Bazzanella, Danilo

Camerlo, Riccardo

English

PORTO Publications Open Repository TOrino

The class of the exceptional sets for a general asymptotic formula

Danilo Bazzanella

Riccardo Camerlo

Crossref

Functiones et Approximatio Commentarii Mathematici

BAZZANELLA, Danilo

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

05 August 2020POLITECNICO DI TORINORepository ISTITUZIONALESome conditional results on primes between consecutive squares / Bazzanella D.. - In: FUNCTIONES ETAPPROXIMATIO COMMENTARII MATHEMATICI. - ISSN 0208-6573. - STAMPA. - 45:2(2011), pp. 255-263.OriginalSome conditional results on primes between consecutive squaresPublisher:PublishedDOI:10.7169/facm/1323705816Terms 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: 11583/2371637 since: 2016-11-09T11:06:54ZFaculty of Mathematics and Computer Science of Adam Mickiewicz UniversitySOME CONDITIONAL RESULTS ON PRIMES BETWEENCONSECUTIVE SQUARESDANILO BAZZANELLAAbstract. A well-known conjecture about the distribution of primes asserts that be-tween two consecutive squares there is always at least one prime number. The proof ofthis conjecture is quite out of reach at present, even under the assumption of the RiemannHypothesis. The aim of this paper is to provide the upper bounds for the exceptional setfor this conjecture under the assumption of some heuristic hypotheses.This is the authors’ post-print version of an article published onFunct. Approx. Comment. Math. 45 (2011), n. 2, 255–263,DOI:10.7169/facm/1323705816.11. IntroductionA well known conjecture about the distribution of primes asserts that for every positiveinteger n, the interval [n2, (n+ 1)2] contains at least one prime. The proof of thisconjecture is quite out of reach at present, even under the assumption of the RiemannHypothesis. However it is not difficult to prove unconditionally that the conjecture holdsfor almost all positive integers n. Indeed, we can prove that almost all intervals of thetype [n2, (n+ 1)2] contain the expected number of primes.This paper is concerned with the exceptional set for the distribution of primes betweentwo consecutive squares, under the assumption of some unproved heuristic hypotheses.The basic idea was to connect the exceptional set for the distribution of primes inintervals of the type [n2, (n+ 1)2] to the exceptional set of the asymptotic formula forthe distribution of primes in short intervals. The properties of the latter set, seeD. Bazzanella and A. Perelli [2], were thus used to obtain the desired results.In a previous paper, see D. Bazzanella [1], the author proved that each of the intervals[n2, (n+ 1)2] ⊂ [1, N ], with at most O(N1/4+ε) exceptions, contained the expectednumber of primes, for every constant ε > 0. Under the assumption of the RiemannHypothesis, the author also proved that each of the intervals [n2, (n+ 1)2] ⊂ [1, N ], with2010 Mathematics Subject Classification. 11NO5.Key words and phrases. distribution of prime numbers, primes between squares.1This version does not contain journal formatting and may contain minor changes with respect tothe published version. The final publication is available at http://dx.doi.org/10.7169/facm/1323705816.The present version is accessible on PORTO, the Open Access Repository of Politecnico di Torino(http://porto.polito.it).12 D. BAZZANELLAat most O(f(N) log2N) exceptions, contained the expected number of primes, for everyreal valued function f(x)→∞ arbitrarily slowly.Under the assumption of a stronger hypotheses than the Riemann Hypothesis, we wouldexpect to obtain smaller bounds for the exceptional set. Infact, assuming a strong formof the Montgomery Conjecture, the author proved that, with n sufficiently large, each ofthe intervals [n2, (n+ 1)2] contained the expected number of primes without exceptions,see [1]. On the other hand, under the assumption of a weaker hypotheses than theRiemann Hypothesis, we would expect to achieve upper bounds of sizes