B. Saffari

C. Jia

D. Bazzanella

D. R. Heath-Brown

G. Halász

G. Yu

M. N. Huxley

R. C. Baker

English

Danilo Bazzanella

CiteSeerX

The exceptional set for the distribution of primes between consecutive powers

BAZZANELLA, Danilo

PORTO@iris (Publications Open Repository TOrino - Politecnico di Torino)

05 August 2020POLITECNICO DI TORINORepository ISTITUZIONALEThe exceptional set for the distribution of primes between consecutive powers / Bazzanella D.. - In: ACTAMATHEMATICA HUNGARICA. - ISSN 0236-5294. - STAMPA. - 116:3(2007), pp. 197-207.OriginalThe exceptional set for the distribution of primes between consecutive powersPublisher:PublishedDOI:10.1007/s10474-007-5174-yTerms of use:openAccessPublisher copyright(Article begins on next page)This article is made available under terms and conditions as specified in the  corresponding bibliographic description inthe repositoryAvailability:This version is available at: 11583/1841396 since: 2016-11-09T10:46:33ZSpringerTHE EXCEPTIONAL SET FOR THEDISTRIBUTION OF PRIMES BETWEENCONSECUTIVE POWERSby DANILO BAZZANELLAAbstractA well known conjecture about the distribution of primes asserts thatbetween two consecutive squares there is always at least one prime num-ber. The proof of this conjecture is quite out of reach at present, evenunder the assumption of the Riemann Hypothesis. This paper is con-cerned with the distribution of prime numbers between two consecutivepowers of integers, as a natural generalization of the afore-mentionedconjecture.Mathematical Subject Classification :11NO5 - Distribution of PrimesThis is the authors’ post-print version of an article published onActa Math. Hungar. 116 (3) (2007), 197–207,DOI:10.1007/s10474-007-5174-y.11This version does not contain journal formatting and may contain minor changeswith respect to the published version. The final publication is available athttp://dx.doi.org/10.1007/s10474-007-5174-y. The present version is accessible on PORTO,the Open Access Repository of Politecnico di Torino (http://porto.polito.it), in com-pliance with the Publisher’s copyright policy as reported in the SHERPA-ROMEO website:http://www.sherpa.ac.uk/romeo/issn/0236-5294/12 D. BAZZANELLA1 IntroductionA well known conjecture about the distribution of primes asserts that all in-tervals of type [n2, (n+ 1)2] contain at least one prime.The proof of this conjecture is quite out of reach at present, even underthe assumption of the Riemann Hypothesis. To get a conditional proof ofthe conjecture we need to assume a stronger hypothesis about the behavior ofSelberg’s integral in short intervals, see Bazzanella [2].This paper is concerned with the distribution of prime numbers betweentwo consecutive powers of integers, as a natural generalization of the aboveconjecture.The well known result of Huxley about the distribution of prime in shortintervals, see [11], implies that, for n→∞, all intervals [nα, (n+ 1)α] containthe expected numbers of prime for α > 125and “almost all” contain the ex-pected numbers of prime for α > 65. By “almost all” we mean that the numberof integers X ≤ n ≤ 2X for which the interval [nα, (n+ 1)α] does not containthe expected number of primes is o(X). Our first result is an unconditionalestimate for the measure of the exceptional set for the distribution of primesbetween two consecutive power of integers.Theorem 1 Let ε > 0. Then for every [nα, (n+1)α] ⊂ [N, 2N ] with O(Nη(α)+ε)exceptions we have the expected number of primes , whereη(α) =85α− 1265< α ≤ 65+ c52α− 1 2716< α ≤ 532672− 9α− 8α23α(α + 12)5326≤ α < 125and c positive suitable constant.For the sake of simplicity we consider the function c(α) only for the extremesand more interesting values of the parameter α. Arguing in the same way wecan obtain the explicit values of the function c(α) for every α.A corollary of this theorem is Theorem 1 of Bazzanella [2], which