Journal of Number Theory

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1 Journal of Number Theory Contents lists available at ScienceDirect Journal of Number Theory Primes of the form αp + β Hongze Li a,haopan b, a Department of Mathematics, Shanghai Jiaotong University, Shanghai , People s Republic of China b Department of Mathematics, Nanjing University, Nanjing 20093, People s Republic of China article info abstract Article history: Received 5 April 2008 Revised 0 March 2009 Availableonline2June2009 Communicated by Carl Pomerance MSC: primary N05 secondary N36, P32 Let β be a real number. Then for almost all irrational α > 0inthe sense of Lebesgue measure, where lim sup πα,β log 2 /, π α,β = { p : bothp αp + β are primes } Elsevier Inc. All rights reserved. Recently Jia [7] solved a conjecture of Long showed that for any irrational number α > 0, there eist infinitely many primes not of the form 2n + 2 αn +, where denotes the largest integer not eceeding. Subsequently, Banks Shparlinski [2] investigated the distribution of primes in the Beatty sequence { αn + β : n }. Motivated by the binary Goldbach conjecture the twin primes conjecture, we have the following conjecture: Conjecture. Let α > 0 be an irrational number β be a real number. Then there eist infinitely many primes p such that αp + β is also prime. On the other h, Deshouillers [3] proved that for almost all in the sense of Lebesgue measure γ > there eist infinitely many primes p of the form [n γ ]. Furthermore, Balog [] showed that for almost all γ > This work was supported by the National Natural Science Foundation of China Grant No * Corresponding author. addresses: lihz@sjtu.edu.cn H. Li, haopan79@yahoo.com.cn H. Pan X/$ see front matter 2009 Elsevier Inc. All rights reserved. doi:0.06/j.jnt

2 H. Li, H. Pan / Journal of Number Theory lim sup {p : bothp p γ are primes} /log 2 γ. In this note we shall show that Conjecture holds for almost all α > 0. Define Theorem. Let β be a real number. Then π α,β = { p : bothp αp + β are primes }. for almost all irrational α > 0. lim sup πα,β log 2 / For a set X R, let mesx denote its Lebesgue measure. Unless indicated otherwise, the constants implied by, O are absolute. Lemma. Let I [0, be an interval. Suppose that b,l > 0.Then mes { α 0, b: {α/l} I } b + l mesi, where {}=. Proof. Without loss of generality, we may assume that I = c, c 2 with 0 c < c 2. Let J ={α 0, b: {α/l} I}. Clearly, Hence, J 0 j b/l j + c l,j + c 2 l. mes J b/l + c 2 c l b + l mesi. Lemma 2. Suppose that b 2 > b > 0 β are arbitrary real numbers. For each 0 < ɛ < /9 all sufficiently large real numbers depending on b,b 2, β ɛ, there eists an eceptional set J E b, b 2 with mes J E = O ɛ such that for any square-free d /3 3ɛ irrational α b, b 2 \ J E, { n : n αn + β 0 mod d } = d 2 p p d + O ɛ /d. 2 Proof. For an irrational α b, b 2,let A ; α = { n αn + β : n } A d ; α = { a A : a 0 mod d }. For a square-free d, we have

3 2330 H. Li, H. Pan / Journal of Number Theory A d ; α = { n /s: αsn + β 0 mod d/s, n, d/s = } s d = μt { n /s: αsn + β 0 mod d/s, t } n s d t d/s = μt { n /s: αsn + β 0 mod td/s }. Suppose that s d t s. Clearly, αsn + β 0 mod td/s { αns 2 /td + βs/td } [0, s/td. Let α = αs 2 /td, β = βs/td, d = td/s y = /s. Clearlyy 2/3+3ɛ d d. Let I a,q = { θ [0, : θq a 2ɛ /y }. Suppose that d 4ɛ q y/ 2ɛ a q with a, q =. If {α } I a,q,then { n y: {an/q + β } [ 2ɛ /q, /d 2ɛ /q } { n y: {α n + β } [0, /d } { n y: {an/q + β } [ 0, /d + 2ɛ /q [ 2ɛ /q, }. Hence, { n y: {α n + β } [0, /d } = y/d + Oq/d + O y 2ɛ /q. Let I d = a q d 4ɛ a,q= I a,q. Clearly, mesi d a q d 4ɛ a,q= mesi a,q d 6ɛ y = td 6ɛ. If {αs 2 /td} / I td/s for each s, t with s d, t s, then Let A d ; α = μt td + O 2ɛ = d 2 /p + O ɛ /d. J d = { α b, b 2 : { αs 2 /td } I td/s for some s, t with s d, t s }. p d

