ON THE DIFFERENCES BETWEEN . - Colgate University

#A3 INTEGERS 12B (2012/13): Integers Conference 2011 ProceedingsON THE DIFFERENCES BETWEEN CONSECUTIVE PRIMENUMBERS, ID. A. Goldston1Department of Mathematics, San José State University, San José, Californiadaniel.goldston@sjsu.eduA. H. LedoanDepartment of Mathematics, University of Tennessee at Chattanooga,Chattanooga, Tennesseeandrew-ledoan@utc.eduReceived: 2/1/12, Accepted: 6/4/12, Published: 10/26/12AbstractWe show by an inclusion-exclusion argument that the prime k-tuple conjecture ofHardy and Littlewood provides an asymptotic formula for the number of consecutiveprime numbers which are a specified distance apart. This refines one aspect of atheorem of Gallagher that the prime k-tuple conjecture implies that the primenumbers are distributed in a Poisson distribution around their average spacing.1. Introduction and Statement of ResultsIn 1976, Gallagher [5], [6] showed that a uniform version of the prime k-tupleconjecture of Hardy and Littlewood implies that the prime numbers are distributedin a Poisson distribution around their average spacing. Specifically, let Pr (h, N )denote the number of positive integers n less than or equal to N such that theinterval (n, n h] contains exactly r prime numbers. Gallagher then proved that anappropriate form of the prime k-tuple conjecture implies, for any positive constantλ and h λ log N as N , thatPr (h, N ) e λλrN.r!In particular, if r 0, then we obtain by an argument using the prime number1 During the preparation of this work, the first author received support from the NationalScience Foundation Grant DMS-1104434.

2INTEGERS: 12B (2012/13)theorem that, as N , pn 1 Npn 1 pn λ log n1 e λN.log N(1)Here, pn is used to denote the nth prime number. The purpose of the present paperis to obtain a refinement of (1), which shows that the Poisson distribution of theprime numbers in short intervals extends down to the individual differences betweenconsecutive prime numbers. To obtain this result, we employ a version of the primek-tuple conjecture formulated as Conjecture H in Section 2, which is equivalent tothe form of the conjecture used by Gallagher.Theorem. Assume Conjecture H. Let d be any positive integer, and let p be a primenumber. Let, further, p 1 , if d is even; 2C2p 2S(d) p d p 2 0,if d is odd;whereC2 1 p 2and define1(p 1)2N (x, d) 0.66016 . . . , 1,pn 1 xpn 1 pn dwhere pn denotes the nth prime number. Then for any positive constant λ and deven with d λ log x as x , we haveN (x, d) e λ S(d)x.(log x)2(2)Here, we note that S(d) is the singular series in the conjectured asymptoticformula for the number of prime pairs differing by d. Our theorem shows that, forconsecutive prime numbers, the Poisson density is superimposed onto this formulafor prime pairs.Our theorem as well as its proof are implicitly contained in the 1999 paper ofOdlyzko, Rubinstein and Wolf [11] on jumping champions.2 Without claiming anyoriginality, we think it is worthwhile to explicitly state and prove (2). More preciseresults when d/ log x 0 will be addressed in a second paper.2 An integer d is called a jumping champion for a given x if d is the most frequently occurringdifference between consecutive prime numbers up to x.

3INTEGERS: 12B (2012/13)2. The Hardy-Littlewood Prime k-Tuple ConjecturesLet H {h1 , . . . , hk } be a set of k distinct integers. Let π(x; H) denote the numberof positive integers n less than or equal to x for which n h1 , . . . , n hk aresimultaneously prime numbers. Then the prime k-tuple conjecture of Hardy andLittlewood [8] is that, for x ,π(x; H) S(H) lik (x),where(3) k 1νH (p)S(H) 1 1 ,pppνH (p) denotes the number of distinct residue classes modulo p occupied by theelements of H, and xdtlik (x) .(4)k2 (log t)Note in particular that, if νH (p) p for some prime number p, then S(H) 0.However, if νH (p) p for all prime numbers p, then S(H) 0 in which case theset H is called admissible. In (3), H is assumed to be admissible, since otherwiseπ(x; H) is equal to 0 or 1.The prime k-tuple conjecture has been verified only for the prime number theorem. That is to say, for the case of k 1. It has been asserted that, in itsstrongest form, the conjecture holds true for any fixed integer k with an error termthat is Ok (x1/2 ε ) at most and uniformly for H [1, x]. (See Montgomery andSoundararajan [9], [10].) However, we do not need such strong conjectures here.Using xkxlik (x) O,(5)(log x)k(log x)k 1obtained from integration by parts, we replace lik (x) by its main term and makethe following conjecture.Conjecture H. For each fixed integer k 2 and admissible set H, we haveπ(x; H) S(H)x(1 ok (1)),(log x)kuniformly for H [1, h], where h λ log x as x and λ is a positive constant.3. Inclusion–Exclusion for Consecutive Prime NumbersThe prime k-tuple conjecture for the case when k 2 provides an asymptoticformula for the number of prime numbers with a given difference d. We need to