states that forevery [n2, (n+1)2] ⊂ [N, 2N ] with O(N1/4+ε) exceptions we have the expectednumber of primes.PRIMES BETWEEN CONSECUTIVE POWERS 3Assuming some hypotheses we can have strong result about the regularity ofdistribution of primes and then a better estimate for the number of exceptionsfor the distribution of primes between two consecutive powers of integers.First we assume the Lindelo¨f hypothesis, which states that, for every η > 0,the Riemann Zeta-function satisfiesζ(σ + it) tη (σ ≥ 12, t ≥ 2).Assuming this hypothesis we can prove that all intervals [nα, (n+ 1)α] containthe expected numbers of prime for α > 2 and “almost all” contain the ex-pected numbers of prime for α > 1, see Bazzanella [1]. Our second result is aconditional estimate for the measure of the exceptional set for the distributionof primes between two consecutive powers of integers under the assumption ofthe Lindelo¨f hypothesis.Theorem 2 Assume the Lindelo¨f hypothesis, let ε > 0 and 1 < α ≤ 2. Thenfor every [nα, (n + 1)α] ⊂ [N, 2N ] with O(N2/α−1+ε) exceptions we have theexpected number of primes.A corollary of this theorem is Theorem 2 of Bazzanella [1], which states thatalmost all intervals [nα, (n + 1)α] contain the expected number of primes forα > 1.Assuming The Riemann Hypothesis we can set our last result.Theorem 3 Assume the Riemann Hypothesis, let 1 < α ≤ 2 and g(x) → ∞arbitrarily slowly.Then for every [nα, (n+ 1)α] ⊂ [N, 2N ] with O(N2/α−1 log2N g(N)) excep-tions we have the expected number of primes.A corollary of this theorem is Theorem 2 of Bazzanella [2], which states thatfor every [n2, (n+ 1)2] ⊂ [N, 2N ] with O(g(N) log2N) exceptions we have theexpected number of primes.2 The necessary lemmasFirst lemma is a result about the structure of the exceptional set for theasymptotic formulaψ(x+ h(x))− ψ(x) ∼ h(x) as x→∞. (1)4 D. BAZZANELLALet X be a large positive number, δ > 0 and let | | denote the modulus ofa complex number or the Lebesgue measure of a set. Let h(x) be an increasingfunction such that xε ≤ h(x) ≤ x for some ε > 0,∆(x, h) = ψ(x+ h(x))− ψ(x)− h(x)andEδ(X, h) = {X ≤ x ≤ 2X : |∆(x, h)| ≥ δh(x)}.It is clear that (1) holds if and only if for every δ > 0 there exists X0(δ) suchthat Eδ(X, h) = ∅ for X ≥ X0(δ). Hence for small δ > 0, X tending to∞ andh(x) suitably small with respect to x, the set Eδ(X, h) contains the exceptions,if any, to the expected asymptotic formula for the number of primes in shortintervals. Moreover, we observe thatEδ(X, h) ⊂ Eδ′(X, h) if 0 < δ′ < δ.We will consider increasing functions h(x) of the form h(x) = xθ+ε(x), withsome 0 < θ < 1 and a function ε(x) such that |ε(x)| is decreasing,ε(x) = o(1) and ε(x+ y) = ε(x) +O( |y|x).A function satisfying these requirements will be called of type θ.First lemma provides the basic structure of the exceptional set Eδ(X, h).Lemma 1 Let 0 < θ < 1, h(x) be of type θ, X be sufficiently large dependingon the function h(x) and 0 < δ′ < δ with δ − δ′ ≥ exp(−√logX). If x0 ∈Eδ(X, h) then Eδ′(X, h) contains the interval [x0−ch(X), x0+ch(X)]∩[X, 2X],where c = (δ − δ′)θ/5. In particular, if Eδ(X, h) 6= ∅ then|Eδ′(X, h)| θ (δ − δ′)h(X).Lemma 1 is part (i) of Theorem 1 of Bazzanella and Perelli, see [3], andessentially says that if we have a single exception in Eδ(X, h), with a fixed δ,then we necessarily have an interval of exceptions in Eδ′(X, h), with δ′ littlesmaller than δ. The interesting consequence of this lemma is that we can usean average estimate to prove the non-existence of the exceptions.PRIMES BETWEEN CONSECUTIVE POWERS 5Remainder lemmas provide conditional and unconditional bounds for the Sel-berg’s integral and the fourth-power integral linked to the distribution ofprimes in short intervals.Lemma 2 Let ε > 0 then exists c > 0 such that∫ 2XX|ψ(x+ δx)− ψ(x)− δx+B(x, θ)|2dx X(11+14θ)/10+εwith16< θ <16+ c,B(x, θ) XθlogX,uniformly for Xθ−1  δ  