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 Advances in Pure Mathematics, 2013, 3, 25-32 http://dx.doi.org/10.4236/apm.2013.37A003 Published Online October 2013 (http://www.scirp.org/journal/apm) Primes in Arithmetic Progressions to Moduli with a Large Power Factor Ruting Guo Network Center, Shandong University, Jinan, China Email: rtguo@sdu.edu.cn Received July 8, 2013; revised September 9, 2013; accepted October 6, 2013 Copyright © 2013 Ruting Guo. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. ABSTRACT Recently Elliott studied the distribution of primes in arithmetic progressions whose moduli can be divisible by high- powers of a given integer and showed that for i nteger and real number Ahere is a Bch that 2 a Tsu0.0BA 112,1,1max max;,,BAyx rqddxqLdqLi yxyqdr qd qL 133exp loglogqx x holds uniformly for moduli that are powers of . In this paper we are able to improve his eywords: Primes; Arithmetic Progressions; Riemann Hypothesis 1. Introduction and Main Results with to count the number of primes in the arithmetic pro- aresult. KLet p denote a prime number. For integer,aq,1aq , we introduce ;,xqmod1pxpa qa gression modaq not exceeding x. For fixed q, we have  1;,xqa xq as x tends to infinity. However the most important thinin this context is the range uniformity for the moduli q in terms of g x. The Siegel-Walfisz Theorem, see for eample [1], shows that this estimate is true only if AqL, where and throughout this paper we denote logxx by L. The Generalized Riemann Hypothesis for let L-functions could give a much better result: non-trivial estimate holds for Dirich 122qxL. Unfortunately the Generalized Riemann Hypos withstood the attack of several generations of researchers and it is still out of reach. However number theorists still want to live a better life with ou t the Generalized Riemann Hypothesis. direction the famous Bombieri-Vinogradov theorem [2, 3], states that Theorem A. For any 0A there exists a constant thesis haTherefore they try to find a satisfactory substitu te. In this 0BBA such that ,1max max;y, ,Ayx aqqQLi yqa xLq where q is the Euler totient function, 12,BQxL and 2d.logyuLi u Reycently in order to study the arithmetic functions on shifted primes, Elliott [4] studied the distribution of primes in arithmetic progressions whose moduli can be divisible by high-powers of a given integer. More pre- cisely, he showed that Theorem B. Let a be an integer, 2a. If 0A, then there is a 0ABB such that 112,1,1max, Li yxrmax ;,BAyx rqddxqLdqyqdqd qL Copyright © 2013 SciRes. APM R. T. GUO 26 33exp loglogqx x holds uniformly for moduli 1that are powers of When result recovers the Bombieri- Vinogradov theorem. And obviously his result gives a n of primes in arithmeticst impa. , his1qdeep insight into the distributio progressions. The moortant thing Elliott concerned in [4] is that in Theorem B the parameter q may reach a fixed power of x. However we want to purse the widest uniformity in q by using some techniques estab- lis l newhed in the study of Waring-Goldbach problems. We shall prove the following resut. Theorem 1.1. Let a be an integer, 2a. If 0A, then there is a 0BA such that B112,1,1Byx rqddxqLdqqd qmax max;,Liyqdr,AL holds uniformly for moduli yx235exp loglogqx x that are powers of When and an odd prime, our result gives that for tmoduli with the form a. 1dhes a e particular q,1,2,3,nqpn ;, 1,AxqrO Lq Li xholds uniformly for moduli 235exp loglog.qx x Then the special case of our result shows that the least prime in these special progressions satisfies min ,Pqr mod qnr52min ,.Pqr q Thisproves a former result given by Barbhudakov [5 result iman, Linnik and Ts], 83min ,Pqr q where ,1,2,3,nqpn. If we focus our attention on the least prime in arithmetic progressions with special moduli, we can prove the following result. rem 1.2. Let be anteger, . If th Theo n ia 2a0A, en there is a 0BBA such that 9120,1,1Byx