [t] and {t} dnot th intgr part, rspctivly, th fractional part of t. Instad of m n mod d w writ for simplicity m nd. Morovr t=xpπit and m,n is th gratst common divisor of m and n. Th lttr p will always dnot prim numbr. W put ψt = {t} /. For any odd q w dnot by q th Jacobi symbol. For any n and q such that n,q = w dnot by n q th invrs of n modulo q. If th valu of th modulus is undrstood from th contxt thn w writ for simplicity n. By Gq,m,n and Gq,m w shall dnot th Gauss sums Gq,m,n = q mx +nx, Gq,m = Gq,m,0. q x=

2 By Kq,m,n w shall dnot th Kloostrman sum Kq,m,n = q x= x,q= mx+n x. q Introduction and statmnt of th rsult Th problm for th conscutiv squar-fr numbrs ariss in 93 whn Carlitz [] provd that n Hµ nµ n+ = p H +O H θ+ε, 3 p whr θ = /3. Formula 3 was subsquntly improvd by Hath-Brown [5] to θ = 7/ and by Russ [8] to θ = /8. Rcntly th author [] showd that for any fixd < c < /3 thr xist infinitly many conscutiv squar-fr numbrs of th form [n c ],[n c ]+. Furthrmor assuming that th Gnralizd Rimann Hypothsis is tru th author [3] provd that thr xist infinitly many conscutiv squar-fr numbrs of th form p+,p+, whr p is prim. On th othr hand in 0 Tolv[0] provd ingniously that thr xist infinitly many squar-fr numbrs of th form x + y +. Mor prcisly h provd th asymptotic formula whr x,y H µ x +y + = ch +O H 4 3 +ε, c = p λp p 4 and Dfin and ΓH = x,y H λq = λq,q,m,n = x,y q x +y + 0q. µ x +y +µ x +y + 4 x,y:6 mx+ny, 5

3 whr th summation is takn ovr th intgrs x, y satisfying th conditions x,y x +y + 0q. 6 x +y + 0q W dfin also λq,q = λq,q,0,0. 7 Motivatd by ths rsults and following th mthod of Tolv [0] w shall prov th following thorm Thorm. For th sum ΓH dfind by 4 th asymptotic formula ΓH = σh +O H 8 5 +ε 8 holds. Hr σ = p λp,+λ,p. 9 p 4 From Thorm it follows that thr xist infinitly many conscutiv squar-fr numbrs of th form x +y +, x +y +. 3 Lmmas Th first lmma w nd givs us th basic proprtis of th Gauss sum. Lmma. For th Gauss sum w hav i If q,q = thn ii If q,m = d thn G,m q +m q,n = Gq,m q,ngq,m q,n. Gq,m,n = { dgq/d,m/d,n/d if d n, 0 if d n. iii If q,m = thn m 4mn Gq,m,n = Gq,. q q iv If q, = thn G q, = q q. 3

4 Proof. S [4] and [6]. Th nxt lmma givs us A. Wil s stimat for th Kloostrman sum. Lmma. Kq,m,n τqq q,m,n. Proof. S [7]. Th nxt lmma is th cntral momnt in th proof of Thorm. Right hr w apply th proprtis of th Gauss sum and A.Wil s stimat for th Kloostrman sum. Lmma 3. Lt 8 q, 8 q and q,q =. Thn for function dfind by 5 th stimation λq,q,m,n 6τ q q,m,n 0 holds. In particular w hav λq,q +ε. Proof. Cas.. Using, 5, 6 and Lmma w gt λq,q,m,n = = = = h q = x,y h q h q h q h q l q l q h h q h,q = q l q h q h,q = q l mx+ny h q h q h q h x +y + q Gq,h q,mgq,h q,n Gq,h q,mgq,h q,n h q h q h q h x +y + G,h q +h q,mg,h q +h q,n Gq,h q,mgq,h q,n Gq,h q,mgq,h q,n. q 4

5 Baring in mind,, and Lmma w obtain r λq,q,m,n = Gl,r q,ml q Gl,r q,nl q l q q l m,n l q q l m,n l = l q q l m,n l q q l m,n = l q q l m,n l r l r,l = G l, l l q q l m,n l r l r,l = G l, l l r l r l r,l = l From 3 and Lmma it follows that thn l l λq,q,m,n r l r,l = Gl,r q,ml q Gl,r q,nl q r 4r qm +n l q l r 4r q m +n l q l Kl,,4q m +n l q Kl,,4q m +n l q. 3 l q q l m,n τ τl l r q,m,n l q q l m,n τl l q r r q,m,n q r τ q q,m,n. 4 Cas. q = h q, whr q and h, and q. Th function λq,q,m,n dfind by 5 is such that, if q q,q q = q,q = q,q = λq q,q q = λ,m,n = q,q,mq q q,nq q q q q λ q,q,mq q q q,nq q q q. 5 5