Xθ−1.Proof.In order to prove the lemma we use the classical explicit formula, see ch.17 of Davenport [5], to writeψ(x+ δx)− ψ(x)− δx = ∑|γ|≤Tx%c%(δ) +O(X log2XT), (2)uniformly for X ≤ x ≤ 2X, where 10 ≤ T ≤ X, % = β + iγ runs over thenon-trivial zeros of ζ(s),c%(δ) =(1 + δ)% − 1%and c%(δ) min(Xθ−1, 1|γ|). (3)ChooseT = X1−θ log3X, (4)We use Theorem 3 of Hala´sz-Tura´n [6], which asserts that there exists aconstant c1 > 0 such thatN(σ, T ) T (1−σ)3/2 log3 11−σ (5)for 1− c1 ≤ σ ≤ 1. From (3) - (5) and Vinogradov’s zero-free region, see ch. 6of Titchmarsh [14], by a standard argument we see that there exists a constantc2 > 0 such that∑|γ|≤T1−c2≤β≤1xρcρ(δ) Xθ−1 log2X max1−c2≤σ≤1XσN(σ, T ) XθlogX(6)6 D. BAZZANELLAuniformly for X ≤ x ≤ 2X.Again by a standard argument, from (3), (4) and Ingham-Huxley densityestimate, we obtain∫ 2XX∣∣∣ ∑|γ|≤T1−c2≤β≤1xρcρ(δ)∣∣∣2dx X2θ−1+ε max1/2≤σ≤1−c2X2σN(σ, T ) X(11+14θ)/10+ε,for θ > 1/6 and sufficiently small. This completes the proof of the Lemma 2.Lemma 3 Let ε > 0 then∫ 2XX|ψ(x+ ∆x)− ψ(x)−∆x+B(x, θ)|4dx Xµ(θ)+εwhereµ(θ) =X(3+5θ)/21127< θ ≤ 2753X(55+60θ2−108θ)/(39−36θ) 2753≤ θ < 712.andB(x, θ) XθlogXuniformly for Xθ−1  ∆ Xθ−1.Proof.We proceed along similar lines of the Lemma 2, using fourth power momentsinstead of mean square estimates. Using The Ingham-Huxley density estimatewe can prove that for every η > 0 we have∑|γ|≤T, β 6∈Ixρcρ(∆) Xθ−1 log2X max1/2≤σ≤1, σ 6∈IXσN(σ, T ) XθlogX(7)withI = [a, b],a = max{12, 3θ − 1− η}PRIMES BETWEEN CONSECUTIVE POWERS 7andb =(10− 9θ)7+ η1127< θ ≤ 275343− θ + η 2753≤ θ < 712,uniformly for X ≤ x ≤ 2X and 11/27 < θ < 7/12.We bound the contribution of the zeros such that β ∈ I by a fourth powermoment and we get∫ 2XX| ∑|γ|≤T, β∈Ixρcρ(ξ)|4dx X4θ−3+ε maxσ∈IX4σN∗(σ, T ). (8)From Theorem 2 of Heath-Brown [8] we haveN∗(σ, T )T (10−11σ)/(2−σ) logk T if 12≤ σ ≤ 23T (18−19σ)/(4−2σ) logk T if 23≤ σ ≤ 34T 12(1−σ)/(4σ−1) logk T if 34≤ σ ≤ 1,(9)where k is an absolute constant. Hence from (8) and (9) we obtain∫ 2XX| ∑|γ|≤T, β∈Ixρcρ(ξ)|4dx Xµ(θ)+ε,withµ(θ) =X(3+5θ)/21127< θ ≤ 2753X(55+60θ2−108θ)/(39−36θ) 2753≤ θ < 712,and then Lemma 3 follows.Lemma 4 Assume the Lindelo¨f hypothesis and let ε > 0. Then there existsa function E(y, T ) such that∫ 2XX∣∣∣∣ψ (y + yT)− ψ(y)− yT+B(y, T )∣∣∣∣4 dy  X4+εT−38 D. BAZZANELLAandB(y, T ) yT ln yuniformly for X ≥ 2, 1 ≤ T ≤ X and X ≤ y ≤ 2X.Lemma 4 is Lemma B of Yu, see [15].Lemma 5 Assume the Riemann hypothesis, then for x ≥ 4∫ 2XX|ψ(x+ θx)− ψ(x)− θx|2dx θX2(log2θ)2,uniformly in 0 < θ ≤ 1.This last lemma is Lemma 5 of Saffari and Vaughan, see [13].3 Proof of the TheoremsWe define H = (n+ 1)α − nα andAδ(N,α) = {N1/α ≤ n ≤ (2N)1/α : |ψ((n+ 1)α)− ψ(nα)−H| ≥ δH}.This set contains the exceptions, if any, to the expected asymptotic formulafor the number of primes in intervals of the type [nα, (n+ 1)α] in [N, 2N ].The main step of the proofs is to connect the exceptional set Aδ(N,α) withthe exceptional set for the distribution of primes in short intervals and showthat|Aδ(N,α)|  Eδ/2(N, h)N1−1/α+ 1 (10)for every δ > 0, α > 1 and h(x) = (x1/α + 1)α − x.In order to prove (10) we let n ∈ Aδ(N,α) and x = nα ∈ [N, 2N ]. Fromthe definition of the set Aδ(N,α) we get|ψ((n+ 1)α)− ψ(nα)−H| ≥ δH,and then|ψ(x+ h(x))− ψ(x)− h(x)| ≥ δh(x),PRIMES BETWEEN CONSECUTIVE POWERS 9which implies x ∈ Eδ(N, h). Using Lemma 1, with δ′ = δ/2, we obtain thatthere exists an effective constant c such that[x, x+ ch(x)] ∩ [N, 2N ] ⊂ Eδ/2(N, h).Let m ∈ Aδ(N,α), m > n. As before we can define y = mα ∈ [N, 2N ] suchthat[y, y + ch(y)] ∩ [N, 2N ] ⊂ Eδ/2(N, h).Choosing c < 1 we findy − x = mα − nα ≥ (n+ 1)α − nα > ch(x),and then[x, x+ ch(x)] ∩ [y, y + ch(y)] = ∅.Hence (10) is proved, since for every n ∈ Aδ(N,α) and x = nα, with at mostone exception, we have[x, x+ ch(x)] ⊂ [N, 2N ].Now we can conclude the proof of the theorems providing a suitable boundfor the measure