CONSECUTIVE PRIMES AND BEATTY SEQUENCES

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1 CONSECUTIVE PRIMES AND BEATTY SEQUENCES WILLIAM D. BANKS AND VICTOR Z. GUO Abstract. Fix irrational numbers α, ˆα > 1 of finite type and real numbers β, ˆβ 0, and let B and ˆB be the Beatty sequences B.= αm + β ) m N and ˆB.= ˆαm + ˆβ ) m N. In this note, we study the distribution of pairs p, p ) of consecutive primes for which p B and p ˆB. Under a strong but widely accepted) form of the Hardy-Littlewood conjectures, we show that {p x : p B and p ˆB} = αˆα) 1 πx) + O xlog x) /+ε). 1. Introduction For any given real numbers α > 0 and β 0, the associated generalized) Beatty sequence is defined by B α,β.= αm + β ) m N, where t is the largest integer not exceeding t. If α is irrational, it follows from a classical exponential sum estimate of Vinogradov [7] that B α,β contains infinitely many prime numbers; in fact, one has # { prime p x : p B α,β } α 1 πx) x ), where πx) is the prime counting function. Throughout this paper, we fix two not necessarily distinct) irrational numbers α, ˆα > 1 and two not necessarily distinct) real numbers β, ˆβ 0, and we denote B.= B α,β and ˆB.= Bˆα, ˆβ. 1.1) Our aim is to study the set of primes p B for which the next larger prime p lies in ˆB. The results we obtain are conditional, relying only on the Hardy-Littlewood conjectures in the following strong form. Let H be a finite subset of Z, and let 1 P denote the indicator function of the primes. The Hardy-Littlewood conjecture for H asserts that the estimate du 1 P n + h) = SH) log u) + H Ox1/+ε ) 1.) n x h H holds for any fixed ε > 0, where SH) is the singular series given by SH).= ) H mod p) H. p p) p Our main result is the following. Date: September 1,

2 W. D. BANKS AND VICTOR Z. GUO Theorem 1.1. Fix irrational numbers α, ˆα > 1 of finite type and real numbers β, ˆβ 0, and let B and ˆB be the Beatty sequences given by 1.1). For every prime p, let p denote the next larger prime. Suppose that the Hardy-Littlewood conjecture 1.) holds for every finite subset H of Z. Then, for any fixed ε > 0, the counting function satisfies the estimate πx; B, ˆB).= {p x : p B and p ˆB} πx; B, ˆB) = αˆα) 1 πx) + O xlog x) /+ε), where the implied constant depends only on α, ˆα and ε. Our results are largely inspired by the recent breakthrough paper of Lemke Oliver and Soundararajan [], which studies the surprisingly erratic distribution of pairs of consecutive primes amongst the φq) permissible reduced residue classes modulo q. In [] a conjectural explanation for this phenomenon is given which is based on the strong form of the Hardy-Littlewood conjectures considered in this note, that is, under the hypothesis that the estimate 1.) holds for every finite subset H of Z.. Preliminaries.1. Notation. The notation t is used to denote the distance from the real number t to the nearest integer; that is, t.= min t n n Z t R). We denote by t and {t} the greatest integer t and the fractional part of t, respectively. We also write et).= e πit for all t R, as usual. Let P denote the set of primes in N. In what follows, the letter p always denotes a prime number, and p is used to denote the smallest prime greater than p. In other words, p and p are consecutive primes with p > p. We also put δ p.= p p p P). For an arbitrary set S, we use 1 S to denote its indicator function: { 1 if n S, 1 S n).= 0 if n S. Throughout the paper, implied constants in symbols O, and may depend where obvious) on the parameters α, ˆα, ε but are absolute otherwise. For given functions F and G, the notations F G, G F and F = OG) are all equivalent to the statement that the inequality F c G holds with some constant c > 0... Discrepancy. We recall that the discrepancy DM) of a sequence of not necessarily distinct) real numbers x 1, x,..., x M [0, 1) is defined by DM).= sup V I, M) M I,.1) I [0,1)

