LARGE SIEVE INEQUALITY WITH CHARACTERS FOR POWERFUL MODULI

Transcription

1 LARGE SIEVE INEQUALITY WITH CHARACTERS FOR POWERFUL MODULI STEPHAN BAIER AND LIANGYI ZHAO arxiv:math/583v2 [mathnt] 26 Mar 26 Abstract In this papr w aim to gnraliz th rsults in [],[2],[9] and dvlop a gnral formula for larg siv with charactrs to powrful moduli that will b an improvmnt to th rsult in [9] kywords: larg siv inquality; powr moduli Mathmatics Subjct Classification 2: N35, L7, B57 Introduction Throughout this papr, w rsrv th symbols c i (i =,2, for absolut positiv constants Larg siv was an ida originatd by J V Linnik [] in 94 whil stuing th distribution of quadratic non-rsidus Rfinmnts of this ida wr mad by many In this papr, w dvlop a larg siv inquality for powrful moduli Mor in particular, w aim to hav an stimat for th following sum ( q k q Q a= n=m+ (a,q= whr k 2 is a natural numbr In th squl, lt With k = in (, it is Z := n=m+ ( a a n q kn a n 2 (2 (Q 2 +NZ 2, This is in fact th consqunc of a mor gnral rsult first introducd by H Davnport and H Halbrstam [7] in which th Fary fractions in th outr sums of ( can b rplacd by any st of wll-spacd points Applying th said mor gnral rsult, ( is boundd abov by (3 (Q k+ +QNZ, and (Q 2k +NZ (s [9] Litratur abound on th subjct of th classical larg siv S [3], [6], [7], [8], [], [2], [3] and [4] In [9] it was provd that th sum ( can b stimatd by ( ( (4 Q k+ + NQ /κ +N /κ Q +k/κ N ε Z, whr κ := 2 k and th implid constant dpnds on ε and k Furthrmor, whn appropriat, som of th constants c i s and th implid constants in in th rmaindr of this papr will dpnd on ε or both ε and k In [] and [2] this bound was improvd for k = 2 Extnding th lmntary mthod in [] to highr powr moduli, w hr stablish th following bound for ( Dat: Novmbr 7, 27

2 2 STEPHAN BAIER AND LIANGYI ZHAO Thorm : W hav q k (5 q Q a= n=m+ (a,q= ( a a n q kn 2 (loglognq k+ (Q k+ +N +N /2+ε Q k Z For k 3 Thormimprovsth classicalbounds (3 as wll as Zhao sbound (4 in th rangn /(2k+ε Q N (κ 2/(2(k κ 2k ε In particular, for k = 3 w obtain an improvmnt in th rang N /6+ε Q N /5 ε W not that for a larg k th xponnt (κ 2/(2(k κ 2k is clos to /(2(k Extnding th Fourir analytic mthods in [2], [9], w stablish anothr bound for cubic moduli which improvs th bounds (3, (4 in th rang N 7/25+ε Q N /3 ε Thorm 2: Suppos that Q N /2 Thn w hav q 3 ( 2 a N ε (Q 4 +N 9/ Q 6/5 Z, if N 7/24 Q N /2, (6 a n q 3n q Q a= n=m+ NQ 6/7+ε Z, if Q < N 7/24 (a,q= Unfortunatly, our Fourir analytic mthod dos not yild any improvmnt if k 4 2 Proof of Thorm Lt S b th st of k-th powrs of natural numbrs Lt Q N St S(Q = S (Q,2Q ] W first not, by classical larg siv, stting Q = N in (2, q ( a (2 a n q n Lt q N a= n=m+ (a,q= S t (Q = {q Æ : tq S(Q } 2 2NZ Lt t = p v pvn n b th prim dcomposition of t Furthrmor, lt vi u i :=, k whr for x Ê, x = min{k : k x} is th ciling of x Morovr, st f t = p u pun n Thrfor, for all q k = q S, q is divisibl by t if and only if q is divisibl by f t Thrfor, w hav whr Morovr w not that and that S t (Q = {q k g t : Q /k /f t < q (2Q /k /f t }, g t := fk t t S t (Q (Q /t,2q /t] (22 S t (Q (2Q /k f t W st for m Æ, l with (m,l = (23 A t (u,m,l = max Q /t y 2Q /t {q S t(q (y,y +u] : q l mod m}

