Higher-Dimensional Kloosterman Sums and the Greatest Prime Factor of Integers of the Form a 1 a 2 a k+1 + 1

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1 Highr-Dimnsional Kloosrman Sums and h Gras Prim Facor of Ingrs of h Form a a a k+ + by Shngli Wu A hsis rsnd o h Univrsiy of Warloo in fulfilmn of h hsis rquirmn for h dgr of Docor of Philosohy in Pur Mahmaics Warloo, Onario, Canada, 007 c Shngli Wu 007

2 I hrby dclar ha I am h sol auhor of his hsis. I auhoriz h Univrsiy of Warloo o lnd his hsis o ohr insiuions or individuals for h uros of scholarly rsarch. ii

3 Absrac L b a osiiv ingr and l A and B b subss of {,,..., }. Wha can b dducd abou h arihmical characr of h ingrs of h form ab + wih a in A and b in B from jus informaion abou h cardinaliis of A and B? In 000, Sárközy and Swar conjcurd, For ach osiiv numbr ɛ hr ar osiiv ral numbrs 0 ɛ and Cɛ such ha if xcds 0 ɛ and A > ɛ and B > ɛ. whr ɛ is a ral numbr wih 0 < ɛ < hn hr ar a in A and b in B wih P ab + > Cɛ. In his hsis w shall discuss rogrss owards his conjcur. W hav imrovd or xndd h work of Sárközy and Swar in his conx in som scial cass. For xaml w rovd L b a osiiv ingr and u S {,,, }. L α b a ral numbr wih α > 0. Whn is larg nough, hn hr ar ingrs a, b S, such ha P ab + > log. α W also considrd h following roblm, L and k b osiiv ingrs and l ɛ b a ral numbr wih 0 < ɛ <. Whn is larg nough, for ach rim wih < k+ k+ ɛ hr xis ingrs a i, i,,, k + wih a i such ha a a a k+ +. iii

4 Acknowldgmns I would lik o hank, firs and formos, my survisor, Camron Swar, for his guidanc and suor hrough my docoral sudis. Escially during h final sag of my hsis. H sn hours and hours o discuss h roblms involvd in my hsis wih m. A svral occasions whn I can no mov on, h oind m in h righ dircion and graly hld m in ging h rsuls in my hsis. H is an English achr wih rsonsibiliy o corrc my slling, grammar, amongs ohr hings. On h ohr hand, I was imrssd by his aiud oward aching. I hav akn fiv courss augh by him. I also wan o hank h rading commi for hir work and h im hy sn on my hsis. I hank hm for roviding m wih hlful suggsions o imrov h hsis. I would lik o hank h saff of h Pur Mahmaics Darmn for hir hl. I would lik o hank Pur Mahmaics Darmn. I is rally a gra xrinc o sudy hr. I would lik o hank all h frinds, for hir hl and advic. Finally I would lik o hank my family for coninud suor. iv

5 Conns Inroducion An Exlici Consan 4 3 Prims in Arihmical Progrssions 3. Our lan Proof of Thorm and is corollary Proof of Thorm 3 and is corollary All h Prims and Highr Dimnsional Kloosrman Sums 9 4. An Obsrvaion Proof of Thorm Proof of Thorm Proof of Thorm 6 and is corollary v

6 oaion L f b any ral or comlx-valud funcion, and l g b a osiiv funcion. W wri f Og if hr xiss a consan C > 0 such ha fx Cgx for all sufficinly larg x in h domain of f. For any ral numbr α, w wri α πiα. Furhrmor, w wri α o dno h disanc of α o h nars ingr, and [α] for h gras ingr lss or qual o α. W wri a, b o dno h gras common divisor of a and b. L S b a subs of {,,, }. L S dno h cardinaliy of S. If hr xis a ral numbr ɛ wih 0 < ɛ < such ha S ɛ Thn w say S is a dns subs of {,,, }. Furhr noaions will b inroducd as ndd. vi

7 Char Inroducion L b a osiiv ingr and l A and B b subss of {,,..., }. A basic qusion of combinaorial numbr hory is h following. Wha can b dducd abou h arihmical characr of h ingrs of h form a+b wih a in A and b in B from informaion abou h cardinaliis of A and B? Thr is an xnsiv liraur addrssing his roblm. For xaml, in 986, by mans of h Hardy- Lilwood mhod, A. Sárközy and C. Swar [4] rovd h following: For any s X l X dno h cardinaliy of X and for any ingr n, largr han on, l P n dno h gras rim facor of n. L ɛ b a osiiv ral numbr and suos ha A > ɛ and B > ɛ.. Thn hr is a osiiv numbr Cɛ, which is ffcivly comuabl in rms of ɛ, such ha if. holds hn hr xis ingrs a in A and b in B wih P a + b > Cɛ.. This rsul is bs ossibl u o a drminaion of Cɛ sinc a + b is a mos. In 99 Ruzsa [0] gav a diffrn roof of.. On migh lik o sudy h mulilicaiv analogus of sum s rsuls. On way of doing his, roosd by Sárközy [6], is o rlac h sums a + b by h numbrs ab + s also [3, 7]. Howvr, i should b nod ha h firs rsul on h arihmic roris of numbrs ab + is du, robably, o Vinogradovs Char V of [3]. L b a rim numbr and n b an ingr corim wih

8 . L n dno h Lgndr symbol of n ovr. Vinogradov sablishd h sima ab + k A B..3 a A b B This rsul can b considrd as h mulilicaiv analogu of a rsul of Fridlandr and Iwanic [] on sums of h form a A b B χa + b whr χ is a non-rincial characr modulo a rim. Pu In 000, Sárközy and Swar [5] rovd h following, Z min A, B..4 Thorm A A. Sárközy, C. Swar For ach osiiv ral numbr ɛ hr ar numbrs ɛ and Cɛ which ar ffcivly comuabl in rms of ɛ such ha if xcds ɛ and Z > Cɛ log,.5 hn hr ar a in A and b in B such ha P ab + > ɛz log..6 In 00, C. Swar [30] sharnd h lowr bound. By making us of simas for Kloosrman sums and Slbrg s ur bound siv, h rovd h following horm, Thorm BC. Swar Thr ar ffcivly comuabl osiiv numbrs c, c and c 3 such ha if xcds c and Z > c,.7 log / log log hn hr ar a in A and b in B such ha P ab + > +c 3Z/..8 This is h bs rsul so far on his roblm. W wan o know wha h bs ossibl lowr bound is for P ab +. In hir join work in 000, Sárközy and Swar [5] mad h following conjcur,