4 H. Li, H. Pan / Journal of Number Theory Applying Lemma, mesj d b 2 b 2 td/s 2 mesi td/s + b 2 <td/s 2 td s mesi td/s = 2 O /3+ɛ. Finally, let J E = d /3 3ɛ J d. Clearly we have mes J E = O ɛ. Lemma 3. Suppose that b 2 > b > 0 β are arbitrary real numbers. For all sufficiently large real numbers depending on b,b 2 β, there eists an eceptional set J E b, b 2 with mes J E = O /00 such that for any irrational α b, b 2 \ J E, { p : both p αp + β } are primes log. 2 3 Proof. Let z = /8 ɛ = /00. Define Let A α ={n αn + β : n }. Clearly, Pz = p p<z SA, z = { a A; a, Pz = }. { p αp + β : z + α z + β p, both p αp + β are primes } is a subset of SA α, z. Furthermore, by Lemma 2, we know that there eists a set J E b, b 2 with mes J E = O ɛ such that for any square-free d /3 3ɛ irrational α b, b 2 \ J E, A d α = d 2 p p d + O ɛ /d, where A d α ={y A α: d y}. Let gm be the completely multiplicative function such that gp = 2/p /p 2 for each prime p. DefineGz = m<z gm. By Selberg s sieve method, m Pz S A α, z A α + O Gz d<z 2 d square-free 3 ωd ɛ /d, where ωd denotes the number of distinct prime divisors of d. Since3 ωd d ɛ, 3 ωd z 2ɛ. d d<z 2

5 2332 H. Li, H. Pan / Journal of Number Theory So it suffices to show Gz log z 2. By Theorem 7.4 of [8], we know Gz = m<z m Pz gm p<z p<z gp = 2/p + /p 2 log z 2. Proof of Theorem. Suppose that b 2 > b > 0. Let { F = α b, b 2 :limsup } πα,β log 2 / < { F n = α b, b 2 :limsup } πα,β log 2 / /n. Clearly F = n> F n. So it suffices to show that mesf n = 0foreveryn >. The measurability of F n will be proven later. Assume on the contrary that there eists n > such that mesf n >0. Let I = c, c 2 be an arbitrary sub-interval of b, b 2.Clearly, c 2 c π α,β dα = c 2 c = p p p = c 2 c αp+β <q αp+β q prime c p+β <q c 2 p+β q prime c p+β<q c 2 p+β q prime p dα mes [ q β/p,q + β/p [c, c 2 ] p + O log p logc p c 2 c log 2 + O, 4 log provided that is sufficiently large depending on b constant in Lemma 3. Let L I = F n I b 2. Suppose that C > is the implied L I,δ = { α I: π α,β δ/log 2}. For any two primes p q, J p,q := { α I: αp + β =q }

6 H. Li, H. Pan / Journal of Number Theory is an interval or empty set. Hence, L I,δ = I k> δ/log 2 p,...,p k are distinct primes q,...,q k are primes is measurable in the sense of Lebesgue measure. By Lemma 3, k J p j,q j j= c 2 πα,β dα O 99/200 c + δ mes L I,δ log + 2 c 2 c C mes L I,δ log, 2 5 provided that is sufficiently large. Combining 4 5, we have mes L I,δ C mesi. 6 C + δ/2 We claim that L I = m>n y y Z L I,/n /m. 7 y Z In fact, for any m > n, if lim sup π α,β /log 2 < n + m, then there eists y 0 such that for any y 0 πα,β n + m log. 2 On the other h, if α y y L I,/n /m, clearly we have By 6 7, we get lim sup π α,β /log 2 n + m. C mesl I lim sup L I,2/3n C + /3n mesi. Since mesf n >0, there eist open intervals I, I 2,... b, b 2 such that F n I k

7 2334 H. Li, H. Pan / Journal of Number Theory But by 6, mesi k C + /4n mesf n. C C mesf n mesl Ik mesi k C + /4n C + /3n C + /3n mesf n. This evidently leads to a contradiction. Remark. In [4] [6], Harman proved that for almost all real α > 0 there are infinitely many pairs of p, q satisfying αp q <ψp, p, q are primes, provided that ψ is a non-increasing positive function 2 p s ψp log p 8 diverges. In fact, Harman [6] established a quantitative version of the above result, on condition that ψn 0, /2 for each n. As an immediate consequence, for almost all α > 0, there eist infinitely many pairs of primes p, q such that αp = q, where is the nearest integer to. For more related results, the readers may refer to [5, Chapter 6]. Acknowledgments We are grateful to Professor Glyn Harman for his very helpful discussions for kindly sending us the copies of Refs. [4] [6]. We also thank Professor Carl Pomerance the anonymous referee for their very useful comments on our paper. References [] A. Balog, On a variant of the Piatetski Shapiro prime number problem, Publ. Math. Orsay [2] W.D. Banks, I.E. Shparlinski, Prime numbers with Beatty sequences, Colloq. Math [3] J.-M. Deshouillers, Nombres premiers de la forme [n c ], C. R. Acad. Sci. Paris, Ser. A [4] G. Harman, Metric diophantine approimation with two restricted variables. III: Two prime numbers, J. Number Theory [5] G. Harman, Metric Number Theory, London Math. Soc. Monogr. Ser., vol. 8, Clarendon Press, Oford, 998. [6] G. Harman, Variants of the second Borel Cantelli lemma their applications in metric number theory, in: R.P. Bambah, et al. Eds., Number Theory, in: Trends Math., Birkhäuser, Basel, 2000, pp [7] C.-H. Jia, On a conjecture of Yiming Long, Acta Arith [8] C.-D. Pan, C.-B. Pan, Goldbach Conjecture, Science Press, Beijing, 992.