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1 This documnt is downloadd from DR-NTU, Nanyang Tchnological Univrsity Library, Singaor Titl Prims in uadratic rogrssions on avrag Authors) Bair, Sthan; Zhao, Liangyi Citation Bair, S & Zhao, L 007) Prims in uadratic rogrssions on avrag athmatisch Annaln, 3384), Dat 007 URL htt://hdlhandlnt/00/4558 Rights

2 PRIES IN QUADRATIC PROGRESSIONS ON AVERAGE STEPHAN BAIER AND LIANGYI ZHAO arxiv:math/ v5 [mathnt] 7 Jan 007 Abstract In this ar, w stablish a thorm on th distribution of rims in uadratic rogrssions on avrag athmatics Subjct Classification 000): L05, L07, L5, L0, L40, N3, N3, N37 Kywords: rims in uadratic rogrssions, rims rrsntd by olynomials Introduction and Statmnts of Rsults It was du to Dirichlt that any linar olynomial rrsnts infinitly many rims rovidd th cofficints ar co-rim Though long bn conjcturd, analogous statmnts ar not known for any olynomial of highr dgr G H Hardy and J E Littlwood [6] conjcturd that ) Λn + k) Sk)x, Λ is th von angoldt function and Sk) is a constant that dnds only on k Thir conjctur is in an uivalnt but diffrnt form as in ) orovr, thy also gav conjcturs rgarding th rrsntation of rims by any uadratic olynomial that may concivably rrsnt infinitly many rims In this ar, w aim to rov that ) holds for almost all suar-fr k y if x log x) A y x or in articular, w shall rov th following Thorm Givn A, B > 0, w hav, for x log x) A y x, ) Λn + k) Sk)x µ k)= ) with bing th Lgndr symbol Sk) = > From th thorm, w hav th following corollary ) yx = O log x) B, ) Corollary Givn A, B, C > 0 and Sk) as dfind in th thorm, w hav, for x log x) A y x, that ) 3) Λn x + k) = Sk)x + O log x) B holds for all suar-fr k not xcding y with at most O ylog x) C) xctions W not hr that if w st Lk) = > ), Dat: Fbruary, 008

3 STEPHAN BAIER AND LIANGYI ZHAO thn Not that Thrfor, w hav 4) Sk)Lk) = > + > ) ) ) ) = > ) ) ) + Sk)Lk) > ) It can asily b shown that th infinit roducts in both th majorant and minorant of th abov convrg absolutly to limits that ar indndnt of k orovr, it is wll-known that 5) Lk) 0 and Lk) log k, sinc Lk) is th valu of a Dirichlt L-function of modulus at most k at s = Thus, th inualitis in 4) and 5) imly that Sk) convrgs and Sk) log k log y log x Th abov inuality shows that th main trms in ) and 3) ar indd dominating for th k s undr considration if B > and that w indd hav an almost all rsult Our starting oint is th idntity 6) Λn + k) = 0 m z Λm)αm) αn + k))dα, z = x + y This idntity is a consunc of th orthogonality rlations for th function z) W us th circl mthod to study th ustion of intrst and mloy mthods dvlod by H ikawa [] in studying th twin rims conjctur on avrag ikawa s rsult on twin rims is an imrovmnt of arlir rsults of D Wolk [4] and A F Lavrik [0] and [9] As usual in th circl mthod, w slit th intgration intrval [0,] into major arcs and minor arcs W sarat th bginning of th so-calld singular sris from th major arcs contribution which will giv ris to th main trm W ar lft with th tail Φk) of th singular sris and crtain othr rror trms from th major arcs Th minor arcs contribution shall turn out to b an rror trm as wll Thn w stimat th scond momnts ovr th suar-fr numbrs k y of all ths rror trms Th scond momnt of Φk) is stimatd using th classical larg siv, larg siv for ral charactrs of Hath-Brown [7], th Pólya-Vinogradov inuality and a Sigl-Walfisz ty stimat To stimat th scond momnts of th othr rror trms, w us Bssl s and Cauchy s inualitis togthr with two imortant Lmmas, Lmma du to Gallaghr [4] and Lmma du to Wolk [4] and ikawa [] For th stimation of th minor arcs contribution w also nd a standard bound for uadratic xonntial sums du to Wyl W us th following standard notations and convntions in numbr thory throughout ar: Th symbol is rsrvd for rims z) = xπiz) = πiz f = Og) mans f cg for som unscifid ositiv constant c f g mans f = Og) Following th gnral convntion, w us ε to dnot a small ositiv constant which may not b th sam at ach occurrnc

