Character sums over generalized Lehmer numbers

Transcription

1 Ma t al. Joural of Iualitis ad Applicatios :270 DOI 0.86/s y R E S E A R C H Op Accss Charactr sums ovr gralizd Lhmr umbrs Yuakui Ma, Hui Ch 2, Zhzh Qi 2 ad Tiapig Zhag 2* * Corrspodc: tpzhag@su.du.c 2 School of Mathmatics ad Iformatio Scic, Shaaxi Normal Uivrsity, Xi a, Shaaxi 709, P.R. Chia Full list of author iformatio is availabl at th d of th articl Abstract Lt > 2 b a itgr, 2 b a fixd itgr with,,ψ b a o-pricipal Dirichlt charactr mod. A uppr boud stimat for charactr sums of th form ψa a C, is giv, whr C, {a a,aa mod, a + a}. MSC: L05; L40; N37 Kywords: Lhmr umbr; charactr sums; Kloostrma sums; uppr boud stimat Itroductio Lt b a odd itgr, c b a fixd positiv itgr with c,.forachitgra with a ada,, it is clar that thr xists o ad oly o itgr b with b suchthatab cmod. If a ad b arofoppositparity,tha is calld a Lhmr umbr. Lt Ac, dotthstofalllhmr umbrs, ad rc, thumbr of Ac,. Lhmr [] posd th problm of fidig r,. Bfor procdig w d to rcall that th otatios U OVadU V ar uivalt to U cv for som costat c >0.Wwrit ρ ad O ρ to idicat that this costat may dpd o th paramtr ρ. mas summig ovr rducd rsidu classs, a dots th multiplicativ ivrs of a modulo ad for a ral x w dot x 2πix, {x} th fractal part of x,ad x mi{{x}, {x}}. I 993, Zhag [2]provdthat r, p α φpα + O p α/2 l 3 p α, 2 r, pl φpl + O pl /2 l 2 pl, 2 whr p, l ar two distict odd prims, α is a positiv itgr, ad φistheulrfuctio. For arbitrary odd itgr 3, h [3]sooobtaid r, φ 2 + O /2 d 2 l 2, whr d is th classical divisor fuctio. 206 Ma t al. This articl is distributd udr th trms of th Crativ Commos Attributio 4.0 Itratioal Lics which prmits urstrictd us, distributio, ad rproductio i ay mdium, providd you giv appropriat crdit to th origial authors ad th sourc, provid a lik to th Crativ Commos lics, ad idicat if chags wr mad.

2 Matal. Joural of Iualitis ad Applicatios :270 Pag 2 of 9 Latr,LuadYi[4] gralizd this problm to icomplt itrvals. I fact, lt 3 b a itgr, 2adc b two fixd itgrs with, c,,0<δ, δ 2, thy dfid r δ, δ 2, c; a δ b δ 2 ab c mod a+b ad got a asymptotic formula as follows:, r δ, δ 2, c; δ δ 2 φ+o /2 d 6 log 2. Rctly, itrstig coctios btw Lhmr umbrs ad charactr sums wr ivstigatd by som scholars. For xampl, for a odd prim p, adafixdprimw lss tha p,lt Bw, p { a a p,aa modp, a amodw }. Th, for ay o-pricipal Dirichlt charactr χ mod w, Ma, Zhag ad Zhag [5] got a uppr boud stimat of charactr sums ovr Bw, pas p a Bw,p χa w p /2+ɛ. At almost th sam tim, Ha ad Zhag [6] obtaidaupprboudstimatofth charactr sums ovr Lhmr umbrs as a A,p χa p 2 a+a χa p /2 l 2 p,. whr χ is a arbitrary o-pricipal charactr modulo a odd prim p. Th rsults of charactr sums ovr othr spcial umbrs or polyomials ca also b foud i [7] ad[8]. For mor proprtis of charactr sums ad thir various applicatios, s [9, 0] ad th rfrcs thri. It sms that. caot b xtdd to arbitrary itgr by thir mthods i [6]. Howvr, rlyig o th mthods i [4],w ca ovrcom th obstacls. Lt 3baitgr, 2 b a fixd itgr with,,ψ b a o-pricipal Dirichlt charactr modulo.if a + a, th a is calld a gralizd Lhmr umbr. Dot th st of all gralizd Lhmr umbrs by C, { a a,aa mod, a + a }. Followig th sam tchiu as i [4], w obtai th followig.