9 Conjcur. A. Sárközy, C. Swar For ach osiiv numbr ɛ hr ar osiiv ral numbrs 0 ɛ and Cɛ such ha if xcds 0 ɛ and. holds, hn hr ar a in A and b in B wih P ab + > Cɛ. Conjcur. A. Sárközy, C. Swar For ach osiiv ral numbr ɛ and ach ingr k wih k >, hr ar osiiv ral numbrs 0 ɛ, k and cɛ, k such ha if xcds 0 ɛ, k and. holds, hn hr ar a in A and b in B and a rim wih k ab + and k > cɛ, k. W will giv an xlici c 3 for Thorm B of Swar. W shall rov ha on may ak c Thorm. L b a osiiv ingr, l A and B b wo subss of {,,, } and u Z min A, B. Thr ar ffcivly comuabl osiiv numbrs c and c such ha if xcds c and.7 holds, hn hr ar a in A and b in B such ha P ab + > Z/..9 W ar unabl o imrov Swar s rsul abou h lowr bound of P ab + for dns subss A and B of h s {,,, }. Howvr w ar abl o giv a much br lowr bound in som scial cass. Insad of looking a h dns subss A and B, w considr a and b boh from {,,, }. In his cas w may considr h rims in h arihmic rogrssion wih m varying from u o. Dfin ψx; q, a mq + n x n amod q Λn, whr Λn is h von Mangold Funcion. I.. { log n if n is a owr of a rim numbr, Λn 0 ohrwis. 3

10 W quo on vrsion of h rim numbr horm for arihmic rogrssions in h following. Thorm C Th rim numbr horm for arihmic rogrssions L x b a osiiv ral numbr, q and a b osiiv ingrs and suos q log x δ,.0 whr δ is a fixd ral numbr such ha 0 < δ <. Thn ψx; q, a x φq + O x clog x,. whr c is an absolu consan. This is a wak rsul bu i is ffciv, in h sns ha, if δ is givn a numrical valu, boh c and h consan imlid by h symbol O can b givn numrical valus. Alying his rsul, w dducd h following, L b a osiiv ingr and l ɛ b a ral numbr such ha 0 < ɛ <. Thr xiss a osiiv ingr such ha whn >, hr xiss a rim numbr and ingrs a and b wih a, b such ha ab + and > log ɛ.. To hav a br sima for h siz of in., w nd br simas for h numbr of rims in an arihmic rogrssion. L χ b a Dirichl characr mod q. Th Dirichl L-funcion is dfind by Lχ, s n χn n s.3 for vry comlx numbr s wih ral ar biggr han. By analyic coninuaion, his funcion can b xndd o a mromorhic funcion dfind on h whol comlx lan. Hr w quo h following rsul which was robably formulad for h firs im by Pilz in 884: Th Gnralizd Rimann Hyohsis GRH assrs ha for vry Dirichl characr χ and vry comlx numbr s wih Lχ, s 0: if h ral ar of s is bwn 0 and, hn i is acually /. 4

11 Th cas χn for all n givs h ordinary Rimann Hyohsis. Thorm D If GRH is ru, hn for q x, whr πx; q, a Lix φq + O x log x.4 Lix x log u du, and πx; q, a { x, a mod q }. By Thorm D, ha is, wih h assumion of GRH, w can rov h following rsul. Thorm. L b a osiiv ingr, l α b a ral numbr wih α > 4, and l q [ ]. Pu x log α q + and x q +. If GRH is ru, w hav πx ; q, πx ; q, 4 log + O W hav h following corollary, log log log +. log α Corollary. L b an ingr and u S {,,, }. L α b a ral numbr wih α > 4. Assum GRH. Whn is larg nough hr ar ingrs a, b S such ha P ab + > log. α o uon choosing suiabl α, h facor may b rmovd. Furhrmor, w may aal o h Bombiri-Vinogradov horm on h avrag of rims in arihmic rogrssions. W hav uncondiionally rovd a slighly wakr rsul han Thorm. I is rovd wihou GRH. W u Ex; q, a ψx; q, a 5 x φq

12 for a, q, w l and Ex; q max Ex; q, a, a a, q E x, q max Ey, q. y x Thorm E Bombiri-Vinogradov L α > 0 b fixd. Thn E x, q x Qlog x 5 q Q rovidd ha x log x α Q x. Proof. S Davnor s Mulilicaiv umbr Thory [0]. 6. Thorm 3. L α and β b wo osiiv ral numbrs wih α < β. Pu x log β + and Thn w hav x log β +. ψx ; q, ψx ; q, log β <q log α β α log log + O log β log β +α 5,.5 If β + 5 < α, hn h main rm in.5 dominas. Tha is, hr is a rim numbr wih > x in an arihmic rogrssion mq + wih m bwn and and q in h rang log < q β log. α 6

13 If is larg nough, hn sinc > x w hav > log β. o β + 5 < α < β so β > 0 so his is a wakr rsul han Corollary which w rovd undr h assumion of GRH. Corollary. L b an ingr and u S {,,, }. L β b a ral numbr wih β > 0. Whn is larg nough hr ar ingrs a, b S such ha P ab + > log. β Equivalnly if β is a ral numbr largr han 0 hn for sufficinly larg P ab + > log. β o again h facor may b rmovd. On h ohr hand, w also discuss an inrsing imlicaion from Swar s roof [30]. To rov Lmma in his ar, Swar inroducd h following s: U { m, n Z Z m, n, mn + }.6 whr Z dnos h s of ingrs. If divids hn w may dcomos {m, n m, n } ino / blocks consising of h Carsian roduc of wo coml ss of rsidus modulo. Thus if divids, U φ/. In gnral w dduc ha U φ + O φ + O..7 By a siml calculaion w find ha h abov sima.7 is dominad by h main rm for u o, which is rivial. To g a sharr sima Swar aald o Wil s simas for Kloosrman sums [33]. For any osiiv ingr n l dn dno h numbr of divisors of n. H rovd, Thorm FC. Swar U φ + O d 3/ log + d log..8 7