3 CONSECUTIVE PRIMES AND BEATTY SEQUENCES where the supremum is taken over all intervals I = b, c) contained in [0, 1), the quantity V I, M) is the number of positive integers m M such that x m I, and I = c b is the length of I. For any irrational number a we define its type τ = τa) by the relation τ.= sup { t R : lim inf n nt an = 0 }. Using Dirichlet s approximation theorem, one sees that τ 1 for every irrational number a. Thanks to the work of Khinchin [1] and Roth [5, 6] it is known that τ = 1 for almost all real numbers in the sense of the Lebesgue measure) and for all irrational algebraic numbers, respectively. For a given irrational number a, it is well known that the sequence of fractional parts {a}, {a}, {a},..., is uniformly distributed modulo one see, for example, [, Example.1, Chapter 1]). When a is of finite type, this statement can be made more precise. By [, Theorem., Chapter ] we have the following result. Lemma.1. Let a be a fixed irrational number of finite type τ. For every b R the discrepancy D a,b M) of the sequence of fractional parts {am+b}) M m=1 satisfies the bound D a,b M) M 1/τ+o1) where the function implied by o ) depends only on a. M ),.. Indicator function of a Beatty sequence. As in 1 we fix possibly equal) irrational numbers α, ˆα > 1 and possibly equal) real numbers β, ˆβ 0, and we set In what follows we denote B.= B α,β and ˆB.= Bˆα, ˆβ. a.= α 1, â.= ˆα 1, b.= α 1 1 β) and ˆb.= ˆα 1 1 ˆβ). It is straightforward to show that 1 B m) = ψ a am + b) and 1 ˆBm) = ψââm + ˆb) m N),.) where for any t 0, 1) we use ψ t to denote the periodic function of period one defined by { 1 if 0 < {x} t, ψ t x).= 0 if t < {x} < 1 or {x} = Modified Hardy-Littlewood conjecture. For their work on primes in short intervals, Montgomery and Soundararajan [4] have introduced the modified singular series for which one has the relation S 0 H).= T H 1) H\T ST ), SH) = T H S 0 T ).

4 4 W. D. BANKS AND VICTOR Z. GUO Note that S ) = S 0 ) = 1. The Hardy-Littlewood conjecture 1.) can be reformulated in terms of the modified singular series as follows: 1 P n + h) 1 ) du = S 0 H) log n log u) + H Ox1/+ε )..) n x h H Lemma.. We have 1 t 1 <t h 1 S 0 {0, t}) h 1/+ε, S 0 {t, h}) h 1/+ε, S 0 {t 1, t }) = 1 h log h + 1 Ah + Oh1/+ε ), where A.= C 0 log π and C 0 denotes the Euler-Mascheroni constant. Proof. Let us denote B.= and S 0 {0, t}), C.= D ±.= 1 t 1 <t h±1 S 0 {t 1, t }) for either choice of the sign ±. Clearly, S 0 {0, h}) + B + C + D = D + and B = From [4, Equation 16)] we derive the estimates D ± = 1 h log h + 1 Ah + Oh1/+ε ). S 0 {t, h}), S 0 {0, h t}) = C. Using the trivial bound S 0 {0, h}) log log h and putting everything together, we finish the proof..5. Technical lemmas. Let νu).= 1 1/ log u. Note that νu) 1 for u. Lemma.. Let c > 0 be a constant, and suppose that f is a function such that fh) h c for all h 1. Then, uniformly for u x and λ R we have h log x) fh)νu) h eλh) = h 1 fh)νu) h eλh) + O c x 1 ). Proof. Write νu) h = e h/h with H.= log νu)) 1. Since H log u for u, for any h > log x) we have h/h h / as u x; therefore, fh)νu) h eλh) h c e h/ x 1 h c e h1/ h/ c x 1, h>log x) h>log x) h>log x) and the result follows.

5 CONSECUTIVE PRIMES AND BEATTY SEQUENCES 5 The next statement is an analogue of [, Proposition.1] and is proved using similar methods. Lemma.4. Fix θ [0, 1] and ϑ = 0 or 1. For all λ R and u, let R θ,ϑ;λ u).= h 1 S λ u).= h 1 When λ = 0 we have the estimates h θ log h) ϑ νu) h eλh), S 0 {0, h})νu) h eλh). R θ,0;0 u) = 1 Γ1 + θ)log u)1+θ + O1), R θ,1;0 u) = 1 log )Γ1 + θ)log u)1+θ + O1), S 0 u) = 1 log u 1 log log u + O1). On the other hand, if λ is such that λ log u) 1, then max { R θ,ϑ;λ u), S λ u) } λ 4. Proof. We adapt the proof of [, Proposition.1]. As in Lemma. we write νu) h = e h/h with H.= log νu)) 1. We simplify the expressions R θ,ϑ;λ u), S λ u) and T λ u) by writing νu) h eλh) = e h/h H λ with H λ.= 1 πiλh. Since Rh/H λ ) = h/h > 0 for any positive integer h, using the Cahen-Mellin integral we have R θ,ϑ;λ u) = h 1 In particular, and h θ log h) ϑ e h/h λ = 1 πi R θ,0;λ u) = θ πi 4+i 4 i 4+i 4 i ) h θ log h) ϑ Γs)Hλ s ds. h 1 h s s ζs θ)γs)h s λ ds.4) 4+i R θ,1;λ u) = R θ,0;λ u) log θ s ζ s θ)γs)hλ s ds..5) πi 4 i When λ 0 we have Rθ,0;λ u) θ 4 H λ 4 ζ4 θ + it)γ4 + it) dt π H λ 4 = H 1 + 4π λ H hence the bound R θ,0;λ u) λ 4 holds if λ log u) 1 since H log u for u. In the case that λ = 0, the stated estimate for R θ,0;0 u) is obtained by shifting the line of integration in.4) to the line {Rs) = 1 } say), taking into account the residues of the poles of the integrand at s = 1 + θ and s = 0. ),