rqddx qLdqqd qmax max;Li yxy, ,Aqdr L holds uniformly for moduli 5312 exp loglogqx x that are powers of Then our resuows that the least prime a. lt shmin ,Pqr in these special progressions modnr q satisfies 12 5min ,PqrIt should be remrk.q aed that the Generalized Riemann Hld allow ypothesis for Dirichlet L-functions wou112AqdxL with no further restrictionpon the natu ure of . Therefore our Theorems 1.1 and 1.2 can be compared with the resu lt under the Geralized Riemann H2. Preliminqenypothesis. ary Reduction Let n denote von Mangoldt’s function, and for mutually prime integers w and r, let ;, .wywrn r modnynrFo 342wx and an integer 1q, define  ,1,11max max;,;,.rqdyxdwdqGwyqdr yqrd Lemma 2.1. For any Then0,1 41 2K, we have  31611exp logloglog2log .KKGxqL Gxxqq xx (1) unteg ers iformly for positive in3exp loglog,3qxxx where 25, if 920 12 and 512, if 14 920 . Here 1q|nq. For Dirichlet characters  an defind real e 0ynyyn ,.n Lemma 2.t (2) 2. Le,y defined as in (2). Then 12 2512max ,cyxxQDQDL4mod ,yxdQ Ddxho ers Lemma 2.3. Let (3) lds uniformly for all integers 1D and real numb2, 1.xQ ,y defined as in (2). Then 11 202mod max ,.cyxdQ DdyxxQD  L ds uniformly for all integers the innall racters 3. Proof of Lemma 2.2 (4) hol and real numbers 1D2, 1.Q Here er sum is taken over hlet chamod Dd . xprimitive DircLet 25XYX Copyright © 2013 SciRes. APM R. T. GUO 27and 110,,MM be positive real numbers such that 151106 10and 2,,2.YMM XMM X  (5) For define where is the Möbius function. Then we define the fu1, ,10j1, if2,, 5,,if 6,,10,jam jmj log ,if1,mj (6) n nctions  ,jjjsmMam mfs m ,,and  110,, ,Fsfsfs (7) where  is a Dirchlet character, s a complex variable. mma 3.1. Let Le,Fs be as in (7), and 1A 0,arbitrary. Then for an and y 12ARX ATX mod|32logdrRRT ) 111022122,rR rTcTXX Xdd (8where is an absolute constant independent of 1,dTFitt0c A, but thmplied in depends on e constant i.A Proof of Lemma 3.1. This lemma with 1d was established in [6], and in this general formWe mention that in general the exponent 3/10 to [7]. X in the oixnvalue of Dsecond termon the right-hand side is the best possiblen considering the lack of sth power mea irchlet L-functions. Now we complete the proof of Lemma 2.2. Proof of Lemma 2. 2. In (5), we take 25,.YxXx Define  ,,jjamfs and ,Fs as above. To go s identity [8], whic and further, we first recall Heath-Browkn’2nz with h states that for any1z1,k   12 21211 21lojjjk 11 gjjjjnnnnnnznnnj kn The for . n2522YxyXx ,,y of which is of t1 11010110111101010,,2:,mMmMymmyamma mmS where  denotes the vector 12 with 10,,,MMMjM as in (5). Obviously some of tervals the in,2jjMM may contain onusing mmation formPropo-sition 5.5 in [1]), and then shifting the contour to the left, we have ly integer 1. By ula with Ty (see Perron’s su   11 21112121121112 1221,d21.2ssLiyLiyiyiy LLiy iyiyyyiyFssOLisOLi  On usinSg the trivial estimate 11011111210,,,Fiyf iyfiy,MLMMx L  the integral on the two horizontal segments above can be estimated as 121 131111022121 1max .Lmax ,LyFiyyyxLxyLxLyThen we ha ve 113221031102211,d122 21d,.21it ityyyyyyFittO xLittyF itxLt   S ,Fs does not depend on , we have yNoting that 3110 111022<1dmax ,,.21xxYyxtyLxFit xt  L(9) hand weOn the other have 2max ,.yY yY (10) From (9) and (10), we have mod max ,yxQ Ddyis a linear combination of terms, each he form 10OL2 2 Axxy ound log 2;, 1,y rDD yyyDwhich is valid for all positive y. W ith these bounds  2logAxx2mod1 22121max; ,log ,logDyxDTAyRDyDrDixxxD xTx   holds uniformly for 2342log,ATx xDx . We set 12.Tx The double-sum does not exceed 12mod1 212 2121ku k12124242 d,2,kkkkDTTxxNuDwhere  1 is the largest value of  taken over all the zeros i in the rectangle  0Re 1,Im2.ssT Supp for the moment that and that there is no zero that is exception al in tha 4.1, then we may take osing Dqde sense of Lemm 134loglog 23loglog 23.c dqTqT InLe view of mma 4.3, typically  1u12111212121112 5212d,,,,,,lo d12,,logd .uuuuxNu DgxNu DNu DxxuxNDcDxxu   with restriction 512 2qx  we have integral is 7812 5exp log,Dx x then the 78 1932 32exp logexp,xxxlog x uniformly for 2T and d . Moreover, 2,her 12, ,NDD2 ls earlier. 