14 Proof. S [30]. This sima is dominad by h main rm for u o 4 3 ɛ. In ohr words, for all h ingrs u o 4 3 ɛ, w can find ingrs m and n wih m, n such ha mn+. This assurs us ha for all h rim numbrs u o 4 3 ɛ, w can find ingrs m and n wih m, n such ha mn +. In aricular, P mn + 4/3 ɛ. Th roof of Thorm F involvs h sima of a scial sum calld a Kloosrman sum. For ach ingr a corim wih l ā dno h ingr from {,, } for which aā mod. For ingrs g, h and, h Kloosrman sum Sg, h; is dfind by Sg, h; a aā mod ga + hā..9 Thy ar namd for h Duch mahmaician Hndrik Kloosrman, who inroducd hm in 96 [8] whn h adad h Hardy-Lilwood circl mhod o solv a roblm of Ramanujan of rrsning sufficinly larg numbrs in h form ax + by + cz + d. In his ar h rovd ha Sg, h; O 3 4 +ɛ g, 4..0 Ths sums urn ou o hav clos conncions wih modular forms, and various analyic numbr hory chniqus ar usd o rovid simas for h cofficins of modular forms saring wih simas for Kloosrman sums. In 93, H. Salié [3] rovd ha if m hn Sg, h; m C m,. whr C is an absolu consan. Bu his argumn gav no informaion for m. In 933, Davnor [7] rovd ha for any rim wih gh,, Sg, h; O 3.. 8

15 In 934, Hass [5] showd ha if h Rimann Hyohsis for algbraic funcion filds in on variabl ovr fini filds is ru, hn Sg, h;..3 In 94,Wil [34] sablishd h ruh of his hyohsis and hrby sablishd.3 and h ublishd h roof [33] in 948. Using h Hass-Wil rsul in.3 and Salié s rsul in., in 96, T. Esrmann [9] s also Hooly [6] rovd ha Sg, h; d g,.4 for all g, h and. In a ar in 958, Mordll [9] inroducd a gnralizaion of h Kloosrman sum and conjcurd ha i saisfis an ur bound analogous o.3. In 977, Dlign [4] sablishd such a bound for h Mordll gnralizd Kloosrman sum. Indd, Dlign s roof aals o his 974 work [5] in which h sablishd h las orion of h Wil Conjcurs [3], i.., h Rimann Hyohsis for algbraic variis ovr fini filds. L l i, i,, k + b ingrs such ha l i,. Th k-dimnsional Kloosrman sum is dfind as follows: Kl,, l k+ ; a,,a k+ a a k+ mod whr a i, i,, k + ar ingrs. Whn k, w hav Sl, l ; a,a a a mod l a + l a l a + + l k+ a k+ a a, l a + l ā whr ā is such ha ā < and aā mod. This is h sandard Kloosrman sum. I is -dimnsional. For h -dimnsional Kloosrman sum, Mordll [9] conjcurd ha for all rims, 9,,

16 Kl, l, l 3 ; c if l l l 3, whr c is an absolu consan. This may b considrd h analogu of h -dimnsional ur bound in.3. Using.3, Carliz [3] rovd ha hr is an absolu consan c such ha Kl, l, l 3 ; c 5 4 if l l l 3. And finally, Dlign rovd [4]s also Bombiri [] and Srr [8] ha if l l k+ hn Kl,, l k+ ; k + k.5 holds for all k,. A rsul analogous o Esrmann s rsul in.4 was givn by Smih [9] in 979 and w shall sa i blow. Smih dducd his rsul from Dlign s Thorm. For ach rim, and any a, a,, a n+ Z n+, hr xiss a uniqu ingr γ 0 such ha a,, a n+ γ b,, b n+ for som b,, b n+ Z n+ Z n+. L s dno h numbr of b i, i,, n + which ar divisibl by so ha 0 s n. For ach α 0, dfin a,, a n+ ; α+ n σna,,a n+ ; α, whr α ifγ α σ n a,, a n+ ; α γ ifγ < α γ ifγ α and s 0 γ + s if γ α and s n. n o ha σ n a,, a n+ ; α α for all a,, a n+. For any q w now dfin a,, a n+ ; q n α q a,, a n+ ; α n. W hav 0

17 Lmma. a,, a n+ ; α n n a, α a n+, α. Proof. L a,, a n+ γ b,, b n+ whr b,, b n+ Z n+ Z n+. If γ α, hn according o h dfiniion of σ n a,, a n+ ; α n, a,, a n+ ; α n n α n a, α an+, α α n+. If γ < α hn a,, a n+ ; α n n γ n, bu a i, α γ for i,, n +. Hnc a,, a n+ ; α n n a, α an+, α. Similarly w hav if γ α and s 0 s is h numbr of b i, i,, n +, such ha b i If γ α and s n a,, a n+ ; α n n < a, α an+, α. a,, a n+ ; α n s γ + n n n nγ n+s. Bu sinc a, α an+, α γ n+ s nγ+γ + s, W onc again hav a,, a n+ ; α n n < a, α an+, α This finishs h roof of Lmma.

18 L n and q b osiiv ingrs and dfin d n+ q {d, d,, d n+ : d i q, d i q, i n + }. Tha is, d n+ q dnos h numbr of rrsnaions of q as a roduc of n + ordrd osiiv facors. Thorm GR. Smih For all n, q, and all a,, a n+ Z n+, w hav Ka,, a n+ ; q q n a,, a n+ ; q n n d n+ q. Proof. S [9]. Alhough h rsul of Dlign.5 is d, w hav no bn abl o find many alicaions of i. In h following, w lookd a a highr dimnsional cas of h s U in.6. By making us of Dlign s and Smih s rsuls, w ar abl o giv muli-dimnsional analogus of Swar s rsul.8. L and k b wo ingrs and l b a osiiv ingr. Dfin U, k {a,, a k+ a,, a k+, a a k+ + }, whr a i, i,, k + ar ingrs. L n l b h binomial cofficin. By making us of Dlign s rsul.5 w hav rovd h following: Thorm 4. L and k b osiiv ingrs and l b a rim. W hav Furhr, if hn U, k [ ] k+ + Ok k log k+. U, k k φ k+ k+ φk k+ k+.