4 PRIES IN QUADRATIC PROGRESSIONS ON AVERAGE 3 Prliminary Lmmas In this sction, w uot lmmas that w shall nd in th roofs of our thorm W bgin with th following Lmma Gallaghr) Lt < < N/ and N < N N For arbitrary comlx numbrs a n, w hav N a n βn) dβ a n dt, β N<n N N / th imlid constant is absolut Proof This is Lmma in [4] in a slightly modifid form max{t,n}<n min{t+ /,N } W shall also nd th following lmma in our stimats of th rror trms Lmma Wolk, ikawa) Lt J, ) = χ mod N N # t<n t+ χn)λn) dt, th # ovr th summation symbol hncforth mans that if χ = χ 0 thn χn)λn) is rlacd by Λn) Lt ε, A and B > 0 b givn If log N) B and N /5+ε N ε, thn w hav J, ) ) Nlog N) A, th imlid constant dnds only on ε, A and B Proof This is from [] and is Lmma thr It can b rovd using th tools in [4] W shall also nd th following wll-known inuality Lmma 3 Bssl) Lt φ,, φ r b orthonormal mmbrs of an innr roduct sac V ovr th comlx numbrs and ξ V Thn R ) ξ, φ r ) ξ r= Proof This is a standard rsult S for xaml [5] for a roof To stimat th contribution on th minor arcs, w nd th following lmma du to Wyl Lmma 4 Wyl) Givn x and with gcda, ) =, w hav th imlid constant is absolut Proof S xrcis on ag 5 of [] a α a +, αn ) log x x / + x) /), W shall also nd th following wll-known rsults in analytic numbr thory Lmma 5 Pólya-Vinogradov) For any non-rincial charactr χ mod ) w hav χn) 6 log <n +N Proof This is uotd from [8] and is Thorm 5 thr

5 4 STEPHAN BAIER AND LIANGYI ZHAO For comltnss, w also uot th classical larg siv inuality for Dirichlt charactrs Lmma 6 Larg Siv) Lt {a n } b an arbitrary sunc of comlx numbrs and Q,, N b intgrs with Q, N > 0 Thn w hav Q +N +N a n χn) Q + N) a n, ϕ) = χ mod n=+ n=+ mans that th sum runs ovr rimitiv charactrs modulo th scifid modulus only Proof S for xaml [], [3], [] or [3] for th roof W shall nd th larg siv for ral charactrs for th stimat of crtain trms in th major arcs contribution Lmma 7 Hath-Brown) Lt and N b natural numbrs and lt a,, a n b arbitrary comlx numbrs Thn m) n a n N) ε + N) a n, n N m n N for any ε > 0, th sums ovr m and n run ovr th suar-fr numbrs Proof This is Thorm in [7] 3 Th ajor Arcs In this sction and nxt, w considr th contribution of th major arcs dfind by 3) = J,a, Q a= gcda,)= [ a J,a = Q, a + ], Q = log x) c, Q = x ε Q for som c > 0 fixd and suitabl If x is sufficintly larg, thn Q > Q and so th intrvals J,a with Q ar disjoint W will assum that this is th cas throughout th sul Lt For α, w writ S α) = m z α = a + β, with β Q Λm)αm) and S α) = αn ) W trat ths sums in a mannr similar to thos tratd in [] W hav 3) S α) = m z Λm) ) a m βm) = m z gcdm,)= ) a Λm) m βm) + Olog z) ) Not that du to th rsnc of Λm), th contribution from th trms with gcdm, ) > only coms from thos m s that ar owrs of rims dividing which can b absorbd into th O-trm abov It is also notworthy that th imlid constant in 3) is absolut It is lmntary to not that if gcdam, ) =, w hav ) a 33) m = ϕ) χ mod χam)τχ),