6 Sinc th proof is lmntary w skip th dtails and lav it to th radr. Using 4, 5 and th trivial stimat λ h,,m,n 4 h w find λ h q,q,m,n = λ h,,mq q h,nq q h λ q,q,m h q q,n h q q 6τ q q q q q q,m,n 6τ q q,m,n. 6 Cas 3. q and q = h q, whr q and h. By 4, 5 and th trivial stimat λ, h,m,n 4 h w gt λq, h q,m,n = λ, h,m h,n h λ q,q,m h q q,n h q q 6τ q q q q q q,m,n 6τ q q,m,n. 7 Now th stimation 0 follows from 4, 6 and 7. As a byproduct of 0 w obtain. Lmma 4. Assum that 8 q, 8 q, q,q = and H 0. Thn for th sums Λ = λq,q,m,0, Λ = λq,q,m,n m mn m H 0 m,n H 0 8 th stimations Λ +ε H ε 0, Λ +ε H ε 0 9 hold. Proof. Using 8 and Lmma 3 w gt Λ +ε,m m m H 0 = +ε Λ 0, 0 whr W hav Λ 0 r r Λ 0 =,m m m H 0 m H 0 m 0r From 0 and follows th first inquality in 9.. m logh 0 r q q H 0 ε. r 6

7 Using 8, and Lmma 3 w obtain Λ +ε,m,n mn m,n H 0 +ε,m,n mn m,n H 0 = +ε Λ 0 q q +ε H ε 0, which provs th scond inquality in 9. Th final lmma w nd givs us important xpansions. Lmma 5. For any H 0, w hav ψt = m H 0 mt πim +O fh 0,t, whr fh 0,t is a positiv, infinitly many tims diffrntiabl and priodic with priod function of t. It can b xpandd into Fourir sris with cofficints b H0 m such that fh 0,t = + m= b H0 mmt, b H0 m logh 0 H 0 for all m and m >H +ε 0 b H0 m H A 0. Hr A > 0 is arbitrarily larg and th constant in th - symbol dpnds on A and ε. Proof. S [9]. 4 Proof of th thorm Using 4 and th wll-known idntity µ n = d n µd w gt µd µd = Γ H+Γ H, ΓH = d,d d,d = x,y H x +y + 0d x +y + 0d 7

8 whr Γ H = d d z d,d = Γ H = d d >z d,d = ΣH,d,d = µd µd ΣH,d,d, 3 µd µd ΣH,d,d, 4 x,y H x +y + 0d x +y + 0d, 5 H z H. 6 Estimation of Γ H At th stimation of Γ H w will suppos that q = d, q = d, whr d and d ar squar-fr, q,q = and d d z. Dnot ΩH,q,q,x = h H h x. 7 Apparntly Using 6, 5 and 7 w obtain ΩH,q,q,x = Hq q +O. 8 ΣH,q,q = x,y:6 On th othr hand 7 givs us [ ] [ ] H y y ΩH,q,q,y = From 9 and 30 w find whr ΣH,q,q = x,y:6 Σ = ΩH,q,q,xΩH,q,q,y. 9 = H y +ψ ψ H ΩH,q,q,x ψ x,y:6 y ΩH,q,q,xψ. W shall stimat th sum Σ. For th purpos w dcompos it as H y. 30 H y +Σ, 3 Σ = Σ +Σ, 3 8

9 whr Σ = x x + 0q x + 0q Σ = x x + 0q x + 0q ΩH,q,q,x ΩH,q,q,x y y 0 y y x q y x q y ψ, 33 y ψ. 34 Firstly w considr th sum Σ. W not that th sum ovr y in 34 dos not contain trms with y = q q and y = q q. Morovr for any y satisfying th congruncs and such that y < q q th numbr q q y satisfis th sam congruncs and w hav y ψ +ψ q q y = 0. Baring in mind ths argumnts for th sum Σ dnotd by 34 w hav that Σ = Nxt w considr th sum Σ. According to th abov considrations th sum ovr y in 33 rducs to a sum with at most two trms corrsponding to y = q q and y = q q. Thrfor Σ ΩH,q,q,x. 36 x x + 0q x + 0q Now taking into account 8, 36 and that th numbr of solutions of th congrunc x + 0q quals to Oq ε w gt Σ H ε Hq q From 3, 35 and 37 it follows Σ H ε Hq q By 3 and 38 w obtain ΣH,q,q = x,y:6 +O H ε H ΩH,q,q,x ψ H y Hq q