3 LARGE SIEVE INEQUALITY WITH CHARACTERS FOR POWERFUL MODULI 3 Lt δ t (m,l b th numbr of solutions x to th congrunc W now us Thorm 2 in [] with Q N: x k g t l mod m Thorm 3: Assum that for all t Æ, m Æ, l, u Ê with t N, m N/t, (m,l =, mq / N u Q /t th conditions ( (24 A t (u,m,l C + S t(q /m u δ t (m,l, Q /t (25 m l= (m,l= δ t (m,l m, (26 δ t (m,l X hold for som suitabl positiv numbrs C and X Thn, (27 q ( 2 a a n q n c C(min{Q X,N}+Q N loglogn + max r S t (Q Z N q S(Q a= n=m+ (a,q= First, w hav to chck th validity of th conditions (24, (25 and (26 Conditions (24 and (25 ar obviously satisfid with C absolut W furthr suppos that (g t,m = for othrwis δ t (m,l = sinc (m,l = Thrfor, w must stimat th numbr of solutions to (28 x k g t l mod m, whr g t is th multiplicativ invrsof g t modulo m By th virtu of th Chins rmaindr thorm, it suffics to stimat th numbr of solutions to (28 with m as a prim powr, say m = p, for p È and Æ Not that th function σ k : ( /p ( /p : x x k is an ndomorphism Hnc it is nough to stimat th siz of its krnl kr(σ k If k = π a πa h h is th prim dcomposition of k, thn Thus, σ k = (29 kr σ k h i= σ ai π i h kr σ πi ai i= Hnc, it suffics to stimat th siz of kr σ π for prim numbrs π For p È, x π mod p has at most π solutions By lmntary rsult (s [5], for xampl, a solution, a mod p with, of th congrunc (2 x π mod p lifts to mor than on solution to x π mod p + only whn p πa π and p + a π If p π, p πa π implis p a, but it is not possibl that p + a π as (a π,a = Thus, in this cas (2 has at most π solutions for all In th following, w considr th cas p = π

4 4 STEPHAN BAIER AND LIANGYI ZHAO By Frmat s littl thorm, thr xists only on solution of th congrunc x π mod π, namly mod π This solution lifts to xactly π solutions to x π mod π 2, namly, +π, +2π,,+(π π mod π 2 Mor gnrally, if a mod π is a solution to (2 x π mod π, thn, if a lifts to solutions to thy ar of th form x π mod π +, (22 a, a+π, a+2π,, a+(π π mod π + Assum thr ar j,j 2 {,,π }, j j 2 such that both a+j π and a+j 2 π lift to solutions modulo π +2 Thn π +2 (a+j π π and π +2 (a+j 2 π π, hnc π (a+j π π (a+j 2 π π = (j j 2 π (a+j π π i (a+j 2 π i is divisibl by π +2 If 2, this implis a mod π, but thn a cannot b a solution to (2 Thrfor, if 2, only on of th solutions (22 lifts to a solution modulo π +2 From this w infr that th numbr of solutions to (2 nvr xcds π 2, i kr σ π π 2 Combining this with (29, w gt kr σ k k 2 Thrfor, by th Chins rmaindr thorm, w obtain i= δ t (m,l k 2ω(m, whr ω(m is th numbr of distinct prim divisors of m Sinc 2 ω(m is th numbr of squar-fr divisors of m, w hav k 2ω(m τ(m 2log 2 k m ε, whr τ(m is th numbr of divisors of m Thus, if m N, (26 holds with Now, by Thorm 3, (23 is majorizd by Th function (min{q N ε,n}+q q X N ε q S(Q a= n=m+ (a,q= ( a a n q n N loglog(n+ max r N G(r = is clarly multiplicativ If r is a prim powr p v, thn ( G(r +k p + p 2 + f t = + k p 2 Q /k ft Z ( + k ( p v k = p ϕ(p v