6 6 W. D. BANKS AND VICTOR Z. GUO Our estimates for R θ,1;λ u) are proved similarly, using.5) instead of.4) and taking into account that ζ s θ) = s 1 θ) 1 + O1) for s near 1 + θ. Next, for all λ R and u, let T λ u).= h 1 S{0, h}) e h/h λ. Since S 0 {0, h}) = S{0, h}) 1 for all integers h, and S{0, h}) = 0 if h is odd, it follows that S λ u) = T λ u) R 0,0;λ u) = T λ u) 1 log u + O1). Hence, to complete the proof of the lemma, it suffices to show that T 0 u) = log u 1 log log u + O1) and T λu) λ 4 if λ log u) 1. As in the proof of [, Proposition.1], we consider the Dirichlet series F s).= h 1 which can be expressed in the form ζs)ζs + 1) F s) = ζs + ) p 1 S{0, h}) h s, ) 1 p 1) + p, p 1) p s+1 + 1) and the final product is analytic for Rs) > 1. Using the Cahen-Mellin integral we have T λ u) = 1 4+i F s)γs)hλ s ds..6) πi For λ 0 we have T λ u) H λ 4 π 4 i F 4 + it)γ4 + it) dt H λ 4 H = 1 + 4π λ H hence T λ u) λ 4 holds provided that λ log u) 1. For λ = 0, we shift the line of integration in.6) to the line {Rs) = 1 } say), taking into account the double pole at s = 0 and the simple pole at s = 1. This leads to the stated estimate for T 0 u). We also need the following integral estimate proof omitted). Lemma.5. For all λ R and x, let I λ x).= eλu) νu) log u du. When λ = 0 we have the estimate I 0 x) = x x log x + O whereas for any λ 0 we have I λ x) λ 1. log x) ), )

7 CONSECUTIVE PRIMES AND BEATTY SEQUENCES 7. Proof of Theorem 1.1 For every even integer h we denote π h x; B, ˆB).= {p x : p B, p ˆB and δ p = h} = 1 B n)1 ˆBn + h)f h n), where f h n).= 1 P n)1 P n + h) Clearly, 0<t<h πx; B, ˆB) = 1 1P n + t) ) = h log x) n x { 1 if n = p P and δ p = h, 0 otherwise. ) π h x; B, ˆB) x + O..1) log x) Fixing an even integer h [1, log x) ] for the moment, our initial goal is to express π h x; B, ˆB) in terms of the function S h x).= n x f h n) recently introduced by Lemke Oliver and Soundararajan [, Equation.5)]. In view of.) we can write π h x; B, ˆB) = n x ψ a an + b)ψâân + h) + ˆb)f h n)..) According to a classical result of Vinogradov see [8, Chapter I, Lemma 1]), for any such that 0 < < 1 and 1 min{a, 1 a} 8 there is a real-valued function Ψ a with the following properties: i) Ψ a is periodic with period one; ii) 0 Ψ a t) 1 for all t R; iii) Ψ a t) = ψ a t) if {t} a or if a + {t} 1 ; iv) Ψ a is represented by a Fourier series Ψ a t) = k Z g a k)ekt), where g a 0) = a, and the Fourier coefficients satisfy the uniform bound For convenience, we denote g a k) min { k 1, k 1} k 0)..) I a.= [0, ) a, a + ) 1, 1), so that Ψ a t) = ψ a t) whenever {t} I a. Defining Ψâ and Iâ similarly with â in place of a, and taking into account the properties i) iii), from.) we deduce that π h x; B, ˆB) = n x Ψ a an + b)ψâân + h) + ˆb)f h n) + OV x)),.4)