6 PRIES IN QUADRATIC PROGRESSIONS ON AVERAGE 5 τχ) := ) n χn) n= is th Gauss sum and ϕ) is th Eulr ϕ function W thus gt that th first trm in 3) is τχ)χa) χm)λm)βm) ϕ) χ mod m z = µ) βm) + Λm) )βm) + τχ)χa) χm)λm)βm) ϕ) ϕ) m z m z χ mod m z χ χ 0 = µ) ϕ) m z = T α) + E α), βm) + ϕ) χ mod τχ)χa) # χm)λm)βm) m z say, µ) is th öbius µ function and th maning of th # ovr th summation symbol is th sam as that in Lmma W arriv at W trat S α) in a similar way, using 33), S α) = a ) n βn ) = ϕ d ) Hnc, w gt th following χ mod χ =χ 0 S α) S α) = T α) + E α) + Olog z) ) n = n d, = d, d = = d ϕ ) + d χ mod χ =χ 0 ϕ ) = T α) + E α), χ mod χ =χ 0 χ mod τχ)χ ad ) gcdn,)=d d gcdd, ), and = gcdd, ) τχ)χ ad ) χ mod χ χ 0 gcdn,)=d τχ)χ ad ) βn ) gcdn,)=d χ n ) βn ) χ n ) βn ), say Lt G = Z/Z) and G = {g : g G} Thn it is asy to obsrv s for xaml ag 44 of [8]) that τχ)χ ad ) = ) b χ ad ) χb) = [ G : G ] ) b b mod, ad b mod gcdb, )= th notation n mod mans that n is congrunt to a suar modulo orovr, w not that ad b mod gcdb, )= ) b = [G : G ] l= gcdl, )= ad l ) = [G : G ] l= gcdl, )= ad l uon noting that ad l ad ad l) mod ), a is th multilicativ invrs of a modulo ),

7 6 STEPHAN BAIER AND LIANGYI ZHAO Consuntly, w hav Furthrmor, w hav 34) T α) = d Λm)αm) m z = ϕ ) l= gcdl, )= αn + k))dα ad l ) gcdn,)=d βn ) T α) + E α) + Olog x) ))T α) + E α)) αk)dα 4 Th Singular Sris W first considr th main trm which will b givn by th following 4) T α)t α)α)dα = µ) a ) ϕ) k Q d and Π,d β) = m z βm) Ga, ) = a= gcda,)= gcdn,)=d l= gcdl, )= βn )β) ad l Th intgral on th right-hand sid of 4) is wll-aroximatd by / 4) βm) βn )β)dβ + O 0 m z /Q) β gcdn,)=d ) Ga, ) ϕ ) gcdn,)=d β Q Π,d β)dβ, βn )β) dβ, w hav usd th bound for th gomtric sum ovr m in th O-trm abov Alying Cauchy s inuality and Parsval s idntity, th O-trm in 4) is 43) / /Q) ) β dβ 0 gcdn,)=d Th first trm in 4) is, by orthogonality of z), 44) = = m z gcdn,)=d m=n +k gcdn,)=d n x/d gcdn,/d)= βn )β) dβ = ϕ/d) x /d d Now combining 4), 4), 43) and 44), 4) bcoms µ) 45) a ) ϕ) k Ga, ) ϕ/d) ϕ x + O ) Q d a= gcda,)= Q x ) d ϕ/d) + O ϕ/d)) = x + O ϕ/d)) Q x d ) )) Not that ϕ/d) is much smallr than and hnc ngligibl in comarison with th O-trm in 45)

8 PRIES IN QUADRATIC PROGRESSIONS ON AVERAGE 7 Now du to th rsnc of µ) in 45), it suffics to considr only thos s that ar suar-fr In that cas, w hav d = d and = = /d Thrfor, 45) bcoms 46) x µ) ϕ) Q a= gcda,)= a ) k d for som fixd c > 0 It can b asily obsrvd that d Thus th first trm in 46) bcoms /d l= gcdl,/d)= /d l= gcdl,/d)= a ) ld) = 47) x µ) ϕ) Σ), Q For rims, w hav Σ) = Σ) = r= a= r= a= gcda,)= a ) ) ld) + O xqlog x) c, r= a ) r a ) k + r ) a ) ) k + r ) = r= r +k) From this it follows that { 0 ) if =, 48) Σ) = if >, ) ) is th Lgndr symbol Hr w hav usd that + is th numbr of solutions to th congrunc rlation x + k 0 mod ) if is an odd rim It can also b sn that Σ) is multilicativ in th following way Givn and with gcd, ) =, w hav Σ ) = a ) k + r ) ) and similarly Σ ) = r = a = gcda, )= r = a = gcda, )= a ) k + r ) ) sinc, by corimality of,, if r and r run ovr all rsidu classs modulo and rsctivly, thn so do r and r rsctivly Hnc Σ )Σ ) = r = a = r = a = gcda, )= gcda, )= fk, a, a,,, r, r )), fk, a, a,,, r, r ) = a + a a r ) + a r )