4INTEGERS: 12B (2012/13)find a corresponding formula where we restrict the count to prime numbers thatare consecutive, and for this one can use the prime k-tuple conjecture with k 3, 4, . . . and inclusion-exclusion to obtain upper and lower bounds for the number ofconsecutive prime numbers with difference d. This method has appeared in a seriesof papers of Brent [1], [2], [3] and was used by Erdős and Strauss [4] and Odlyzko,Rubinstein and Wolf [11] in their study of jumping champions.We consider a special type of tuple Dk for whichD2 {0, d}and, for k 3,Dk {0, d1 , . . . , dk 2 , d}.Here, we require that d is even. We want to count the number of consecutive primenumbers which do not exceed x and have difference d, namely N (x, d), and for thiswe do inclusion-exclusion with π2 (x, d) 1,p xp p dwhere p is also a prime number and, for k 3,πk (x, d1 , . . . , dk 2 , d) 1.p xp p dp pj dj , 1 j k 2Inserting the expected main term, we obtainπ2 (x, d) S(d) li2 (x) R2 (x, d)(6)πk (x, d1 , . . . , dk 2 , d) S(Dk ) lik (x) Rk (x, Dk ).(7)and, for k 3,We now carry out the inclusion-exclusion. We trivially haveN (x, d) π2 (x, d).The consecutive prime numbers that differ by d are those prime numbers p and p satisfying p p d such that there is no third prime number p with p p p.We can exclude these non-consecutive prime numbers differing by d by removingall triples of this form, although this will exclude the same non-consecutive pair ofprime numbers more than once if there are quadruples of prime numbers such thatp p p p. Hence, writing p p d , we obtain the lower bound N (x, d) π2 (x, d) π3 (x, d , d).1 d d

5INTEGERS: 12B (2012/13)We next obtain an upper bound by including the quadruples eliminated in theprevious step and continue in this fashion to get, for R 1,Q2R 1 (x, d) N (x, d) Q2R (x, d),(8)where, for N 2,QN (x, d) π2 (x, d) N ( 1)kk 3πk (x, d1 , . . . , dk 2 , d).1 d1 . dk 2 dWe use the convention here that an empty sum has the value zero.To evaluate QN (x, d), we require a special type of singular series average considered by Odlyzko, Rubinstein and Wolf. Let, for k 3, Ak (d) S(Dk ).(9)1 d1 . dk 2 dOdlyzko, Rubinstein and Wolf [11] proved that, for k 3,Ak (d) S(d)wheredk 2 Ek (d),(k 2)!(10) dk 2Ek (d) Ok.(11)log log d(See, also, Goldston and Ledoan [7].) Thus, on substituting (6), (7), (9) and (10),we find that, for N 2,QN (x, d) S(d) li2 (x) N ( 1)k Ak (d) lik (x) R2 (x, d)k 3 N ( 1)kk 3 S(d) 1 d1 . dk 2 d 2xRk (x, Dk ) N 2 k x N 1 ddtdt ( 1)k Ek (d)k! log t(log t)2(logt)k2k 0 R2 (x, d) N k 3( 1)kk 3 1 d1 . dk 2 dRk (x, Dk ),where we used (4) in the second line.We can extract a main term independent of N out of the first term on the farright-hand side above by using Taylor’s theorem. With the remainder expressed inLagrange’s form, we have that, for M 0 and x 0,e x M 1e ξ( x)k ( x)M 1 ,k!(M 1)!k 0