19 I follows from a ar of Fouvry and Kaz [0] ha if on has U, k k+ k+ k + O k k log k+ + k log k. So our rsul is an imrovmn of hirs. W hav a mor xlici main rm and as a consqunc w hav a mor rcis rsul. W shall g asymoic simas for U, k by using Smih s rsul on highr-dimnsional Kloosrman sums. Howvr, w shall firs look a h cas whn is a rim owr. Thorm 5. L, k and m b osiiv ingrs and l b a rim. Thn w hav U m, k φ m [ ] k+ + O d k+ m m k log k+ m. Th main rm dominas for m < k+ k+ ɛ. Scondly w hav h gnral rsul. I is wakr han Thorm 4 and Thorm 5. W shall discuss his lar in h roof. Thorm 6. L, k and b osiiv ingrs, hn U, k k+ k k+ φk + O d log + k k+ d k+ dk+ log k+. Hr onc again, h main rm dominas for < k+ k+ ɛ. W hav h following corollary, Corollary 3. L, k b ingrs. L ɛ b a ral numbr wih 0 < ɛ <. For ach osiiv ingr wih < k+ k+ ɛ, w can find ingrs a,, a k+ wih a i, i,,, k + such ha a a k+ +. 3

20 Char An Exlici Consan As mniond abov, our firs goal is o find an xlici c 3 in Swar s horm. For comarison, w ra Swar s horm hr again. Thorm B C. Swar Thr ar ffcivly comuabl osiiv numbrs c, c and c 3 such ha if xcds c and Z > c, log / log log hn hr ar a in A and b in B such ha P ab + > +c 3Z/. Swar s roof mloyd a sragy firs inroducd by C. Hooly [7] for his roof ha P n + xcds n 0 for infinily many ingrs n. Mor xacly, h usd simas of Kloosrman sums and Slbrg s ur bound siv. Th alicaion of h siv mhod is similar o ha of Gravs [] in ha hy siv a subs of Z Z no a s of ingrs. I shall ra mos of his roof. Howvr w nd o xand h roof o g an xlici rsul. W hav Thorm L b a osiiv ingr, l A and B b wo subss of {,,, } and u Z min A, B. Thr ar ffcivly comuabl osiiv numbrs c and c such ha if xcds c and.7 holds, hn hr ar a in A and b in B such ha P ab + > Z/.. 4

21 Dfin E by and u E ab +. a A,b B E ord E,.3 whr h roduc is akn ovr all rims u o and ord E dnos h highs owr of ha divids E. Lmma. L ɛ > 0, hr xiss a osiiv numbr 0 ɛ, which is ffcivly comuabl in rms of ɛ, such ha for > 0, Proof. S lmma in [5]. log E < + ɛz log..4 L and b osiiv ingrs and l z b an ingr wih z. W u U, z o b h following s { m, n m, n, mn +, all rim facors of mn + } xcd z In [30], Swar gav an sima of U, z by mans of Slbrg s ur bound siv. Bfor w sa his rsul, w nd h following: L > 0 b an ingr. W dfin f d for ach ingr d by f d d d..5 Obsrv ha f d is mulilicaiv and, for ach osiiv ingr n, u Sinc g n f n n..6 f { if f if, 5

22 w s ha So g n n n g n n n + n n + n + n 4 n + n 4 < n n 4 n + n >4 n > >4 < n n 4 4 n + π 6 + n n..8 Lmma 3. For ach ingr z and ach ingr > 0, w u V z n z µ n g n..9 W hav V z >.66 log z.0 π rovidd ha z is larg nough. 6

23 Proof. From.8 and.9 w g V z n z µ n g n n z µ n.87743n,. By Abl s summaion formula, w furhr hav V z > µ n n n z z µ n u n + µ n z n z u z π z + O z 6 u + O u π z u π log z π + O z.66 log z,. π rovidd ha z is larg nough. Hr w hav usd h known rsul ha S horm 334 in [4] µ n 6 π x + O x. n x W can now sa Swar s ur bound for U, z. Lmma 4. L ɛ > 0 and l, and z b osiiv ingrs wih > 3 and z. Thn U, z φ V z + O ɛ z 3 +ɛ..3 Proof. S lmma in [30]. ow w ar rady o rov Thorm. L ɛ > 0 and l 0,,,, dno osiiv numbrs which ar ffcivly comuabl in rms of ɛ. Dfin E by. and E by.3 and u E E/E. Th roof rocds by a comarison of simas for E. 7

24 Proof. o ha Thus by.7, for > 0. By Lmma, E a A a ɛz 0 ɛz for >. Hnc, for >, b B b ɛz 0 0 A ɛz ɛz 0 + ɛz ɛz B 0 0 ɛ 0 0 Z, log E > ɛz log, log E < + ɛz log L P dno h gras rim facor of E, hn E whr log E > ɛz log..4 G m,n P Pu P Y and no ha ord G log Obsrv ha < Y < Y ord G.5 mn +..6 m,n mn+ < Y mn+ log m,n mn+ log log log m,n mn+ log..7 O oz log.8 8

25 and so, by.4,.5,.6,.7 and.8, for >, log > 3ɛZ log..9 < Y m,n mn+ mn+> log Pu S m,n mn+ < Y mn+> log. If Y > 0 hn P Y /0 hn h rsul holds and so w may suos ha Y 0. Thn by.9 log Y log < + S > 3ɛZ log,.0 and so, for > 3 S > Y log < 9.5 Z.. For ach ral numbr z wih z S U, U, z.. L c 4, c 5, dno ffcivly comuabl osiiv numbrs. By Lmma 3, w hav V z >.66 π log z..3 W now aly Lmma 4 wih z 7 and ɛ.3, ha for > 3 and <, Y log 0 o conclud from. and 9

26 and so S φ V z + O ɛ z 3 +ɛ < <.66 π 7 log z φ + O ɛ z 3 +ɛ.65 π S < Y log < log φ 7π.65 log φ 7π.65 log Y log < φ.4 o ha S [] x φ 6 log x π log x + A + O x whr A is a consan. Thrfor w hav 7π 6 S <.65 log π Y log < < rovidd ha is larg nough. log 3 log Y + log log + OY 4 log Y +. log log.5.65 log W now can choos suiabl c in.7 such ha log log <.65 log Z..6 Hnc h firs rm in.4, from.,.4 and.5, saisfis 4 9 log Y >.65 log Z 0

27 and his givs us whr > Y > Z/, This finishs h roof of Thorm.