of the exceptional set Eδ/2(N, h). Let g(x) a function suchthat g(x)→∞ arbitrarily slowly, for x→∞. Then we subdivide [N, 2N ] into g(N) intervals of type Ij = [Nj, Nj + Y ] withN ≤ Nj < 2N and Y  Ng(N).For every x ∈ Eδ/2(N, h) we have|ψ(x+ h(x))− ψ(x)− h(x)|  N1−1/αand then|Eδ/2(N, h)|N2−2/α ∫|ψ(x+ h(x))− ψ(x)− h(x)|2dxEδ/2(N,h)∑j∫|ψ(x+ h(x))− ψ(x)− h(x)|2dxEjδ/2(N,h),10 D. BAZZANELLAwhere Ejδ/2(N, h) = Eδ/2(N, h) ∩ [Nj, Nj + Y ]. Choosing Tj = N1/αj /α we candeduce|Eδ/2(N, h)|N2−2/α ∑j∫ ∣∣∣∣∣ψ(x+ xTj )− ψ(x)− xTj∣∣∣∣∣2dxEjδ/2(N,h)≤∑j∫ 2NN∣∣∣∣∣ψ(x+ xTj )− ψ(x)− xTj∣∣∣∣∣2dx.Hence|Eδ/2(N, h)|  N−2+2/α∑j∫ 2NN∣∣∣∣∣ψ(x+ xTj )− ψ(x)− xTj∣∣∣∣∣2dx. (11)Follow the same line, but choosing the function g(x)  N ε, for every ε > 0,we can also obtain|Eδ/2(N, h)|  N−2+2/α∑j∫ 2NN∣∣∣∣∣ψ(x+ xTj )− ψ(x)− xTj +B(x, Tj)∣∣∣∣∣2dx,(12)and|Eδ/2(N, h)|  N−4+4/α∑j∫ 2NN∣∣∣∣∣ψ(x+ xTj )− ψ(x)− xTj +B(x, Tj)∣∣∣∣∣4dx,(13)for every function B(x, T ) such thatB(x, T ) xT logNuniformly for 1 ≤ T ≤ N and N ≤ x ≤ 2N .Lemma 2 and (12) imply that|Eδ/2(N, h)|  N1/2+3/(5α)+ε. (14)Lemma 3 and (13) lead to|Eδ/2(N, h)|  N τ(α)+ε, (15)PRIMES BETWEEN CONSECUTIVE POWERS 11withτ(α) =32α2716≤ α ≤ 532636 + 24α− 5α23α(α + 12)5326≤ α < 125Theorem 1 is consequence of (10), (14) and (15).We now assume the Lindelo¨f hypothesis. Lemma 4 and (13) yields|Eδ/2(N, h)|  N−4+4/α∑jN4+εT−3j  N1/α+2ε. (16)From (10) and (16) we can conclude|Aδ(N,α)|  Eδ/2(N, h)N1−1/α+ 1 N1/α+2εN1−1/α+ 1 N2/α−1+2ε,for every 1 < α ≤ 2 and δ > 0. This completes the proof of Theorem 2.Lastly we assume the Riemann hypothesis. Lemma 5 and (11) lead to|Eδ/2(N, h)|  N−2+2/α∑jT−1j N2 log2(2Tj) N1/α log2N g(N). (17)From (10) and (17) we can conclude|Aδ(N,α)|  Eδ/2(N, h)N1−1/α+ 1 N1/α log2N g(N)N1−1/α+ 1 N2/α−1 log2N g(N),for every 1 < α ≤ 2 and δ > 0. Thus Theorem 3 is proved.References[1] D. Bazzanella, Primes between consecutive powers, preprint 2005.[2] D. Bazzanella, Primes between consecutive square, Arch. Math. 75(2000), 29-34.12 D. BAZZANELLA[3] D. Bazzanella and A. Perelli, The exceptional set for the number of primesin short intervals, J. Number Theory 80 (2000), 109-124 .[4] R. C. Baker, G. Harman and J. Pintz, The difference between consecutiveprimes II, Proc. London Math. Soc. (3) 83 (2001), no. 3, 532–562.[5] H. Davenport, Multiplicative Number Theory, volume GTM 74, Springer- Verlag, 1980. second edition.[6] G.Hala´sz and P.Tura´n, On the distribution of roots of Riemann zeta andallied functions, J. Number Theory 1 (1969), 121-137.[7] D.R. Heath-Brown, The difference between consecutive primes II, J.London Math. Soc. (2), 19 (1979), 207–220.[8] D.R. Heath-Brown, Zero density estimates for the Riemann zeta-functionand Dirichlet L-function, J. London Math. Soc. (2), 19 (1979), 221–232.[9] D.R. Heath-Brown, The difference between consecutive primes IV, A trib-ute to Paul Erdo¨s (ed. A Baker, B. Bolloba´s, and A. Hajnal, CambridgeUniversity Press, 1990), 277–278.[10] D.R. Heath-Brown, The difference between consecutive primes IV, A trib-ute to Paul Erdo¨s (ed. A Baker, B. Bolloba´s, and A. Hajnal, CambridgeUniversity Press, 1990), 277–278.[11] M.N.Huxley, On the difference between consecutive primes, Invent. Math.15 (1972), 164–170.[12] C. Jia, Almost all short intervals containing prime numbers, Acta Arith.76 (1996), no. 1, 21–84.[13] B. Saffari and R. C. Vaughan, On the fractional parts of x/n and relatedsequences II, Ann. Inst. Fourier, Grenoble, 27 (1977), 1-30.[14] E.C.Titchmarsh, The Theory of the Riemann Zeta-function, second ed.,Oxford U.P., 1986.[15] G. Yu, The differences between consecutive primes, Bull. London Math.Soc. 28 (1996), no. 3, 242–248.PRIMES BETWEEN CONSECUTIVE POWERS 13Danilo BazzanellaDipartimento di MatematicaPolitecnico di TorinoCorso Duca degli Abruzzi, 2410129 Torino, Italye-mail : danilo.bazzanella@polito.itweb: http://staff.polito.it/danilo.bazzanella