10 Procding in th sam way with th sum ΩH,q,q,x w find ΣH,q,q = H H x H H y ψ ψ x,y:6 +O H ε Hq q Baring in mind 5, 7 and 40 w gt ΣH,q,q = H λq,q q q H Σ H,q,q +Σ H,q,q +O H ε Hq q +, 4 whr Σ H,q,q = x,y:6 Σ H,q,q = x,y:6 H x ψ H x ψ, 4 H y ψ. 43 Firstly w considr th sum Σ H,q,q. Using 5 and Lmma 5 with H 0 = H w obtain Σ H,q,q = Σ H,q,q +O Σ H,q,q, 44 whr Σ H,q,q = Σ H,q,q = Formula 45 and Lmma 4 giv us x,y:6 m H H x m πim mh λq,q, m,0 =, 45 πim m H f H, H x. 46 x,y:6 Σ H,q,q H ε. 47 0

11 In ordr to stimat Σ H,q,q w us 5, 46 and Lmmas 3, 4, 5 and gt Σ H,q,q = H x b H 0+ b H m m +O x,y:6 m H +ε mh = b H 0λq,q + b H m λq,q, m,0 +O m H +ε H ε ++H ε λq,q, m,0 m H +ε H ε ++H ε λq,q,m,0 m m H +ε H ε +H ε. 48 From 44, 47 and 48 it follows Σ H,q,q H ε +H ε. 49 Nxt w considr th sum Σ H,q,q. Baring in mind 43, 46, 48 and Lmmas 4, 5 w find Σ H,q,q = m+nh mx+ny +O H ε Σ H,q,q = x,y:6 m, n H m, n H m, n H πi mn m+nh πi mn λq,q, m, n+o λq,q,m,n mn +H ε +H ε H ε +H ε H ε +H ε. 50 Taking into account 4, 49 and 50 w gt ΣH,q,q = H λq,q +O H ε Hq q q q +q q +H. 5 From, 3, 6 and 5 w obtain Γ H = H d d z d,d = µd µd λd,d d 4 d 4 +O H H ε +z +H z 3 = σh +O H ε H +z +H z 3, 5

12 whr σ = d,d = d,d = µd µd λd,d d 4 d 4 It is asy to s that th product 9 and th sum 53 coincid. Estimation of Γ H Using 4 w writ Γ H logh D d <D D d <D k H +d kd + 0d. 53, 54 x,y H x +y =kd whr D,D H +, D D z On th on hand 54 giv us Γ H H ε H ε H ε On th othr hand 54 implis D d <D k H +D D d <D k H +D D d <D k H +D D d <D τkd + l H +D kd +=ld H +ε D. 56 Γ H H ε H ε H ε D d <D l H +D D d <D l H +D D d <D l H +D D d <D τld k H +D kd =ld H +ε D. 57 By it follows Γ H H +ε z. 58

13 Th nd of th proof Baring in mind, 5, 58 and choosing z = H 4 5 w obtain th asymptotic formula 8. Th thorm is provd. Acknowldgmnts. Th author is spcially gratful to Prof. Tolv for numrous dirctions and advics for fiv yars. Th door to Prof. Tolv offic was always opn whnvr I had a qustion about my rsarch. This papr is ddicatd to Tolv s birthday annivrsary. Rfrncs [] L. Carlitz, On a problm in additiv arithmtic II, Quart. J. Math., 3, 93, [] S. I. Dimitrov, Conscutiv squar-fr numbrs of th form [n c ],[n c ]+, JP Journal of Algbra, Numbr Thory and Applications, 40, 6, 08, [3] S. I. Dimitrov, Conscutiv squar-fr numbrs of th form p +,p +, Far East Journal of Mathmatical Scincs, 07,, 08, [4] T. Estrmann, A nw application of th Hardy-Littlwood-Kloostrman mthod, Proc. London Math. Soc.,, 3, 96, [5] D. R. Hath-Brown, Th Squar-Siv and Conscutiv Squar-Fr Numbrs, Math. Ann., 66, 984, [6] L. K. Hua, Introduction to Numbr Thory, Springr, Brlin, 98. [7] H. Iwanic, E. Kowalski, Analytic numbr thory, Colloquium Publications, 53, Am. Math. Soc., 004. [8] T. Russ, Pairs of k-fr Numbrs, conscutiv squar-full Numbrs, arxiv:.350v [math.nt]. [9] D. I. Tolv, On th xponntial sum with squarfr numbrs, Bull. Lond. Math. Soc., 37, 6, 005, [0] D. I. Tolv, On th numbr of pairs of positiv intgrs x,y H such that x +y + is squarfr, Monatsh. Math., 65, 0,

14 S. I. Dimitrov Faculty of Applid Mathmatics and Informatics Tchnical Univrsity of Sofia 8, St.Klimnt Ohridski Blvd. 756 Sofia, BULGARIA -mail: 4