9 8 STEPHAN BAIER AND LIANGYI ZHAO Not that th abov is a + a a + a ) r + r ) mod ) As a and a run ovr th rimitiv rsidu classs modulo and rsctivly, a + a runs ovr th rimitiv rsidu classs modulo ; and as r and r run ovr th rsidu classs modulo and rsctivly, r + r runs ovr th rimitiv rsidu classs modulo Thrfor, w hav that Q Σ )Σ ) = Σ ) In othr words, Σ) is multilicativ This fact, togthr with 48), givs that if is suar-fr, thn { 0 ) if, Σ) = if, ) is now th Jacobi symbol From th abov, w infr that 47) is 49) x ) µ) = Sk)x + Ox Φk) ), ϕ) Sk) = = µ) ϕ) ), Φk) := µ) ϕ) >Q ) It is asy to show that th so-calld singular sris Sk) can b rwrittn as an Eulr roduct: ) 40) Sk) = > W now infr from 46), 49) and 40) that 4) T α)t α)α)dα = Sk)x + O x Φk) + ) xqlog x) c 5 Th Estimat of th Scond omnt of Φk) In this sction, w stimat th scond momnt ovr th suar-fr numbrs k y of th tail Φk) of th singular sris, that is, w stimat Φk) µ k)= Throughout this sction, all sums ovr ar rstrictd to odd Th abov sum is majorizd by 5) ) µ) + ) µ) ϕ) + ) µ) Q < U ϕ), U< v U ϕ) > v U µ k)= with ral numbrs U > Q and v to b chosn latr Th first trm in 5) is tratd as ) µ) = ϕ) Q < U Q <, U y Q < U µ )µ ) ϕ )ϕ ) µ ) ϕ ) + ) ) Q <, U µ )µ ) ϕ )ϕ ) / gcd, ) )

10 Th first trm abov is and th scond trm, by Lmma 5, is Q <, U with som c > 0 Thrfor, w hav 5) PRIES IN QUADRATIC PROGRESSIONS ON AVERAGE 9 y Q y log x) c ϕ )ϕ ) log ) log U Q < U µ) ϕ) ) Q < U ϕ) y log x) c + Ulog U)5 Ulog U) c, To stimat th scond trm in 5), w us both th classical larg siv inuality, Lmma 6, and th larg siv for ral charactrs of Hath-Brown [7], Lmma 7 Using Cauchy s inuality, w hav ) µ) [v+] v T r, ϕ) Lmma 7 givs that µ k)= T r := U< v U µ k)= r U< r U µ) ϕ) r= T r Uy) ε rε + U ε y ε+ rε ) Summing th abov ovr r with r U y +ε w gt that 53) T r Uy) ε rε + U ε y ε+ r R r R rε ) RUy R ) ε + U ε y ε+, r r U y +ε It is now asily obsrvd that 53) is y R = [log +ε ] U 54) y 3ε + U ε y ε+ Using th classical larg siv inuality, Lmma 6, w hav 55) T r r U + y ) r U W nd not worry about th rimitivity of charactrs that is ruird by Lmma 6, sinc it is wll-known that if k 3 mod 4) and is suar-fr thn ) ) ) is rimitiv and = k k ) ) is rimitiv; if k mod 4) and is suar-fr thn ) ; and if k is suar-fr and vn thn th Jacobi symbol is of conductor k/ S, for xaml, 5 of [] Hnc ach rimitiv charactr aars at most a boundd numbr of tims in T r W may brak th summation ovr k into thr ics according to whthr k, or 3 mod 4) and thn aly Lmma 6 to ach of th rsulting ics Summing 55) ovr r with y +ε < r U and r [v + ], w obtain that 56) r [v+] y +ε < r U T r v + y ε