28 Char 3 Prims in Arihmical Progrssions 3. Our lan As w mniond bfor, w ar unabl o imrov h lowr bound for h gras rim facor of ab + wih a and b from wo dns subss of {,,, } aar from h consan in h xonn. I may b worhwhil o mnion hr ha C. Hooly [7] rovd ha P n + xcds n 0 infinily many ims. Lar, i was rfind o n 0 by J. Dshouillrs and H. Iwanic [6]. Howvr, vn if h conjcur ha hr ar infinily many rims of h form n + is ru, i dos no imly h conjcur of Sárközy and Swar. Th conjcur will b difficul o rov. This lads us o look a h cas whn A B {,,, }. In ohr words, w wan o know wha kind of lowr bound w could hav for P ab + whn a and b ar boh from h s {,,, }. W hn can immdialy look a h rims in h arihmical rogrssion mq + whr q is fixd such ha q and m,,,. L ɛ b a fixd ral numbr such ha 0 < ɛ < and q [log ɛ ]. L x q +, hn q < log x ɛ, By., h rim numbr horm for arihmic rogrssions, w hav

29 ψx; q, ψ x ; q, x φq x φq + O x clog x x φq + O x clog x whr c is an absolu consan. o ha h main rm dominas h rror rm, so w dduc ha hr xiss a las on rim of h form aq + bwn x and x. To imrov h lowr bound for h siz of h rim of h form ab +, w nd a br sima for ψx; q, a. On way of doing his is wih h assumion of h GRH. Th ohr subsiu is by using h Bombiri-Vinogradov horm. 3. Proof of Thorm and is corollary Thorm L and q b osiiv ingrs and l α b a ral numbr wih α > 4 and q [ ]. Pu x log α q + and x q +. If GRH is ru, w hav πx ; q, πx ; q, 4 log + O log log log + log α Proof. Undr GRH, if q x, by horm D, w hav, πx; q, a Lix φq + Ox log x 3. W shall now sima h numbr of rims in h arihmic rogrssion mq + whr < m. By 3. w hav πq + ; q, Liq + φq + Oq logq 3. 3

30 and Hnc from 3. and 3.3, π q + ; q, Li q + φq πq + ; q, π q + ; q, Liq + Li q + + Oq logq φq q+ du q+ log u + Oq logq φq u log u q+ + q+ q+ q+ log u + Oq logq φq q+ q+ logq+ log q log q q q + O log q φq q + O q+ φq q log q q log q log q + Oq logq Oq logq + Oq logq log q + O + Oq logq φq q φq log q + O q + Oq φqlog q logq log q + O q + Oq φqlog q logq. 3.4 For any osiiv ingr q, hr xiss an absolu consan C such ha φq C q. o ha sinc q [ ], w may dduc ha log log q log α This comls h roof. πx ; q, πx ; q, 4 log + O log log log +. log α 4

31 This imlis ha hr is a rim lying bwn [ ]q + and q +. i.. whn is larg, hr is a rim numbr such ha W hav h following corollary: q +. Corollary L b an ingr and u S {,,, }. L α b a ral numbr wih α > 4. Assum GRH. Whn is larg nough, hr ar ingrs a, b S such ha P ab + > log. α By choosing a suiabl α, w may rmov h facor. 3.3 Proof of Thorm 3 and is corollary Thorm 3 L α and β b wo osiiv numbrs wih α < β. Pu x log +, β and Thn w hav x log β +. ψx ; q, ψx ; q, log β <q log α β α log log + O log β log β +α Proof. L x and x b as in h samn of h horm. W hav ψx ; q, ψx ; q, ψx ; q, x φq + x φq ψx ; q, x x φq + ψx ; q, x φq + x φq ψx ; q, 5

32 Thrfor for < q, log β log α ψx ; q, ψx ; q, log β <q log α + + log β <q log α x x φq ψx ; q, log β <q log α x log β <q log α x φq φq ψx ; q,, 3.6 whr log β <q log α x x φq > x x log β <q log α β αx x log log + O q log β. 3.7 By h Bombiri-Vinogradov horm, Thorm E, w hav x ψx ; q, φq log β <q log α x ψx ; q, φq log β <q log α O x log log x 5 α 3.8 6

33 and O ψx ; q, log β <q log α ψx ; q, log β <q log α x log α log x 5 x φq x φq. 3.9 Thus by 3.6, and 3.9, This finishs h roof. ψx ; q, ψx ; q, log β <q log α > β αx x log log + O β α log log + O log β x o ha by 3.0 h main rm is of h siz β α and h rror rm is of h siz log log x 5 α log β +α 5 log log log β log β +α 5. So long as w choos α, β wih α < β and α + β 5 > β,. 3.0 i.. α > β + 5, 7

34 w hav ha 3.0 is dominad by h main rm. If w hn also choos β so ha β + 5 < β,i.., β > 0, W obain h following corollary. Corollary L b a osiiv ingr and u S {,,, }. L β b a ral numbr wih β > 0, whn is larg nough, hn hr ar ingrs a, b S, such ha P ab + > log. β Equivalnly if β is a ral numbr largr han 0 hn for sufficinly larg P ab + > log. β 8