Bazzanella, Danilo

PORTO Publications Open Repository TOrino

The exceptional set for the distribution of
primes between consecutive powers
Danilo Bazzanella
Dipartimento di Matematica, Politecnico di Torino, Italy
danilo.bazzanella@polito.it
Autor’s version
Published in Acta Math. Hungar. 116 (3) (2007), 197-207.
DOI: 10.1007/s10474-007-5174-y
The original publication is available at
http://www.springerlink.com/content/x32182n4m1573m56/
THE EXCEPTIONAL SET FOR THE
DISTRIBUTION OF PRIMES BETWEEN
CONSECUTIVE POWERS
by DANILO BAZZANELLA
Abstract
A well known conjecture about the distribution of primes asserts that
between two consecutive squares there is always at least one prime num-
ber. The proof of this conjecture is quite out of reach at present, even
under the assumption of the Riemann Hypothesis. This paper is con-
cerned with the distribution of prime numbers between two consecutive
powers of integers, as a natural generalization of the afore-mentioned
conjecture.
Mathematical Subject Classification :
11NO5 - Distribution of Primes
1 Introduction
A well known conjecture about the distribution of primes asserts that all in-
tervals of type [n2, (n+ 1)2] contain at least one prime.
The proof of this conjecture is quite out of reach at present, even under
the assumption of the Riemann Hypothesis. To get a conditional proof of
the conjecture we need to assume a stronger hypothesis about the behavior of
Selberg’s integral in short intervals, see Bazzanella [2].
1
2 D. BAZZANELLA
This paper is concerned with the distribution of prime numbers between
two consecutive powers of integers, as a natural generalization of the above
conjecture.
The well known result of Huxley about the distribution of prime in short
intervals, see [11], implies that, for n→∞, all intervals [nα, (n+ 1)α] contain
the expected numbers of prime for α > 12
5
and “almost all” contain the ex-
pected numbers of prime for α > 6
5
. By “almost all” we mean that the number
of integers X ≤ n ≤ 2X for which the interval [nα, (n+ 1)α] does not contain
the expected number of primes is o(X). Our first result is an unconditional
estimate for the measure of the exceptional set for the distribution of primes
between two consecutive power of integers.
Theorem 1 Let ε > 0. Then for every [nα, (n+1)α] ⊂ [N, 2N ] with O(Nη(α)+ε)
exceptions we have the expected number of primes , where
η(α) =