1log AqdRxx ogD Prachar [11], Satz 3.3, p. 220, aAltogetwith for which ,and the corresponding function the same uniformity in d. If there is an exceptional zero mod qd ,1112log4ca ,Ls is attached to a real character induced by a itive character primmod ,D ,d thenis a disor with D viof some4ad aows that there is no character e line-send anication of fuher L-function that n th0 applrt,DDLemma 4.2 shformed with a realmodhas a real zero ogment 4 ,a 1112log4Re 1,Imcas s  unless that character is also induced by. In particul, le by mod D arD will be divisib .D For those mle of D we moduli qd for which 4ad is not ay choose the same a multip as before and recover the above estima for .qdR teHence   1,log logAAdqdxqxwhere '' icates that the moduli are not divisible by the (possibly non-existent) modulus D. A theorem of Siegel shows that foere is a positive constant max;,yqdr1 yxdqdyxxindr any th 0 c so that an L-ffromed with a real chunction aracter mod D has n o zero on theline-segment ()0;cD se1R1,Im  cf. Prachar [ 1 1], Satz 8. 2, p .144. Unles s 112log ,Dc x this again allows the argument to pceed. We may therefore assume that  ro 2ADxlogthe above summ and remove the restriction from ation over at an expense o ''f d 240modloglog .log Adad Dxxxx xqdqD qx  A modified version of this argument delivers the bound   1Ao exceptition we see that max;,,logyx yqrqqx and in this case there is nonal zero. By substrac yx1lo AGxqxg indeed holds for every fixed 0.A Since log ,qq an application of Lemma 2.1 6,BAshows that with  Copyright © 2013 SciRes. APM R. T. GUO Copyright © 2013 SciRes. APM 32  ,1log1max max;,,logBrqdy xqd xxAyyqdr qdxqx of whuniformly for moduli 3exp loglogqx x that are powers aere 25, if 12 and 512, if 920. Replacing ;,yqd r in this bound by ;, logpyyqdr pmoprdqd introduces an error  1212log612 1logloglog ,BmBqd x xAqd xxpxxq xthe congruence condition  haviignored. Employing the Brun-Titchmarsh bound valid unifromly for 12logmodmmxpxpr qd modmpr qdng been  1;,log ,yDry Dy341,,1,DyrD bution to the sum in the theorem s from that e rangeAyy  occur in th We see that the contrithat arise maximag is 00loxx 12 11logdy y10001log .dx qAyq qxq x e confine our attention to maximaover the range Integration bys that We may therefor 0.yyx parts show 0max;,1max;yyxLi yyDr DyD D The theorems hold with 0logyyxDx6.0;, , .Li yyyDr r  BA 8. Acknowledgements This work is su (Gpported by IIFSDUrant No2012JC020). Th e author would like to thank to ProfJianya Liu and Guangshi Lü for their encouragements. REFERENCES [1] H. Iwaniec and E. Kowalski, “Analy tic Number Theory,” American Mathematical Society, P r o v i d e n c e , 2004. [2] rge Sieve,” , Vol. 12, 0025579300005313 E. Bo mb ie r i, “ On the LaMathematikaNo. 2, 1965, pp. 210-225. http://dx.doi.org/10.1112/S [3] A. I. Vinogradov, “The Density Hypothesis for Dirichlet L-Series,” Izvestiya Rossiiskoi Akademii Nauk SSSR. Seriya Matematicheskaya, Vol. 29, 1965, pp. 903-934. http://dx.doi.org/10.1007/s11139-006-0250-4 [4] P. D. T. A. Elliott, “Primes in Progressions to Moduli with a Large manujan Journal, Vol. 13, No. 1h a t in Four - ford), Vol. Power Factor,” The Ra-3, 2007, pp. 241-251. [5] M. B. Barban, Y. V. Linink and N. G. Chudakov, “On Prime Numbers in an Arithmetic Progression witPrime-Power Difference,” Acta Arithmetica, Vol. 9, No . 4, 1964, pp. 375-390. [6] J. Y. Liu and M. C. 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Heath-Brown, “Zero-Free Regions for DiriL-Functions and the Least Prime in an Archlet ithmetic Pro- gressions,” Proceedings of the London Mathematical So- ciety, Vol. 64, No. 2, 1992, pp. 265-338. http://dx.doi.org/10.1112/plms/s3-64.2.265 . essor