35 Char 4 All h Prims and Highr Dimnsional Kloosrman Sums 4. An Obsrvaion To rov Lmma 3, Swar inroducd h following s: U { m, n Z Z m, n, mn + }, 4. If divids hn w may dcomos {m, n m, n } ino / blocks consising of h Carsian roduc of wo coml ss of rsidus modulo. Thus if divids, U φ/. In gnral w dduc ha U φ + O φ + O. By a siml calculaion w find ha h abov sima is dominad by h main rm for u o, which is rivial. To g sharr simas Swar aald o Wil s simas for Kloosrman sums [33]. For any osiiv ingr n l dn dno h numbr of divisors of n. H rovd, Thorm FC. Swar U φ + O d 3/ log + d log. 9

36 Proof. S [30]. This sima is dominad by h main rm for u o 4 3 ɛ. In ohr words, for all h ingrs u o 4 3 ɛ, w can find ingrs m and n wih m, n such ha mn +. In aricular, if is a rim numbr, w immdialy g h following from Thorm F. U + O log + log. This assurs us ha for all h rim numbrs u o 4 3 ɛ, w can find ingrs m and n wih m, n such ha mn +. In ohr words, w hav P mn + 4/3 ɛ. for som ingrs m and n wih m, n. W shall rov h highr-dimnsional analogu of his rsul in h following scions. 4. Proof of Thorm 4 L and k b wo ingrs and l b a osiiv ingr. Rcall ha U, k {a,, a k+ a,, a k+, a a k+ + }, whr a i, i,, k + ar ingrs. Thorm 4 L and k b osiiv ingrs and l b a rim. W hav Furhr, if hn U, k U, k [ ] k+ + Ok k log k+. k φ k+ k+ φk k+ k+. 30

37 Acually Thorm 4 can b dducd from our nx rsul Thorm 5. Bu Thorm 4 can b rovd by using Dlign s rsul dircly whras for h roof of Thorm 5 Smih s rsul is ndd. Also, h rsul is much simlr in h cas > and h argumn is asir. oic in aricular ha if > hn U, k k+ + Ok k log k+. Proof. Th simls cas is whn, for ach k-ul a, a,, a k wih a,, a k+ and a a a k. Thn hr xiss xacly on choic of a k+ modulo. Hnc hr xis choics of a k+ such ha a a a k+ mod. Thus for, U, k k φ k+ k+ φk. In h following, w shall considr h rmaining cass, > and < bu. Rcall ha m g gn m { m m n 0 m n. 4. W hav U, k k+ b,,b k+ l,,l k+ a,,a k+ a a k+ mod l a b + + l k+ a k+ b k+. This is bcaus for ach k+-ul a, a,..., a k+ such ha a a a k+ mod, if hr is a k+-ul b, b,, b k+ such ha a i b i, i,,, k +, 4.3 3

38 hn h sums involving his k+-ul b, b,, b k+ conribu k+ o U, k and h k+-uls no saisfying 4.3 conribu 0 o U, k. Th rms wih l l l k+ conribu o U, k. Thus w hav k+ k k+ U, k k+ k+ k + R, 4.4 whr R dnos h sum ovr hos l i s such ha no all l i s ar qual o. L R i b h subsums of R such ha i of h l j s qual. Thn w hav R R R k W now will giv an sima for ach R i, i 0,,, k. If no l i quals, w hav R 0 k+ l,,l k+ a,,a k+ a a k+ mod b,, b k+ k+ l a b + + l k+ a k+ b k+ l,,l k+ a,,a k+ a a k+ mod b,,b k+ k+ l b l k+ b k+ l,,l k+ a,,a k+ a a k+ mod b,,b k+ l b l k+ b k+ l a + + l k+ a k+ l a + l k+ a k+ 3

39 k+ l,,l k+ b,,b k+ Kl,, l k+ ; l b + + l k+ b k+. By Dlign s horm.5, w hav R 0 O k k k+ O k k k+ O k k k+ O k k log k+ l,,l k+ l,,l k+ l l k+ 0 l,,l k+ l l k+ 0 b,,b k+ b,,b k+ l whr w hav usd h wll known inqualiy s M l b + + l k+ b k+ l b + + l k+ b k+ l k+, 4.5 gs O g, for g. L R,i dno h subsum in R such ha only l i, hn R R, + R, + + R,k

40 ow assum ha only l. W hav R, k+ l,,l k+ l b,,b k+ k+ a,,a k+ a a k+ mod l a b + + l k+ a k+ b k+ l,,l k+ a,,a k+ a a k+ mod b,,b k+ l b l k+ b k+ l a + + l k+ a k+. o ha for fixd a,, a k+, h soluion for xa a k+ mod, wih x <, is uniqu if h a i, i,, k + ar corim o. Hnc assuming ha >, w hav R, k+ l,,l k+ a,,a k+ a a k+ mod b,,b k+ k+ a k+ l b l k+ b k+ l,,l k+ a l k+a k+ l a + + l k+ a k+ l a l3 a 3 a 3 b,,b k+ l b l k+ b k+ 34

41 k+ l,,l k+ k k+ k k+ k k+ k+ k+. k b,,b k+ b,,b k+ l b,,b k+ b,,b k+ l,,l k+ l b k l b l k+ b k+ l b l k+ b k+ l k+ lk+ b k+ Similarly w hav Thus by 4.6 w g R, R,3 R,k+ k+ k+. k + k+ R. 4.7 k+ If wo of h l i s ar qual o hn, sinc h sum is symmric, w may assum ha l l. Thus, R k+ k+ b,,b k+ k+ k+ b,,b k+ l 3,,l k+ a,,a k+ a a k+ mod l3 a 3 b l k+ a k+ b k+ l 3,,l k+ a,,a k+ a a k+ mod l 3b l k+ b k+ l3 a l k+ a k+ 35

42 k+ k+ b,,b k+ k+ k+ k + a 3 l 3,,l k+ l3 a 3 a k+ l k+a k+ l 3b l k+ b k In gnral, if j of hs l i s ar qual o, w hav R j k+ k + j. 4.9 k+ j Combining all h rsuls for h R i, by 4.5, 4.7, 4.8 and 4.9 w hav R k+ k+ k k + j + Ok k log k. 4.0 j j Thus, by 4.4 and 4.0, if >, U, k k+ k k+ k + k+ k + j + Ok k k+ log k+ j j k+ k+ k + j + Ok k k+ log k+ j j k+ k+ k + j + Ok k k+ log k+ j j k+ k+ k + j + Ok k k+ log k+ j j0 k+ + k+ + Ok k k+ log k+ k+ k+ k+ + Ok k log k+. 36