8
5α
− 1
2
6
5
< α ≤ 6
5
+ c
5
2α
− 1 27
16
< α ≤ 53
26
72− 9α− 8α2
3α(α + 12)
53
26
≤ α < 12
5
and c positive suitable constant.
For the sake of simplicity we consider the function c(α) only for the extremes
and more interesting values of the parameter α. Arguing in the same way we
can obtain the explicit values of the function c(α) for every α.
A corollary of this theorem is Theorem 1 of Bazzanella [2], which states that for
every [n2, (n+1)2] ⊂ [N, 2N ] with O(N1/4+ε) exceptions we have the expected
number of primes.
Assuming some hypotheses we can have strong result about the regularity of
distribution of primes and then a better estimate for the number of exceptions
for the distribution of primes between two consecutive powers of integers.
First we assume the Lindelo¨f hypothesis, which states that, for every η > 0,
the Riemann Zeta-function satisfies
ζ(σ + it) tη (σ ≥ 1
2
, t ≥ 2).
PRIMES BETWEEN CONSECUTIVE POWERS 3
Assuming this hypothesis we can prove that all intervals [nα, (n+ 1)α] contain
the expected numbers of prime for α > 2 and “almost all” contain the ex-
pected numbers of prime for α > 1, see Bazzanella [1]. Our second result is a
conditional estimate for the measure of the exceptional set for the distribution
of primes between two consecutive powers of integers under the assumption of
the Lindelo¨f hypothesis.
Theorem 2 Assume the Lindelo¨f hypothesis, let ε > 0 and 1 < α ≤ 2. Then
for every [nα, (n + 1)α] ⊂ [N, 2N ] with O(N2/α−1+ε) exceptions we have the
expected number of primes.
A corollary of this theorem is Theorem 2 of Bazzanella [1], which states that
almost all intervals [nα, (n + 1)α] contain the expected number of primes for
α > 1.
Assuming The Riemann Hypothesis we can set our last result.
Theorem 3 Assume the Riemann Hypothesis, let 1 < α ≤ 2 and g(x) → ∞
arbitrarily slowly.
Then for every [nα, (n+ 1)α] ⊂ [N, 2N ] with O(N2/α−1 log2N g(N)) excep-
tions we have the expected number of primes.
A corollary of this theorem is Theorem 2 of Bazzanella [2], which states that
for every [n2, (n+ 1)2] ⊂ [N, 2N ] with O(g(N) log2N) exceptions we have the
expected number of primes.
2 The necessary lemmas
First lemma is a result about the structure of the exceptional set for the
asymptotic formula
ψ(x+ h(x))− ψ(x) ∼ h(x) as x→∞. (1)
Let X be a large positive number, δ > 0 and let | | denote the modulus of
a complex number or the Lebesgue measure of a set. Let h(x) be an increasing
function such that xε ≤ h(x) ≤ x for some ε > 0,
∆(x, h) = ψ(x+ h(x))− ψ(x)− h(x)
and
Eδ(X, h) = {X ≤ x ≤ 2X : |∆(x, h)| ≥ δh(x)}.
4 D. BAZZANELLA
It is clear that (1) holds if and only if for every δ > 0 there exists X0(δ) such
that Eδ(X, h) = ∅ for X ≥ X0(δ). Hence for small δ > 0, X tending to∞ and
h(x) suitably small with respect to x, the set Eδ(X, h) contains the exceptions,
if any, to the expected asymptotic formula for the number of primes in short
intervals. Moreover, we observe that
Eδ(X, h) ⊂ Eδ′(X, h) if 0 < δ′ < δ.
We will consider increasing functions h(x) of the form h(x) = xθ+ε(x), with
some 0 < θ < 1 and a function ε(x) such that |ε(x)| is decreasing,
ε(x) = o(1) and ε(x+ y) = ε(x) +O
( |y|
x
)
.
A function satisfying these requirements will be called of type θ.
First lemma provides the basic structure of the exceptional set Eδ(X, h).
Lemma 1 Let 0 < θ < 1, h(x) be of type θ, X be sufficiently large depending
on the function h(x) and 0 < δ′ < δ with δ − δ′ ≥ exp(−√logX). If x0 ∈
Eδ(X, h) then Eδ′(X, h) contains the interval [x0−ch(X), x0+ch(X)]∩[X, 2X],
where c = (δ − δ′)θ/5. In particular, if Eδ(X, h) 6= ∅ then
|Eδ′(X, h)| θ (δ − δ′)h(X).
Lemma 1 is part (i) of Theorem 1 of Bazzanella and Perelli, see [3], and
essentially says that if we have a single exception in Eδ(X, h), with a fixed δ,
then we necessarily have an interval of exceptions in Eδ′(X, h), with δ
′ little
smaller than δ. The interesting consequence of this lemma is that we can use
an average estimate to prove the non-existence of the exceptions.
Remainder lemmas provide conditional and unconditional bounds for the Sel-
berg’s integral and the fourth-power integral linked to the distribution of
primes in short intervals.
Lemma 2 Let ε > 0 then exists c > 0 such that∫ 2X
X
|ψ(x+ δx)− ψ(x)− δx+B(x, θ)|2dx X(11+14θ)/10+ε
with
1
6
< θ <
1
6
+ c,
PRIMES BETWEEN CONSECUTIVE POWERS 5
B(x, θ) X
θ
logX
,
uniformly for Xθ−1  δ  Xθ−1.
Proof.
In order to prove the lemma we use the classical explicit formula, see ch.
17 of Davenport [5], to write
ψ(x+ δx)− ψ(x)− δx = ∑
|γ|≤T
x%c%(δ) +O(
X log2X
T
), (2)
uniformly for X ≤ x ≤ 2X, where 10 ≤ T ≤ X, % = β + iγ runs over the
non-trivial zeros of ζ(s),
c%(δ) =
(1 + δ)% − 1
%
and c%(δ) min(Xθ−1, 1|γ|). (3)
Choose
T = X1−θ log3X, (4)
We use Theorem 3 of Hala´sz-Tura´n [6], which asserts that there exists a
constant c1 > 0 such that
N(σ, T ) T (1−σ)3/2 log3 11−σ (5)
for 1− c1 ≤ σ ≤ 1. From (3) - (5) and Vinogradov’s zero-free region, see ch. 6
of Titchmarsh [14], by a standard argument we see that there exists a constant
c2 > 0 such that
∑
|γ|≤T
1−c2≤β≤1
xρcρ(δ) Xθ−1 log2X max
1−c2≤σ≤1
XσN(σ, T ) X
θ
logX
(6)
uniformly for X ≤ x ≤ 2X.
Again by a standard argument, from (3), (4) and Ingham-Huxley density
estimate, we obtain∫ 2X
X
∣∣∣ ∑
|γ|≤T
1−c2≤β≤1
xρcρ(δ)
∣∣∣2dx X2θ−1+ε max
1/2≤σ≤1−c2
X2σN(σ, T ) X(11+14θ)/10+ε,
6 D. BAZZANELLA
for θ > 1/6 and sufficiently small. This completes the proof of the Lemma 2.
Lemma 3 Let ε > 0 then∫ 2X
X
|ψ(x+ ∆x)− ψ(x)−∆x+B(x, θ)|4dx Xµ(θ)+ε
where
µ(θ) =

X(3+5θ)/2
11
27
< θ ≤ 27
53
X(55+60θ
2−108θ)/(39−36θ) 27
53
≤ θ < 7
12
.
and
B(x, θ) X
θ
logX
uniformly for Xθ−1  ∆ Xθ−1.
Proof.
We proceed along similar lines of the Lemma 2, using fourth power moments
instead of mean square estimates. Using The Ingham-Huxley density estimate
we can prove that for every η > 0 we have
∑
|γ|≤T, β 6∈I
xρcρ(∆) Xθ−1 log2X max
1/2≤σ≤1, σ 6∈I
XσN(σ, T ) X
θ
logX
(7)
with
I = [a, b],
a = max
{
1
2
, 3θ − 1− η
}
and
b =

(10− 9θ)
7
+ η
11
27
< θ ≤ 27
53
4
3
− θ + η 27
53
≤ θ < 7
12
,
uniformly for X ≤ x ≤ 2X and 11/27 < θ < 7/12.
PRIMES BETWEEN CONSECUTIVE POWERS 7
We bound the contribution of the zeros such that β ∈ I by a fourth power
moment and we get∫ 2X
X
| ∑
|γ|≤T, β∈I
xρcρ(ξ)|4dx X4θ−3+ε max
σ∈I
X4σN∗(σ, T ). (8)
From Theorem 2 of Heath-Brown [8] we have
N∗(σ, T )