5 Hnc, for all r Æ w hav LARGE SIEVE INEQUALITY WITH CHARACTERS FOR POWERFUL MODULI 5 (24 G(r ( k r (loglogr k ϕ(r Hnc (23 is (loglognq k+ ( N +Q /k (min{q N ε,n}+q Th abov is always majorizd by (loglognq k+( Q +/k +N /2+ε Q Summing ovr all rlvant adic intrvals and combining with (2, w s that ( is majorizd by Thrfor, our rsult follows (loglognq k+ (Q k+ +N +N /2+ε Q k Z 3 Proof of Thorm 2 3 Rduction to Fary fractions in short intrvals As in [], [2], our starting point is th following gnral larg siv inquality Lmma : Lt (α r r Æ b a squnc of ral numbrs Suppos that < /2 and R Æ Put (3 K( := max α Ê R r= α r α whr x dnots th distanc of a ral x to its closst intgr Thn R S(α r 2 c K( (N + Z r= In th squl, w suppos that S is th st of cubs of natural numbrs and that α,,α R is th squnc of Fary fractions a/q with q S(Q, a q and (a,q =, whr Q W furthr suppos that α Ê and < /2 Put I(α := [α,α+ ] and P(α :=, q S (Q,2Q ] (a,q= a/q I(α Thn w hav K( = max P(α α Ê Thrfor, th proof of Thorm 2 rducs to stimating P(α As in [] and [2], w bgin with an ida of D Wolk [8] Lt τ b a positiv numbr with (32 τ In [] and [2] w put τ := /, but in fact our mthod works for all τ satisfying (32 W will latr fix τ in an optimal mannr In th said arlir paprs, τ = / was th optimal choic By Dirichlt s approximation thorm, α can b writtn in th form whr α = b r +z, (33 r τ, (b,r =, z rτ Thus, it suffics to stimat P(b/r+z for all b,r,z satisfying (33

6 6 STEPHAN BAIER AND LIANGYI ZHAO W furthr not that w can rstrict ourslvs to th cas whn (34 z If z <, thn Furthrmor, w hav P(α P ( ( b b r +P r + τ 2 rτ Thrfor this cas can b rducd to th cas z = Morovr, as P(α = P( α, w can choos z positiv So w can assum (34, without any loss of gnrality Summarizing th abov obsrvations, w dduc Lmma 2: W hav (35 K( 2max r Æ r τ max b (b,r= max P z /(τr ( b r +z 32 Estimation of P(b/r+z - first way W now prov a first stimat for P (b/r+z by using som rsults in [] In th squl, w suppos that th conditions (32, (33 and (34 ar satisfid By inquality (4 in [], w hav (36 P ( b r +z +6 <m 4rzQ /t (m,r/t= ( Q A t tz, r t, bm, whr A t (u,m,l is dfind as in (23 and b is th multiplicativ invrs of b modulo r By th rsults of sction 2, for S th st of cubs, th conditions (24, (25 and (26 with X = ε ar satisfid for all t Æ, m Æ, l, u Ê with t τ, m τ/t, (m,l =, mq /τ u Q /t Conditions (24 and (26 imply (37 From (36, (37 and w driv <m 4rzQ /t (m,r/t= (38 P A t ( Q tz, r t, bm C t (+ p + p 2 + = p r p r ( b r +z ( + t S t(q rz 4rzQ X p p = r ϕ(r c 2loglogr, +c 3 Q X rzloglogr+ S t (Q Furthrmor, by (22 and (24, w hav S(Q (loglogr 3 Q /3 Thus, from (38 and th fact that r τ = /2, w obtain Proposition : Lt S b th st of cubs of natural numbrs Suppos that th conditions (32, (33 and (34 ar satisfid Thn w hav ( b (39 P r +z +c 4 ε( Q 4/3 +Q rz t

7 LARGE SIEVE INEQUALITY WITH CHARACTERS FOR POWERFUL MODULI 7 33 Estimation of P(b/r+z - scond way W now prov a scond stimat for P (b/r+z by xtnding th Fourir analytic mthods in [2], [9] to cubic moduli Th following bound for P(b/r+z can b provd in th sam mannr as Lmma 2 in [2] Lmma 3: Lt S b th st of cubs of natural numbrs Suppos that Q (3 δ Q z Thn, ( b (3 P r +z c 5 + 2Q Π(δ,y, δ Q whr I(δ,y = [y /3 c 6 δ/q 2/3,y /3 +c 6 δ/q 2/3 ], J(δ,y = [(y 4δrz,(y +4δrz] and (32 Π(δ,y = W shall prov th following q I(δ,y m J(δ,y m Proposition 2: Lt S b th st of cubs of natural numbrs Suppos that th conditions (32, (33 and (34 ar satisfid Thn w hav ( b (33 P r +z c 7 ε( Q 4/3 +Q /3 r /3 z + /2 (rz /2 To driv Proposition 2 from Lmma 3, w nd th following standard rsults from Fourir analysis Lmma 4: (Poisson summation formula, [5] Lt f(x b a complx-valud function on th ral numbrs that is picwis continuous with only finitly many discontinuitis and for all ral numbrs a satisfis f(a = ( lim 2 x a f(x+ lim x a +f(x Morovr, suppos that f(x c 8 (+ x c for som c > Thn, th Fourir transform of f(x n f(n = n ˆf(n, whr ˆf(x := f(y(xy, Lmma 5: (s [9], for xampl For x Ê\{} dfin ( 2 sinπx π2 (x :=, and ( := lim (x = 2x x 4 Thn (x for x /2, and th Fourir transform of th function (x is ˆ(s = π2 4 max{ s,} Lmma 6: (s Lmma 3 in [9] Lt F : [a,b] Ê b twic diffrntiabl Assum that F (x u > for all x [a,b] Thn, b if(x dx c 9 u a