3 Matal. Joural of Iualitis ad Applicatios :270 Pag 3 of 9 Thorm Lt 3 b a itgr, 2 b a fixd itgr with,,ψ b a opricipal Dirichlt charactr mod. Th w havth uppr boud stimat a C, ψa a+a ψa /2 d 5 log 2. Lt 3 b a odd itgr, 2 i th thorm, w may immdiatly obtai th followig. Corollary Lt ψ b a o-pricipal Dirichlt charactr modulo. Th w hav a A, ψa 2 a+a ψa /2 d 5 log 2. Lt b a odd prim p, 2iCorollary, th. ca b dducd dirctly as follows. Corollary 2 Lt ψ b a o-pricipal Dirichlt charactr modulo p. Th w hav ψa p /2 log 2 p. a A,p 2 Som lmmas To prov th thorm, w d th followig svral lmmas. First w d a uppr boud stimat of th gral Kloostrma sum Sm,, χ; as follows. Lmma Lt b a positiv itgr ad χ a Dirichlt charactr mod. Th for ay itgrs m ad, w hav Sm,, χ; /2 m,, /2 d, whr Sm,, χ; is dfid by Sm,, χ; a mod ma + a χa. Proof S Lmma of [7]. Lmma 2 Lt b a positiv itgr, χ 0 b th pricipal Dirichlt charactr mod, ψ b a o-pricipal charactr mod, r, r 2 b itgrs with r, r 2.Th w hav Gr, ψgr 2, χ 0 /2 r, r 2,. Proof ByLmma2ofChaptr.2i[], w hav Gr 2, χ 0 μ φφ r 2,, r 2, r 2, whrwhavusdthfactφ/φt /t if t.

4 Matal. Joural of Iualitis ad Applicatios :270 Pag 4 of 9 Not that ψ is a o-pricipal charactr mod, w oly d to cosidr th followig cass. If r,,whav Gr, ψ ψr G, ψ G, ψ /2. If r, >,adψ is a primitiv charactr mod,whav Gr, ψ ψr G, ψ /2. If r, >,adψ is a o-primitiv charactr mod, th Lmma 5 of Chaptr.2 i [] idicats that thr xists o ad oly o such that, withχ th primitiv charactr mod corrspodig χ.thus Gr, ψ χ r χ r, /2 r,. r, μ r, φφ r, τ χ Combiig th abov, w hav Gr, ψgr 2, χ 0 /2 r, r 2,. Lmma 3 Lt 3 b a itgr, χ, ψ b Dirichlt charactrs mod suchthatψ χ 0 ad ψψ χ 0. Th w hav th stimat Gr, χψgr 2, χ φ /2 r, /2 r 2, /2 d. χ mod Proof Combiig Lmmas ad 2,whav Gr, χψgr 2, χ χ mod χ mod χ mod Gr, χψgr 2, χ Gr, ψgr 2, χ 0 Gr, χ 0 Gr 2, ψ ar br2 χψa χb Gr, ψgr 2, χ 0 Gr, χ 0 Gr 2, ψ φ ψa ab mod ar + br 2 φsr, r 2, ψ; Gr, ψgr 2, χ 0 Gr, χ 0 Gr 2, ψ φ /2 r, r 2, /2 d+ /2 r, r 2, φ /2 r, /2 r 2, /2 d.

5 Matal. Joural of Iualitis ad Applicatios :270 Pag 5 of 9 Lmma 4 Lt 0<ρ 2, x 0, x,...,x k b a suc of ral umbrs such that x k x k ρ, x k x k, ad x 0 mi{ x,..., x k }. Th w hav K k x k ρ logk +. Proof S Lmma 2 of Chaptr 5. i []. Lmma 5 Lt 3 b a itgr, ψ b a charactr mod, 2 b a fixd itgr with,,l b a itgr with l. Th w hav Proof Th rlatios a + bl ψa /2 φd 2 log. l, r,, imply that Ad also l r 0. ψa Gr, ψ ar Gr, ψ ar. r r Thus a + bl ψa al ψa r Gr, ψ Gr, ψ r bl ar al bl l r bl bl f l, r,, Gr, ψ r l, r a whr f l, r,, l r.

6 Matal. Joural of Iualitis ad Applicatios :270 Pag 6 of 9 Apply th uppr boud Gr, ψ /2 r,, w hav f l, r,, Gr, ψ r r l /2 r /2 r /2 d d< /2 d d< r, r l r, si π r l /2 d r r,d d k r l /2 d d< μk m kd r r, r l d mkd l. m d m, md l Now writ k /d h 0 0,whr 0, h 0, 0,whav kd 0 h 0 0. Th Lmma 4 d implis mi kd So w gt m jkd mi m j h 0 0 f l, r,, Gr, ψ r r l /2 d d k d< /2 d d< 0 if i j, i, j kd. d k 0 log kd + d log 3/2 d 2 log. 2. Thus a + bl χ a /2 φd 2 log. 3 Proof of th thorm I this sctio, w shall complt th proof of th thorm. Proof of th thorm From th orthogoality rlatio for Dirichlt charactrs mod ad th trigoomtric sum idtity, w ca gt ψa a C, ψa ψa a+a ψa a+b ab mod ψa