T (10−11σ)/(2−σ) logk T if 1
2
≤ σ ≤ 2
3
T (18−19σ)/(4−2σ) logk T if 2
3
≤ σ ≤ 3
4
T 12(1−σ)/(4σ−1) logk T if 3
4
≤ σ ≤ 1,
(9)
where k is an absolute constant. Hence from (8) and (9) we obtain
∫ 2X
X
| ∑
|γ|≤T, β∈I
xρcρ(ξ)|4dx Xµ(θ)+ε,
with
µ(θ) =

X(3+5θ)/2
11
27
< θ ≤ 27
53
X(55+60θ
2−108θ)/(39−36θ) 27
53
≤ θ < 7
12
,
and then Lemma 3 follows.
Lemma 4 Assume the Lindelo¨f hypothesis and let ε > 0. Then there exists
a function E(y, T ) such that
∫ 2X
X
∣∣∣∣ψ (y + yT
)
− ψ(y)− y
T
+B(y, T )
∣∣∣∣4 dy  X4+εT−3
and
B(y, T ) y
T ln y
uniformly for X ≥ 2, 1 ≤ T ≤ X and X ≤ y ≤ 2X.
Lemma 4 is Lemma B of Yu, see [15].
8 D. BAZZANELLA
Lemma 5 Assume the Riemann hypothesis, then for x ≥ 4
∫ 2X
X
|ψ(x+ θx)− ψ(x)− θx|2dx θX2
(
log
2
θ
)2
,
uniformly in 0 < θ ≤ 1.
This last lemma is Lemma 5 of Saffari and Vaughan, see [13].
3 Proof of the Theorems
We define H = (n+ 1)α − nα and
Aδ(N,α) = {N1/α ≤ n ≤ (2N)1/α : |ψ((n+ 1)α)− ψ(nα)−H| ≥ δH}.
This set contains the exceptions, if any, to the expected asymptotic formula
for the number of primes in intervals of the type [nα, (n+ 1)α] in [N, 2N ].
The main step of the proofs is to connect the exceptional set Aδ(N,α) with
the exceptional set for the distribution of primes in short intervals and show
that
|Aδ(N,α)|  Eδ/2(N, h)
N1−1/α
+ 1 (10)
for every δ > 0, α > 1 and h(x) = (x1/α + 1)α − x.
In order to prove (10) we let n ∈ Aδ(N,α) and x = nα ∈ [N, 2N ]. From
the definition of the set Aδ(N,α) we get
|ψ((n+ 1)α)− ψ(nα)−H| ≥ δH,
and then
|ψ(x+ h(x))− ψ(x)− h(x)| ≥ δh(x),
which implies x ∈ Eδ(N, h). Using Lemma 1, with δ′ = δ/2, we obtain that
there exists an effective constant c such that
[x, x+ ch(x)] ∩ [N, 2N ] ⊂ Eδ/2(N, h).
Let m ∈ Aδ(N,α), m > n. As before we can define y = mα ∈ [N, 2N ] such
that
[y, y + ch(y)] ∩ [N, 2N ] ⊂ Eδ/2(N, h).
PRIMES BETWEEN CONSECUTIVE POWERS 9
Choosing c < 1 we find
y − x = mα − nα ≥ (n+ 1)α − nα > ch(x),
and then
[x, x+ ch(x)] ∩ [y, y + ch(y)] = ∅.
Hence (10) is proved, since for every n ∈ Aδ(N,α) and x = nα, with at most
one exception, we have
[x, x+ ch(x)] ⊂ [N, 2N ].
Now we can conclude the proof of the theorems providing a suitable bound
for the measure of the exceptional set Eδ/2(N, h). Let g(x) a function such
that g(x)→∞ arbitrarily slowly, for x→∞. Then we subdivide [N, 2N ] into
 g(N) intervals of type Ij = [Nj, Nj + Y ] with
N ≤ Nj < 2N and Y  N
g(N)
.
For every x ∈ Eδ/2(N, h) we have
|ψ(x+ h(x))− ψ(x)− h(x)|  N1−1/α
and then
|Eδ/2(N, h)|N2−2/α 
∫
|ψ(x+ h(x))− ψ(x)− h(x)|2dx
Eδ/2(N,h)
∑
j
∫
|ψ(x+ h(x))− ψ(x)− h(x)|2dx
Ej
δ/2
(N,h)
,
where Ejδ/2(N, h) = Eδ/2(N, h) ∩ [Nj, Nj + Y ]. Choosing Tj = N1/αj /α we can
deduce
|Eδ/2(N, h)|N2−2/α 
∑
j
∫ ∣∣∣∣∣ψ(x+ xTj )− ψ(x)− xTj
∣∣∣∣∣
2
dx
Ej
δ/2
(N,h)
10 D. BAZZANELLA
≤∑
j
∫ 2N
N
∣∣∣∣∣ψ(x+ xTj )− ψ(x)− xTj
∣∣∣∣∣
2
dx.
Hence
|Eδ/2(N, h)|  N−2+2/α
∑
j
∫ 2N
N
∣∣∣∣∣ψ(x+ xTj )− ψ(x)− xTj
∣∣∣∣∣
2
dx. (11)
Follow the same line, but choosing the function g(x)  N ε, for every ε > 0,
we can also obtain
|Eδ/2(N, h)|  N−2+2/α
∑
j
∫ 2N
N
∣∣∣∣∣ψ(x+ xTj )− ψ(x)− xTj +B(x, Tj)
∣∣∣∣∣
2
dx,
(12)
and
|Eδ/2(N, h)|  N−4+4/α
∑
j
∫ 2N
N
∣∣∣∣∣ψ(x+ xTj )− ψ(x)− xTj +B(x, Tj)
∣∣∣∣∣
4
dx,
(13)
for every function B(x, T ) such that
B(x, T ) x
T logN
uniformly for 1 ≤ T ≤ N and N ≤ x ≤ 2N .
Lemma 2 and (12) imply that
|Eδ/2(N, h)|  N1/2+3/(5α)+ε. (14)
Lemma 3 and (13) lead to
|Eδ/2(N, h)|  N τ(α)+ε, (15)
with
τ(α) =