43 If < and hn R k + b,,b k+ k+ k + k+ b,,b k+ k + k+ a k+ l,,l k+ l a,,a k+ a a k+ mod l a b + + l k+ a k+ b k+ l,,l k+ a,,a k+ a a k+ mod l b l k+ b k+ l k+a k+ k + k+ l,,l k+ a l,,l k+ b,,b k+ l a + + l k+ a k+ l a l3 a 3 a 3 l b l k+ b k+ k b,,b k+ l b l k+ b k+ k + k k+ b,,b k+ l,,l k+ k + k k+ b l k + k + k+ k + k k+ k + k+ b b [ lb l b l k+ b k+ k b b k [ ] [ ] k ] k

44 Similarly, w hav R φ k+. k + R k k k+ φk k + k [ ] k 4. [ ]. 4.3 Thus, by4,4, 4.5, 4.,4. and 4.3 w hav ha if < and, hn U, k φ k+ k + j φ j k+ j j φ φ + k+ [ [ ] k+ +Ok k log k+ [ ] k+ + Ok k log k+. This finishs h roof of horm 4. ] k+ j + Ok k log k+ [ ] k+ 4.3 Proof of Thorm 5 L and k b wo ingrs and l b a osiiv ingr. Again rcall U, k {a,, a k+ a,, a k+, a a k+ + }, whr a i, i,, k + ar ingrs. W hav Thorm 5 L, k and m b osiiv ingrs and l b a rim, hn w hav U m, k φ m [ ] k+ + O d k+ m m k log k+ m. W rmark ha whn n w rcovr h firs ar of Thorm 4 sinc d k+ k +. 38

45 Proof. By 4. w hav U m, k m k+ b,,b k+ m m l,,l k+ a,,a k+ a a k+ mod m l a b + + l k+ a k+ b k+ m. 4.4 This is bcaus for ach k+-ul a, a,..., a k+ such ha a a a k+ mod m, if hr is a k+-ul b, b,, b k+ such ha a i b i mod m, i,,, k +, 4.5 hn h sums involving his k+-ul b, b,, b k+ conribu m k+ o U m, k, and h k+-uls ha do no saisfy 4.5 conribu 0. Th rms wih l l l k+ m conribu o h righ hand sid of h 4.4. Thus w hav whr U m, k m k+ φk m k+ 4.6 m k+ φk m k+ + R, 4.7 R R 0 + R + + R k 4.8 and R i dnos h sum ovr hos rms in U m, k such ha i of l j s ar qual o m, h ohrs sum ovr o m. I is asir o comu R, R,, and R k han o comu R 0 sinc in h formr cass w do no hav o dal wih h condiions a a k+ mod m. 39

46 W firs comu R. o ha i is symmric ovr l i, i,, k +. W g R k+ m k+ b,,b k+ k+ m k+ b,,b k+ m m l,,l k+ a,,a k+ a a k+ mod m l a b + + l k+ a k+ b k+ m m m l,,l k+ a,,a k+ a a k+ mod m l b + + l k+ b k+ m l a + + l k+ a k+ m o ha a a k+, m, onc a,, a k+ ar givn, w hn hav a uniqu choic for a m and sinc R is symmric ovr l i and also symmric ovr b j w may comu R in h following way. W hav R k+ m k+ b,,b k+ k+ m k+ b,,b k+ m m l,,l k+ a a, m l a m l b + + l k+ b k+ m m a k+ a k+, m l k+a k+ m l, m m m µ l, m l k+, m m µ l k+, m l l b + + l k+ b k+ m, l k+ whr 40

47 m a a, m la m m a a, m l, m l a l, m m l, m m l, m a a, m m l, m µ l, m. l a l, m m l, m Hr w hav usd h wll known fac ha h sum of h n-h rimiiv roos of uniy quals µn. S [4], Thorm 7. If l m, hn m µ l, m {, m l 0, m l. Thus, R k+ m k+ b,,b k+ k+ m l m l m m l k+ m l k+ l b + + l k+ b k+ m k+ k m k b,,b k+ k+ m k+ k m k l l k+ l b + + l k+ b k+ l b m l k b. 4

48 o ha l b l b + b b b b [ ] [ ] [ ]. 4.9 By 4.9 and h abov rsul for R w furhr hav, R k+ [ ] k m k+ k m k k+ [ ] k m k+ m k. 4.0 To comu R i, i,, k +. W firs choos l l i m. Dno h sum by R i. W hav R i i φ i m m k+ b i+,,b k+ φi m i m k+ m l k+ m l i+,,l k+ m a i+,,a k+ a i+ a k+, l i+b i+ + + l k+ b k+ m m b i+,,b k+ l i+ l k+, m µ m l k+, m li+ a i+ + + l k+ a k+ l i+, m m µ l i+, m l k+b k+ m m l i+b i+ m i m k+ φi m m l k+ i lb. b By 4.9, w hav 4

49 R i i m k+ φi m m k+ i [ ] k+ i. 4. Combining all h cass of diffrn choics of i of h l j s ha ar qual o m, by 4. w hav k + i R i i m k+ φi m m k+ i [ ] k+ i. 4. I rmains o comu R 0, whr no l i quals m. W shall aly Smih s horm on simas of h highr-dimnsional Kloosrman sums and Lmma. W hav R 0 m k+ b,,b k+ m k+ m b,,b k+ m m l,,l k+ a,,a k+ a a k+ mod m l a + + l k+ b k+ m m l,,l k+ Kl, l,, l k+ ; m l b + + l k+ b k+ m l a + + l k+ a k+ m By Thorm G and Lmma w g Kl,, l k, l k+ ; m m k l,, l k, l k+ ; m k k d k+ m m k l, m lk+, m dk+ m

50 Thus, by R 0 from h abov and 4.3, w hav R 0 m k m k+ b,,b k+ d k+ m d k+ m m l k+ m l,,l k+ l,, l k, l k+ ; m k k d k+ m l b + + l k+ b k+ m m k m k+ l k+, m m k m k+ m l m l l, m b b k+ l, m o ha if α dos no qual an ingr, hn αb α. Thus, b l k+b k+ m b l b m lb m k+ 4.4 m l m l l, m b l, m l m lb m 44