6INTEGERS: 12B (2012/13)where ξ lies in the open interval joining 0 and x. Hence, we have k x N x 21 ddt ddt exp2k!logt(logt)logt(logt)222k 0 N 1x1ddt Olog t(log t)22 (N 1)! x ddt explogt(logt)22 N 113dx O ,(log x)2N N log xby the Stirling formula 1M 1/2 M(M 1)! 2πMe1 O.MTherefore, we have proved the following lemma.Lemma. For N 2, we haveQN (x, d) S(d)I(x, d) 2 N O ( 1)k Ek (d)k 3 ( 1)kk 3Nx 1 d1 . dk 2 d1 N whereI(x, d) 2xexpRk (x, Dk ) N 1x(log x)2 3dN log x dt R2 (x, d)(log t)k dlog t ,dt.(log t)24. Proof of the TheoremIf we had imposed the additional condition that n hj is less than or equal to x,for j {1, . . . , k}, in the definition of π(x; H) in Section 2, we would have thatπ2 (n, d) π(x; D2 ) and, for k 3, πk (x, d1 , . . . , dk 2 , d) π(x; Dk ). However, thiscondition has no effect on Conjecture H, since with H [1, h] the condition removesat most h tuples, which are absorbed into the error term. Thus, assuming thatConjecture H holds true for k N , we have that R2 (x, d) o(S(d)x/(log x)2 ),

7INTEGERS: 12B (2012/13)Rk (x, Dk ) ok (S(Dk )x/(log x)k ) and, by (11), Ek (d) ok (dk 2 ). Then by thelemma and since S(d) 1 for d even, we have, for x and d λ log x, xxQN (x, d) S(d)I(x, d) oN e2λ oS(d)(log x)2(log x)2 N x ok S(Dk )(log x)kk 3 1 d1 . dk 2 d N 113λx O (log x)2N N x2λ S(d)I(x, d) oN e S(d)(log x)2 N4λx O.N(log x)2Finally, on letting N tend to infinity sufficiently slowly, the theorem follows fromthe estimatexI(x, d) e λ(log x)2and (8).To prove this last estimate, we let d d/ log x and apply (5) to obtain the upperbound xdtxx d d I(x, d) e e O,2(log x)2(log x)32 (log t)and the lower bound x ddtI(x, d) exp2log x log log xx/ log x (log t) log log xx exp d 1 Oli2 (x) li2log xlog x xd log log x e d1 O.(log x)2log xHence, we have d I(x, d) e xd log log x1 O,(log x)2log xand the required estimate now follows since, if d λ log x, d λ as x . Hence,the proof of the theorem is completed.Acknowledgment. The authors would like to express their sincere gratitude tothe referee for his comments on the earlier version of this paper.

INTEGERS: 12B (2012/13)8References[1] R. P. Brent, The distribution of small gaps between successive primes, Math. Comp. 28(1974), 315–324.[2] R. P. Brent, Irregularities in the distribution of primes and twin primes, in Collection ofarticles dedicated to Derrick Henry Lehmer on the occasion of his seventieth birthday, Math.Comp. 29 (1975), 43–56.[3] R. P. Brent, Correction to: Irregularities in the distribution of primes and twin primes(Math. Comp. 29 (1975), 43–56), Math. Comput. 30 (1976), no. 133, 198.[4] P. Erdős and E. G. Straus, Remarks on the differences between consecutive primes, Elem.Math. 35 (1980), no. 5, 115–118.[5] P. X. Gallagher, On the distribution of primes in short intervals, Mathematika 23 (1976),4–9.[6] P. X. Gallagher, Corrigendum: On the distribution of primes in short intervals (Mathematika23 (1976), 4–9), Mathematika 28 (1981), no. 1, 86.[7] D. A. Goldston and A. H. Ledoan, The jumping champion conjecture, submitted for publication (2012), available at http : // arxiv.org / pdf / 1102.4879v1.[8] G. H. Hardy and J. E. Littlewood, Some problems of ‘Partitio numerorum’; III: On theexpression of a number as a sum of primes, Acta Math. 44 (1922), no. 1, 1–70. Reprintedas pp. 561–630 in Collected Papers of G. H. Hardy, Vol. I (including joint papers with J.E. Littlewood and others; edited by a committee appointed by the London MathematicalSociety), Clarendon Press, Oxford University Press, Oxford, 1966.[9] H. L. Montgomery and K. Soundararajan, Beyond pair correlation, in: Erdős and his mathematics, I (Budapest, 1999), 507–514; Bolyai Soc. Math. Stud. 11, János Bolyai Math. Soc.,Budapest, 2002.[10] H. L. Montgomery and K. Soundararajan, Primes in short intervals, Comm. Math. Phys.252 (2004), no. 1-3, 589–617.[11] A. Odlyzko, M. Rubinstein and M. Wolf, Jumping champions, Experiment. Math. 8 (1999),no. 2, 107–118.

Department of Mathematics, San Jos e State University, San Jos e, California daniel.goldston@sjsu.edu A. H. Ledoan Department of Mathematics, University of Tennessee at Chattanooga, Chattanooga, Tennessee andrew-ledoan@utc.edu Received: 2/1/12, Accepted: 6/4/12, Published: 10/26/12 Abstract