8 8 W. D. BANKS AND VICTOR Z. GUO where V x) is the number of positive integers n x for which {an + b} I a or {ân + h) + ˆb} Iâ. Since I a and Iâ are unions of intervals with overall measure 4, it follows from the definition.1) and Lemma.1 that V x) x + x 1 1/τ+o1) x )..5) Now let K 1 be a large real number, and let Ψ a,k be the trigonometric polynomial given by Ψ a,k t).= g a k)ekt). Using.) it is clear that the estimate k K Ψ a t) = Ψ a,k t) + OK 1 1 ).6) holds uniformly for all t R. Defining Ψâ,K in a similar way, combining.6) with.4), and taking into account.5), we derive the estimate where Therefore π h x; B, ˆB) = Σ h + O x + x 1 1/τ+ε + K 1 1 x ), Σ h.= n x Ψ a,k an + b)ψâ,k ân + h) + ˆb)f h n) = n x = π h x; B, ˆB) = k, l K g a k)ekan + b))gâl)elân + h) + ˆb))f h n) k, l K g a k)ekb)gâl)elˆb) elâh) n x k, l K eka + lâ)n)f h n). g a k)ekb)gâl)elˆb) elâh) eka + lâ)u) ds h u)) + O x + x 1 1/τ+ε + K 1 1 x ),.7) which completes our initial goal of expressing π h x; B, ˆB) in terms of the function S h. To proceed further, it is useful to recall certain aspects of the analysis of S h that is carried out in []. First, writing 1 P n).= 1 P n) 1/ log n, up to an error term of size Ox 1/+ε ) the quantity S h x) is equal to 1 P n) + 1 ) 1 P n + h) + 1 ) log n log n n x = 1 log n A {0,h} T [1,h 1] 1) T n x 0<t<h ) A 1 1 log n 1 1 ) log n 1 P n + t) ) h 1 T t A T 1 P n + t);

9 CONSECUTIVE PRIMES AND BEATTY SEQUENCES 9 see [, Equations.5) and.6)]. By the modified Hardy-Littlewood conjecture.) the estimate n) n xlog ) x c 1 P n + t) = log u) c d 1 P n + t) t H n u t H = S 0 H) log u) c H du + Ox 1/+ε ) holds uniformly for any constant c > 0; consequently, up to an error term of size Ox 1/+ε ) the quantity S h x) is equal to where A {0,h} T [1,h 1] 1) T S 0 A T ) νu).= 1 1 log u log u) T νu) h 1 T du, u > 1) note that νu) is the same as αu) in the notation of []). For every integer L 0 we denote D h,l u).= 1) T S 0 A T )νu) log u) T νu) h, so that A {0,h} T [1,h 1] A + T =L) h+1 S h x) = L=0 νu) 1 log u) D h,l u) du. We now combine this relation with.7), sum over the even natural numbers h log x), and apply.1) to deduce that the quantity πx; B, ˆB) is equal to h log x) h+1 L=0 k, l K g a k)ekb)gâl)elˆb) elâh) eka + lâ)u) νu)log u) D h,lu) du up to an error term of size x log x) + x + x 1 1/τ+ε + K 1 1 x ) log x). Choosing.= log x) 6 and K.= log x) 1 the combined error is Ox/log x) ), which is acceptable. Next, arguing as in [] and noting that g a k)gâl) log log x), k, l K one sees that the contribution to πx; B, ˆB) coming from terms with L does not exceed Ox/log x) 5/ ). Since D h,1 is identically zero as S 0 vanishes on singleton sets), this leaves only the terms with L = 0 or L =. The function D h,