7 Matal. Joural of Iualitis ad Applicatios :270 Pag 7 of 9 φ φ φ φ φ χ mod a+b χ mod χ mod l l : E E 2 E 3. ψaχab ψaχab ψaχab a + bl l a + bl l a + bl ψa a + bl ψb First of all, w shall stimat E. Makig us of Lmma 3,wgt E φ φ φ χ mod χ mod χ mod r 2 l l ψaχab a + bl l al bl χψa χb r Gr, χψ ar al Gr 2, χ br 2 bl φ 2 φ 2 χ mod χ mod l r Gr, χψ Gr 2, χ l r l r r 2 a Gr, χψgr 2, χ r 2 l r 2 b f l, r,, f 2 l, r 2,, l r l r 2

8 Matal. Joural of Iualitis ad Applicatios :270 Pag 8 of 9 φ 2 φ /2 r, /2 r 2, /2 d l r r 2 l r l r 2 d 3/2 d 3/2 r, /2 r 2, /2 l r r 2 l r l r l r 2 r, /2 l r 2. Similar to 2., w hav r r, /2 l r d d< d /2 k d log log d /2 d 2 log. d k d< Th E d 3/2 2 d 4 log 2 /2 d 5 log Scod, w stimat E 2. By Lmma 5,whav E 2 φ /2 φd 2 log /2 d 2 log. 3.2 I th sam way w ca gt th stimat E 3 /2 d 2 log. 3.3 Combiig 3., 3.2, ad 3.3, w obtai th rsult. Comptig itrsts Th authors dclar that thy hav o comptig itrsts. Authors cotributios HC ad ZZQ draftd th mauscript. YKM ad TPZ participatd i its dsig ad coordiatio ad hlpd to draft th mauscript. All authors rad ad approvd th fial mauscript. Author dtails School of Scic, Xi a Tchological Uivrsity, Xi a, Shaaxi 7002, P.R. Chia. 2 School of Mathmatics ad Iformatio Scic, Shaaxi Normal Uivrsity, Xi a, Shaaxi 709, P.R. Chia. Ackowldgmts This work is supportd by th Natioal Natural Scic Foudatio of Chia No , th Natural Scic Foudatio of Shaaxi Provic of Chia No. 206JM07, th Scitific Rsarch Program Fudd by Shaaxi Provicial Educatio Dpartmt No. 6JK373 ad th Fudamtal Rsarch Fuds for th Ctral Uivrsitis No. GK Th authors wat to xprss thir grat thaks to th aoymous rfr for his/hr hlpful commts ad suggstios. Th first ad th fourth authors also gratfully ackowldg th support, hospitality, ad xcllt coditios of th School of Computr Scic ad Egirig, School of Mathmatics ad Statistics of UNSW durig thir visits. Rcivd: 2 July 206 Accptd: 4 Octobr 206 Rfrcs. Guy, RK: Usolvd Problms i Numbr Thory, pp Sprigr, Nw York Zhag, WP: O a problm of D. H. Lhmr ad its gralizatio. Compos. Math. 863, Zhag, WP: O a problm of D. H. Lhmr ad its gralizatio II. Compos. Math. 9,

9 Matal. Joural of Iualitis ad Applicatios :270 Pag 9 of 9 4. Lu, YM, Yi, Y: O th gralizatio of th D.H. Lhmr problm. Acta Math. Si. Egl. Sr. 258, Ma, R, Zhag, YL, Zhag, GH: O a kid of Dirichlt charactr. Abstr. Appl. Aal. 203, Articl ID Ha, D, Zhag, WP: Uppr boud stimat of charactr sums ovr Lhmr s umbr. J. Iual. Appl. 203, Xi, P, Yi, Y: O charactr sums ovr flat umbrs. J. Numbr Thory 305, R, GL, H, DD, Zhag, TP: O crtai charactr sums. Quast. Math. to appar 9. Alka, M, Simsk, Y: Gratig fuctio for -Eulria polyomials ad thir dcompositio ad applicatios. Fixd Poit Thory Appl. 203, Shparliski, IE: Op problms o xpotial ad charactr sums. I: Sr. Numbr Thory Appl., vol. 6, pp World Scitific, Hacksack, NJ 200. Pa, CD, Pa, CB: Goldbach Cojctur. Scic Prss, Bijig 98