11 0 STEPHAN BAIER AND LIANGYI ZHAO Lt W = v U For th third trm in 5), w gt ) µ) 57) x c ) log W ϕ) >W if k y log W) /ε by using a similar argumnt as in th roof of th classical Sigl-Walfisz thorm s [], Satz 333) W not hr that if k is vn, thn th rstriction of bing odd on th sum ovr on) th lft-hand sid of 57) can b rmovd with no chang to th valu of th sum, du to th fact that is a charactr modulo ) k; and if k is odd, thn th sum on th lft-hand sid) of 57) rmains unaltrd if th charactr is rlacd by th charactr modulo k inducd by and th sum ovr is xtndd ovr all s, both vn and odd In both cass, a Sigl-Walfisz ty argumnt yilds th stimat in 57) Now w st x y ε/) v = log, U = y U and assum without loss of gnrality that ε /5 Thn w hav v y ε/, W = x y ε/), y = log W) /ε, and from 54), 56) and 57), w gt that th sum of scond and third trms in 5) is v y 3ε + U ε y ε+ + v + y ε) y ε/ Thrfor, combining 5) and th abov, w gt that 5) is 58) y log x) c 6 Th Error Trms from th ajor Arcs W considr th scond momnt ovr k of th rmaining trms in 34) trm by trm First, by Bssl s inuality, Lmma 3, w hav T α)e α)α)dα T α)e α) dα su T α) E α) dα α 6) z E α) dα Now, to stimat th scond factor of th abov, w hav 6) E α) dα = Ω β) := d φ ) χ mod χ χ 0 Q a= gcda,)= τχ)χad ) β Q gcdn,)=d Ω β)dβ, χ n ) βn ) Alying Cauchy s inuality to Ω β) aftr braking th sum ovr n into dyadic intrvals of th form N < n N N x and using th fact that τχ) =, w gt that in ordr to stimat 6), it suffics to stimat 63) log x) c3 χ n ) βn ) dβ, Q d β Q N<n N χ mod χ χ 0 gcdn,)=d

12 PRIES IN QUADRATIC PROGRESSIONS ON AVERAGE for som constant c 3 > 0 W now aly Gallaghr s lmma, Lmma, to th intgral abov W gt that, if x is sufficintly larg, 63) is majorizd by log x)c3 x Q χ n ) dt Q d x max{t,n}<n min{t+q/,n } χ mod χ χ 0 gcdn,)=d Hr w hav usd that Q = log x) c and Q = x ε Th charactr sum abov can b rwrittn in th form χ n ) = χ n ) max{t,n}<n min{t+q/,n } gcdn,)=d = max{t,n}/d<n min{t+q/,n }/d gcdn, )= max{t,n}/d<n min{t+q/,n }/d χ n ), χ is th non-trivial) charactr modulo inducd by th charactr χ mod W now aly Pólya- Vinogradov s stimat, Lmma 5, to th abov charactr sum Thn, collcting all contributions, w arriv at th stimat 64) E α) dα xq log x) c4, for som fixd c 4 > 0 Consuntly, w infr from 6) and 64) that 65) T α)e α)α)dα x5 log x) c4 Q Now onc again, by Bssl s inuality, Lmma 3, w hav E α)t α)α)dα E α)t α) dα su T α) E α) dα α 66) x E α) dα W now nd to stimat th intgral in 66) involving E α) Braking th sum ovr m in E α) into dyadic intrvals of th form < m z and alying Cauchy s inuality, w gt that in ordr to stimat th intgral in ustion, it suffics to stimat # τχ)χa) dβ Q a= gcda,)= β Q ϕ) χ mod χm)λm)βm) <m which is rcisly th sam as th trm B on ag of [] Using th stimat for B in [] which is obtaind by using Lmma and Lmma, w gt that this xrssion is boundd by 67) ϕ) Q) J, Q/) + Q 3 Qlog x) zlog z) c A, Q J, ) is dfind in Lmma Now from th stimats in 66) and 67), w conclud that 68) E α)t α)α)dα x zlogz) c A x4 log x), c5 for any c 5 > 0