10 10 W. D. BANKS AND VICTOR Z. GUO splits naturally into four pieces according to whether A =, {0}, {h} or {0, h}. Consequently, up to Ox/log x) 5/ ) we can express the quantity πx; B, ˆB) as 5 j=1 k, l K eka + lâ)u) g a k)ekb)gâl)elˆb) νu)log u) F j,lu) du,.8) where taking into account Lemma.) we have written with h log x) F 1,l u).= h 1 F,l u).= h 1 F,l u).= F 4,l u).= F 5,l u).= elâh)d h,l u) = νu) h elâh), 5 F j,l u) + Ox 1 ) j=1 S 0 {0, h})νu) h elâh), 1) νu) log u 1) νu) log u h 1 h 1 1 νu) log u) h 1 S 0 {0, t})νu) h elâh), S 0 {t, h})νu) h elâh), 1 t 1 <t h 1 S 0 {t 1, t })νu) h elâh). First, we show that certain terms in.8) make a negligible contribution that does not exceed Ox/log x) / ε ). For any l 0, using Lemma.4 with λ = lâ we have F 1,l u) = R 0,0;lâ u) l 4 provided that lâ log u) 1, and for this it suffices that u expˆα). Thus, k, l K l 0 eka + lâ)u) νu)log u) F 1,lu) du 1 + l 4 x log x). In view of.), the contribution to.8) from terms with j = 1 and l 0 is ) g a k) l l 4 x x log log x x log x) log x) log x). / ε Similarly, for l 0 and u expˆα) we have F,l u) = S lâ u) l 4 by Lemma.4, so the contribution to.8) from terms with j = and l 0 is also Ox/log x) / ε ).

11 CONSECUTIVE PRIMES AND BEATTY SEQUENCES 11 For any l Z, by Lemma. and Lemma.4 we have max { F,l u), F4,l u) } 1 h 1/+ε/ νu) h log u) 1/+ε/, log u hence for j =, 4 we see that h 1 eka + lâ)u) νu)log u) F j,lu) du x log x) / ε/. By.), it follows that the contribution to.8) from terms with j =, 4 is x xlog log x) g log x) / ε/ a k)gâl) log x) x / ε/ log x). / ε k, l K Finally, for any l Z and u expˆα), by Lemma. and Lemma.4 we have 1 F 5,l u) = 1 νu) log u) h log h + 1Ah + Oh1/+ε/ ) ) νu) h elâh) h 1 = 1 R 1,1;lâu) + 1 AR 1,0;lâu) + OR 1/+ε/,0;0 u)) νu) log u) λ 4 + log u) /+ε/ log u), and arguing as before we see that the contribution to.8) coming from terms with j = 5 does not exceed Ox/log x) / ε ). Applying the preceding bounds to.8) we see that, up to Ox/log x) / ε ), the quantity πx; B, ˆB) is equal to â j=1, k K g a k)ekb) ekau) νu)log u) F j,0u) du, where we have used the fact that gâ0) = â. By Lemma.4 we have and therefore, F,0 u) = h 1 F 1,0 u) = h 1 ekau) νu)log u) F j,0u) du = 1 νu) h = R 0,0;0 u) = 1 log u + O1) S 0 {0, h})νu) h = S 0 u) = 1 log u 1 log log u + O1); ) ekau) x log log x νu) log u du + O log x) j = 1, ). Consequently, up to Ox/log x) / ε ) we can express the quantity πx; B, ˆB) as â ekau) g a k)ekb) du..9) νu) log u k K

12 1 W. D. BANKS AND VICTOR Z. GUO To complete the proof of Theorem 1.1, we apply Lemma.5, which shows that the term k = 0 in.9) contributes aâ x ) ) x x log x + O = αˆα) 1 πx) + O log x) log x) to the quantity πx; B, ˆB) and thus accounts for the main term), whereas the terms in.9) with k 0 contribute altogether only a bounded amount. References [1] A. Khinchin, Zur metrischen Theorie der diophantischen Approximationen. Math. Z ), no. 4, [] L. Kuipers and H. Niederreiter, Uniform distribution of sequences. Pure and Applied Mathematics. Wiley-Interscience, New York-London-Sydney, [] R. J. Lemke Oliver and K. Soundararajan, Unexpected biases in the distribution of consecutive primes. Proc. Natl. Acad. Sci. USA, to appear. arxiv: [4] H. L. Montgomery and K. Soundararajan, Primes in short intervals. Comm. Math. Phys ), no. 1-, [5] K. Roth, Rational approximations to algebraic numbers. Mathematika 1955), 1 0. [6] K. Roth, Corrigendum to Rational approximations to algebraic numbers. Mathematika 1955), 168. [7] I. M. Vinogradov, A new estimate of a certain sum containing primes Russian). Rec. Math. Moscou, n. Ser. 44) 197), no. 5, English translation: New estimations of trigonometrical sums containing primes. C. R. Dokl.) Acad. Sci. URSS, n. Ser ), [8] I. M. Vinogradov, The method of trigonometrical sums in the theory of numbers. Dover Publications, Inc., Mineola, NY, 004. Department of Mathematics, University of Missouri, Columbia MO, USA. address: bankswd@missouri.edu Department of Mathematics, University of Missouri, Columbia MO, USA. address: zgbmf@mail.missouri.edu