51 m l m m l m d m m d m m l l, m l, m l m l l, m d [ m d ] l l, m d d l ld O m log m d m dm O m log m O m log m. n d n Thus, by 4.4 w g R 0 O d k+ m m k log k+ m. 4.5 Combining all h simas for R 0, R,, and R k, by 4.7, 4.8,4.0, 4. and 4.5, w hav 45

52 U m, k φ k m m k+ k k + + i m k+ i k+ m k+ i i [ ] k+ i φ i m + O d k+ m m k log k+ m k+ k + i m k+ i φ i m m k+ i i0 [ ] k+ i + O d k+ m m k log k+ m k+ [ k + i m k+ i φ i m m k+ i i0 [ ] k+ φ m + O d k+ m m k log k+ m [ ] k+ [ ] k+ φ m +O d k+ m m k log k+ m. ] k+ i This finishs h roof of horm Proof of Thorm 6 and is corollary Thorm 6 L, k and b osiiv ingrs, hn w hav U, k k+ k k+ φk + O d log + k k+ d k+ dk+ log k This sima of U, k is dominad by h main rm for k comaraivly much smallr han and for of h siz lss han k+ k+ ɛ. Bu in comaring o Thorm 4 and Thorm 5, h main rm in Thorm 6 is lss xlici. This 46

53 is du o h difficulis involvd in simaing h sums in U, k whn has mor han on rim facor. Mor imoranly, h firs rm in h rror rm of 4.6, which vry likly can b calculad xlicily, would domina h scond rror rm of 4.6 for small. So w indd hav a wakr rsul hr for h gnral cas. Proof. Rcall ha W hav m g gn m { m m n 0 m n. U, k k+ l,,l k+ b,,b k+ a,,a k+ a a k+ mod l a b + + l k+ a k+ b k+ 4.7 This is again bcaus for ach k+-ul a, a,..., a k+ such ha a a a k+ mod, if hr is a k+-ul b, b,, b k+ such ha a i b i, i,,, k +, 4.8 hn h sums involving his k+-ul b, b,, b k+ conribu k o U, k, and h k-uls no saisfying 4.6 conribu 0. Taking l l k+, w g h main rm for U, k, which is Thus w hav by 4.5, k+ k+ φk. U, k k+ k+ φk + R,

54 whr R dnos sum of hos rms such ha no all of h l i s ar qual o. Ling R i b h sum of hos rms in R such ha i of h l j s ar qual o, hn w hav R R 0 + R + + R k W will comu R firs, whr hr is only on l i qual o. For h cas ha l i w dno i by R i. Hnc and R R + + R k+ 4.3 R k+ k+ l,,l k+ a,,a k+ a a k+ mod b,,b k+ l a b + + l k+ a k+ b k+ l a + + l k+ a k+ l,,l k+ b,,b k+ a,,a k+ a a k+, l b + + l k+ b k+ k+ l,,l k+ b,,b k+ a a, l a l b + + l k+ b k+ o ha Ramanujan s sum qualss [4] a a, la a a, l a l, l, l, l, a a, a k+ a k+,. l a l, l, lk+ a k+ l, µ, l, 48

55 and if α dos no qual an ingr, hn αb α, b whr α dnos h disanc bwn α and h nars ingr. W g R k+ l, µ l 3, µ l, l 3, l,,l k+ l k+, µ l k+, b,,b k+ l k+, l b l 3, l b l 3 b 3 l k+, l k+b k+ l k+ b k+ k l k+, l l l b + + l k+ b k+ l 3b 3 o ha l l and if l, hn l, l,. W furhr hav k R l, k k+ l d l l,d k d l d l l, d O dk log k. Similarly w comu R,,, R,k+. Thus k + R O dk log k. 49

56 To comu R i, i,, k. W firs choos l l i. Dno h sum by R i. W hav R i φi i k+ b i+,,b k+ φi i k+ l k+ l i+,,l k+ l i+ l k+, µ a i+,,a k+ l i+k+, l i+b i+ + + l k+ b k+ l i+, µ l k+, l i+, li+ a i+ + + l k+ a k+ l i+b i+ l k+b k+ i O k+ φi k+ i d k+ i log k+ i i O dk+ i log k+ i. Combining all h cass of diffrn choics of i of h l j s ha ar qual o, w hav k + i R i O i dk+ i log k. I rmains o comu R 0, 50

57 Rcall ha R 0 k+ l,,l k+ a,,a k+ a a k+ mod b,,b k+ k+ l b + + l k+ b k+ l,,l k+ b,, b k+ l a + + l k+ a k+ Kl,, l k+, l k+ ; l b + + l k+ b k+. Thus by Thorm G, w g R 0 k+ l,,l k+ k l,, l k, l k+ ; k k d k+ l l k l k+. o ha by Lmma W furhr hav l,, l k, l k+ ; k k l, lk+, R 0 k+ l,,l k+ k l, lk+, dk+ l l k l k+ d k+ k+ k+ l, l l l k+ l k+ l k+,. 5

58 o also ha l l, l d l l,d d l d d l l, d ld O log d d O log d n n Od log. Thus w hav R 0 O k k+ log k+ d k+ dk+. Combining h sima for R 0, R,, R k, w hav U, k k+ k + k+ φk + O dk log k k + i + + i dk+ i log k+ i k + k + + k d log + k k+ log k+ d k+ dk+ k+ k k+ φk + O d log + k k+ d k+ dk+ log k+. This finishs h roof of Thorm 6. Corollary 3 L, k b ingrs. L ɛ b a ral numbr wih 0 < ɛ <. For ach ingr wih < k+ k+ ɛ, w can find ingrs a,, a k+ wih a i, 5

59 i,,, k +, such ha a a k+ +. Proof. o ha d O ɛ, d k+ Od k+ O k+ɛ, k O ɛ and log k+ O ɛ. Th rsul hn follows by comaring h main rm and h rror rm of Thorm 6. o ha could b ihr a rim or owr of a rim numbr. 53

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