13 STEPHAN BAIER AND LIANGYI ZHAO W now obsrv that by Cauchy s inuality and th stimats in 64) and 67) 69) E α)e α) αk)dα y E α) dα E α) dα yx3 Q Thus, combining 65), 68) and 69), w obtain that 60) T α)e α) + T α)e α) + E α)e α)) α)dα x5 log x) c4 x 4 yx3 Q + + c5 log x) Q 7 Th inor Arcs It still rmains to considr th contribution from th minor arcs [ m = Q, + ], Q is dfind in 3) W aim to hav an stimat of th scond momnt ovr k y of th minor arcs contributions In articular, w nd to stimat 7) Λm)αm) αn + k) ) dα m m z m Using Bssl s inuality, Lmma 3, and Parsval s inuality, th abov is 7) S α)s α) dα su S α) S α) dα su S α) z log z, α m α m th S α) and S α) ar th sums ovr m and n in th intgrand of 7), rsctivly W not that by Dirichlt aroximation and th fact that α m, a Q α a + Q for som a and with Q < Q and gcda, ) = Thrfor, w gt, by Lmma 4, S α) log x x / + x) /) and hnc su S α) log x) x Q + Qx ) α m Thus w infr that 7), and hnc 7) is majorizd by 73) x4 log x) c 3 + log x)3 Qx 3 8 Proof and Discussion of th Thorm Using Cauchy s inuality, combining 34), 4), 58), 60) and 73) and rcalling Q = x ε, w obtain th thorm W would lik to not that th rsult could b imrovd substantially if w assum th Gnralizd Rimann Hyothsis GRH) for Dirichlt L-functions Indd, undr GRH, Lmma holds for a much largr rang of, namly for x δ with som ositiv δ orovr, a bttr majorant for J, ) would b tru undr th GRH or rcisly, w would hav in Lmma a saving of a ositiv owr of N rathr than a saving of a owr of logarithm as w hav now Ths imrovmnts would imly that th thorm holds in a much widr y-rang, namly x δ y x, for som ositiv δ 0

14 PRIES IN QUADRATIC PROGRESSIONS ON AVERAGE 3 Acknowldgmnts This ar was writtn whn th first and scond-namd authors hld ostdoctoral fllowshis at th Dartmnt of athmatics and Statistics at Qun s Univrsity and th Dartmnt of athmatics at th Univrsity of Toronto, rsctivly Th authors wish to thank ths institutions for thir financial suort orovr, this work was startd whn th scond-namd author was visiting La Cntr d Rchrchs athématius CR) in Univrsité d ontréal as a gust rsarchr during th Thm Yar in Analysis in Numbr Thory H would lik to thank th CR for thir financial suort and warm hositality during his lasant stay in ontral Rfrncs [] J Brüdrn, Einführung in di analytisch Zahlnthori, Sringr-Vrlag, Brlin tc, 995 [] H Davnort, ultilicativ Numbr Thory, Third Edition, Graduat Txts in athmatics, Sringr-Vrlag, Barclona, tc, 000 [3] P X Gallaghr, Th larg siv, athmatika 4 967), 4 0 [4], A larg siv dnsity stimat nar σ =, Invnt ath 970), [5] P R Halmos, Finit-Dimnsional Vctor Sacs, Scond Edition, D Van Nostrand, Nw York, 958 [6] G H Hardy and J E Littlwood, Som roblms of artitio numrorum ; III: On th xrssion of a numbr as sum of rims, Acta ath 44 9), no 3, 70 [7] D R Hath-Brown, A man valu stimat for ral charactr sums, Acta Arith 7 995), no 3, [8] H Iwanic and E Kowalski, Analytic Numbr Thory, Amrican athmatical Socity Collouium Publications, vol 53, Amrican athmatical Socity, Providnc, 004 [9] A F Lavrik, On th distributions of k-twin rims, Sov ath, Dokl 960), translation from Dokl Akad Nauk SSSR 3, ) [0], On th twin rim hyothsis of th thory of rims by th mthod of I Vinogradov, Sov ath, Dokl 960), translation from Dokl Akad Nauk SSSR 3, ) [] H ikawa, On rim twins, Tsukuba J ath 5 99), no, 9 9 [] H L ontgomry, Toics in ultilicativ Numbr Thory, Lctur Nots in athmatics, Sring-Vrlag, Barclona, tc, 97 [3], Th analytic rincils of larg siv, Bulltin of th Amrican athmatical Socity July), no 4, [4] D Wolk, Übr das rimzahl-zwillingsroblm, ath Ann ), Dartmnt of athmatics and Statistics, Qun s Univrsity Univrsity Av, Kingston, ON K7L 3N6 Canada sbair@mastunsuca Dartmnt of athmatics, Univrsity of Toronto 40 Saint Gorg Strt, Toronto, ON 5S E4 Canada lzhao@mathtorontodu

INCOMPLETE KLOOSTERMAN SUMS AND MULTIPLICATIVE INVERSES IN SHORT INTERVALS. xy 1 (mod p), (x, y) I (j)

INCOMPLETE KLOOSTERMAN SUMS AND MULTIPLICATIVE INVERSES IN SHORT INTERVALS T D BROWNING AND A HAYNES Abstract W invstigat th solubility of th congrunc xy (mod ), whr is a rim and x, y ar rstrictd to li