8 8 STEPHAN BAIER AND LIANGYI ZHAO Lmma 7: (s Lmma 43 in [4] Lt F : [a, b] Ê b twic continuously diffrntiabl Assum that F (x u > for all x [a,b] Thn, b if(x dx c u W shall also nd th following stimats for cubic xponntial sums Lmma 8: (s [], [7] Lt c Æ, k,l with (k,c = Thn, a c ( kd 3 +ld c c /2+ε (l,c c d= Furthrmor, c ( kd 3 c c 2/3 c d= Proof of Proposition 2: W put (34 δ := Q z By Lmma 5, (32 can b stimatd by (35 Π(δ,y ( q y /3 q 2c 6 δ/q 2/3 m ( m yrz 8δrz Using Lmma 4 aftr a linar chang of variabls, w transform th innr sum on th right-hand sid of (35 into ( m yrz = 8δz ( jbq 3 +jyz ˆ(8jδz 8δrz r j m Thrfor, w gt for th doubl sum on th right-hand sid of (35 ( q y /3 ( m yrz q 2c 6 δ/q 2/3 8δrz (36 m = 8δz j (jyzˆ(8jδz d= ( jbd 3 k k d mod ( k y /3 2c 6 δ/q 2/3 whr := r/(r,j and j := j/(r,j Again using Lmma 4 aftr a linar chang of variabls, w transform th innr sum on th right-hand sid of (36 into ( k y /3 (37 = 2c 6δ ( (l d y/3 2c 6 lδ ˆ 2c 6 δ/q 2/3 Q 2/3 l Q 2/3 k k d mod,

9 LARGE SIEVE INEQUALITY WITH CHARACTERS FOR POWERFUL MODULI 9 From (36 and (37, w obtain (38 δ 2Q Q q 6c 6δz Q 2/3 ( q y /3 2c 6 δ/q 2/3 m ( ˆ(8jδz 2c 6 lδ ˆ j l Q 2/3 ( m yrz 8δrz d= ( jbd 3 +ld 2Q Q (jyz l y/3 Applying th Lmmas 5 and 8 to th right-hand sid of (38, and taking r / by (32 and (33 into account, w dduc (39 2Q ( q y /3 δ q c 6 δ/q 2/3 Q c 2δz ε Q 2/3 If j, thn j /(8δz If j = and l, thn m ( m yrz 8δrz l (Q 2/3 /(2c 6δ l 2Q Q (l, 2Q Q 2Q (jyz j z Q (jyz l y/3 + (jyz l y/3 c 3Q l 2/3 j /(8δz by Lmma 6 (tak into account that = if j = If j and l, thn Lmma 7 yilds 2Q (jyz l y/3 c 5/6 4 Q l Q Thrfor, th right-hand sid of (39 is majorizd by (32 c 5 δ ε zq /3 + +z Q 2/3 j 3 j /(8δz l Q 2/3 /(2c 6δ l + zq/6 j /(8δz 3 l Q 2/3 /(2c 6δ Now, w stimat th sums in th last lin of (32 Using (32, (33 and (34, w obtain (32 l c 6 ε l Q 2/3 /(2c 6δ Using th dfinition of, (32, (33 and (34, w obtain (322 j /(8δz j 3 = 3 3 t r j /(8δz (r,j=t j c 7 ε 3 t 2/3 c 8 2ε r /3 r 2Q (jyz Q (l, l