3
2α
27
16
≤ α ≤ 53
26
36 + 24α− 5α2
3α(α + 12)
53
26
≤ α < 12
5
Theorem 1 is consequence of (10), (14) and (15).
PRIMES BETWEEN CONSECUTIVE POWERS 11
We now assume the Lindelo¨f hypothesis. Lemma 4 and (13) yields
|Eδ/2(N, h)|  N−4+4/α
∑
j
N4+εT−3j  N1/α+2ε. (16)
From (10) and (16) we can conclude
|Aδ(N,α)|  Eδ/2(N, h)
N1−1/α
+ 1 N
1/α+2ε
N1−1/α
+ 1 N2/α−1+2ε,
for every 1 < α ≤ 2 and δ > 0. This completes the proof of Theorem 2.
Lastly we assume the Riemann hypothesis. Lemma 5 and (11) lead to
|Eδ/2(N, h)|  N−2+2/α
∑
j
T−1j N
2 log2(2Tj) N1/α log2N g(N). (17)
From (10) and (17) we can conclude
|Aδ(N,α)|  Eδ/2(N, h)
N1−1/α
+ 1 N
1/α log2N g(N)
N1−1/α
+ 1 N2/α−1 log2N g(N),
for every 1 < α ≤ 2 and δ > 0. Thus Theorem 3 is proved.
References
[1] D. Bazzanella, Primes between consecutive powers, preprint 2005.
[2] D. Bazzanella, Primes between consecutive square, Arch. Math. 75
(2000), 29-34.
[3] D. Bazzanella and A. Perelli, The exceptional set for the number of primes
in short intervals, J. Number Theory 80 (2000), 109-124 .
[4] R. C. Baker, G. Harman and J. Pintz, The difference between consecutive
primes II, Proc. London Math. Soc. (3) 83 (2001), no. 3, 532–562.
[5] H. Davenport, Multiplicative Number Theory, volume GTM 74, Springer
- Verlag, 1980. second edition.
12 D. BAZZANELLA
[6] G.Hala´sz and P.Tura´n, On the distribution of roots of Riemann zeta and
allied functions, J. Number Theory 1 (1969), 121-137.
[7] D.R. Heath-Brown, The difference between consecutive primes II, J.
London Math. Soc. (2), 19 (1979), 207–220.
[8] D.R. Heath-Brown, Zero density estimates for the Riemann zeta-function
and Dirichlet L-function, J. London Math. Soc. (2), 19 (1979), 221–232.
[9] D.R. Heath-Brown, The difference between consecutive primes IV, A trib-
ute to Paul Erdo¨s (ed. A Baker, B. Bolloba´s, and A. Hajnal, Cambridge
University Press, 1990), 277–278.
[10] D.R. Heath-Brown, The difference between consecutive primes IV, A trib-
ute to Paul Erdo¨s (ed. A Baker, B. Bolloba´s, and A. Hajnal, Cambridge
University Press, 1990), 277–278.
[11] M.N.Huxley, On the difference between consecutive primes, Invent. Math.
15 (1972), 164–170.
[12] C. Jia, Almost all short intervals containing prime numbers, Acta Arith.
76 (1996), no. 1, 21–84.
[13] B. Saffari and R. C. Vaughan, On the fractional parts of x/n and related
sequences II, Ann. Inst. Fourier, Grenoble, 27 (1977), 1-30.
[14] E.C.Titchmarsh, The Theory of the Riemann Zeta-function, second ed.,
Oxford U.P., 1986.
[15] G. Yu, The differences between consecutive primes, Bull. London Math.
Soc. 28 (1996), no. 3, 242–248.
Danilo Bazzanella
Dipartimento di Matematica
Politecnico di Torino
Corso Duca degli Abruzzi, 24
10129 Torino, Italy
e-mail : danilo.bazzanella@polito.it
web: http://staff.polito.it/danilo.bazzanella


Almost all short intervals containing prime numbers,

Multiplicative Number Theory,

On the dierence between consecutive primes,

On the distribution of roots of Riemann zeta and allied functions,

On the fractional parts of x/n and related sequences II,

Primes between consecutive powers, preprint

Primes between consecutive square,

The dierence between consecutive primes II,

The dierence between consecutive primes IV, A tribute to Paul Erd

The dierences between consecutive primes,

The exceptional set for the number of primes in short intervals,

Zero density estimates for the Riemann zeta-function and Dirichlet L-function,

AKJournals (Akadémiai Kiadó)

Crossref

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The exceptional set for the distribution of primes between consecutive powers

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