10 STEPHAN BAIER AND LIANGYI ZHAO For A, w hav Thrfor, (323 l A j /(8δz (l, l t t l Q 2/3 /(2c 6δ Using th dfinition of, w obtain (324 = r j /(8δz l A/t t A ε A lt t (l, c 9 ε Q /3 l δ j /(8δz (r,j=t r 8δz j /(8δz t c 2 r 3/2 δz Combining Lmma 3 and (39-324, w obtain ( b (325 P r +z c 7 3ε( +δzq /3 +δq 2/3 r /3 +δ /2 Q /2 r From (34 and (325, w infr th dsird stimat Not that th first trm in th right-hand sid of (325 can b absorbd into th last trm on th right-hand sid of (33 by (34 34 Final proof of Thorm 2 Combining Propositions,2 and (33, w obtain ( b (326 P r +z c 2 ε( { Q 4/3 +min Q rz,q /3 r /3 z } + /2 τ /2 If thn If thn z /2 Q /3 r 2/3, { min Q rz,q /3 r /3 z } = Q rz Q 2/3 /2 r /3 z > /2 Q /3 r 2/3, { min Q rz,q /3 r /3 z } = Q /3 r /3 z Q 2/3 /2 r /3 From th abov inqualitis and (33, w dduc { (327 min Q rz,q /3 r /2 z } Q 2/3 /2 r /3 Q 2/3 /2 τ /3 Combining (326 and (327, w gt ( b (328 P r +z c 22 ε( Q 4/3 τ ε +Q 2/3 /2 τ /3+ε + /2 τ /2 Now w choos τ := N 6/5 Q 4/5, if N 7/8 Q N 3/2, Q 4/7, if Q < N 7/8, N, if N 7/8 Q N 3/2, and := Q 8/7, if Q < N 7/8 Thn th condition (32 is satisfid in ach cas, and from (328 and Lmmas,2, w obtain ( q 3 ( (329 a 2 N ε Q 4/3 +N 9/ Q 2/5 Z, if N 7/8 Q N 3/2, S q 3 Q /3 q (2Q /3 a= (a,q= NQ 2/7+ε Z, if Q < N 7/8 [ W can divid th intrval [,Q] into O(logQ subintrvals of th form Q /3,(2Q ], /3 whr Q Q 3 Hnc, th rsult of Thorm 2 follows from (329

11 LARGE SIEVE INEQUALITY WITH CHARACTERS FOR POWERFUL MODULI Acknowldgmnts This papr was writtn whn th first-namd author hld a postdoctoral position at th Harish-Chandra Rsarch Institut at Allahabad (India and th scond-namd author was supportd by a postdoctoral fllowship at th Univrsity of Toronto Th authors wish to thank ths institutions for thir financial support Rfrncs [] S Bair, On th larg siv with a spars st of moduli, prprint [2] S Bair, Th larg siv with squar moduli, prprint [3] E Bombiri and H Davnport, Som inqualitis involving trigonomtrical polynomials, Annali Scuola Normal Suprior - Pisa 23 ( [4] J Brüdrn, Einführung in di analytisch Zahlnthori, Springr-Vrlag, Brlin ct, 995 [5] D Bump, Automorphic Forms and Rprsntations, Cambridg Stud Adv Math 55, Cambridg Univ Prss, Cambridg, 996 [6] H Davnport, Multiplicativ Numbr Thory, Third Edition, Graduat Txts in Mathmatics, 74, Springr-Vrlag, Nw York, tc, 2 [7] H Davnport and H Halbrstam, Th valus of a trigonomtric polynomial at wll spacd points, Mathmatika 3 ( , Corrigndum and addndum, Mathmatika 4 ( [8] P X Gallaghr, Th larg siv, Mathmatika 4 ( [9] SW Graham, G Kolsnik, Van dr Corput s Mthod of Exponntial Sums, Cambridg Univrsity Prss, Cambridg ct, 99 [] J V Linnik, Th larg siv, Dokla Akad nauk SSSR 36 ( [] L K Hua, On xponntial sums, Sci Rcord (957-4 [2] H L Montgomry, Topics in Multiplicativ Numbr Thory, Lctur Nots in Mathmatics, 227, Spring-Vrlag, Brlin, tc, 97 [3] H L Montgomry, Th analytic principls of larg siv, Bull Amr Math Soc, 84 ( [4] H L Montgomry and R C Vaughan, Th Larg Siv, Mathmatika 2 ( [5] I Nivn, HS Zuckrman, HL Montgomry, An introduction to th Thory of Numbrs, John Wily & Sons, Nw York, 99 [6] RC Vaughan, Th Har-Littlwood Mthod, Cambridg Univrsity Prss, Cambridg, 997 [7] IM Vinogradov, Th Mthod of Trigonomtrical Sums in th Thory of Numbrs, Trav Inst Math Stkloff 23, 947 [8] D Wolk, On th larg siv with prims, Acta Math Acad Sci Hungar 22 (97/ [9] L Zhao, Larg siv inquality with charactrs to squar moduli, Acta Arith 2 ( Stphan Bair Harish-Chandra Rsarch Institut, Chhatnag Road, Jhusi, Allahabad 2 9, India sbair@mrirntin Liangyi Zhao Dpartmnt of Mathmatics, Univrsity of Toronto, Saint Gorg Strt, Toronto, ON M5S 3G3, Canada lzhao@mathtorontodu