betweenO(f(N) log2N) and O(N1/4+ε).In order to estimate some sums which arose in our argument we employed the countingfunctions N(σ, T ) and N∗(σ, T ). The former is defined as the number of zeros ρ = β + iγof the Riemann zeta function which satisfy σ ≤ β ≤ 1 and |γ| ≤ T , while N∗(σ, T ) isdefined as the number of ordered sets of zeros ρj = βj + iγj (1 ≤ j ≤ 4), each counted byN(σ, T ), for which |γ1 + γ2 − γ3 − γ4| ≤ 1. As D. Bazzanella and A. Perelli [2] weconsidered the heuristic assumption that there exists a constant T0 such that(1.1) N∗(σ, T ) N(σ, T )4TT εfor every T ≥ T0 and arbitrarily small ε > 0, which is close to being the best possible, inview of the trivial estimateN∗(σ, T ) N(σ, T )4T.Recall that the Ingham–Huxley density estimate [5, Theorem 11.1] implies(1.2) N(σ, T )T 3(1−σ)/(2−σ)+ν12≤ σ ≤ 34T 3(1−σ)/(3σ−1)+ν34≤ σ ≤ 1,for arbitrarily ν > 0. Thus from (1.1) it follows that(1.3) N∗(σ, T )T (10−11σ)/(2−σ)+ε12≤ σ ≤ 34T (13−15σ)/(3σ−1)+ε34≤ σ ≤ 1315.Remark. The estimate of N∗(σ, T ) proved by D. R. Heath-Brown, see [5, Lemma 12.7],implies that the upper bound (1.3) holds for 1/2 ≤ σ ≤ 2/3.Assuming that (1.3) holds also for σ > 2/3 we obtain the first result.PRIMES BETWEEN CONSECUTIVE SQUARES 3Theorem 1.1. Let ε > 0 be arbitrarily small and assume (1.3). Then each of the intervals[n2, (n + 1)2] ⊂ [1, N ], with at most O(N1/5+ε) exceptions, contains the expected numberof primes.If we assume the Lindelo¨f hypothesis, which states that, for every η > 0, the RiemannZeta-function satisfiesζ(σ + it) tη (1/2 ≤ σ ≤ 1, t ≥ 2),we can get the following stronger result.Theorem 1.2. Let ε > 0 be arbitrarily small and assume the Lindelo¨f hypothesis. Theneach of the intervals [n2, (n + 1)2] ⊂ [1, N ], with at most O(N ε) exceptions, contains theexpected number of primes.Finally we assume the Density Hypothesis, which states that for every η > 0 thecounting function N(σ, T ) satisfiesN(σ, T ) T 2(1−σ)+η (1/2 ≤ σ ≤ 1),and we thus obtain our last result.Theorem 1.3. Let ε > 0 be arbitrarily small, assume the Density Hypothesis and (1.1).Then each of the intervals [n2, (n+1)2] ⊂ [1, N ], with at most O(N ε) exceptions, containsthe expected number of primes.2. Definitions and preliminary lemmasWe will always assume that n, x and N are sufficiently large as prescribed by thevarious statements, and ε > 0 is arbitrarily small and not necessarily the same at eachoccurrence. As in [2] we define a set related to the asymptotic formula(2.1) ψ(x+ h(x))− ψ(x) ∼ h(x) (x→∞)asEδ(N, h) = {N ≤ x ≤ 2N : |ψ(x+ h(x))− ψ(x)− h(x)| ≥ δh(x)},where h(x) is an increasing function such that xε ≤ h(x) ≤ x for some ε > 0. It is clearthat (2.1) holds if and only if for every δ > 0 there exists N0(δ) such that Eδ(N, h) = ∅for every N ≥ N0(δ). Hence for small δ > 0, N tending to ∞ and h(x) which is suitablysmall with respect to x, the set Eδ(N, h) contains the exceptions, if any, to the expectedasymptotic formula for the number of primes in short intervals. Our first lemma isconcerned with the structure of the exceptional set above.4 D. BAZZANELLALemma 2.1. Let δ > 0, xε ≤ h(x) ≤ x, for some ε > 0 and N be sufficiently largedepending on the function h(x). If x0 ∈ Eδ(N, h) then there exists an effective constant c,depending on δ and h(x), such that[x0, x0 + c · h(N)] ∩ [N, 2N ] ⊂ Eδ/2(N, h).Lemma 2.1 may be proved along the same lines as Theorem 1 of [2], and essentially saysthat if we have a single exception in Eδ(N, h), with a fixed δ, then we necessarily havean interval of exceptions in Eδ/2(N, h). Moreover we define a set related to theasymptotic formula(2.2) ψ((n+ 1)2)− ψ(n2) ∼ 2n (n→∞)asAδ(N) = {√N ≤ n ≤√2N : |ψ((n+ 1)2)− ψ(n2)− (2n+ 1)| ≥ δ(2n+ 1)},that contains the exceptions, if any, to the expected asymptotic formula for the numberof primes in intervals of the type [n2, (n+ 1)2] ⊂ [N, 2N ]. The main tool of the proofs isthe following lemma.Lemma 2.2. For h(x) = 2√x+ 1 and every δ > 0 we have|Aδ(N)| δ |Eδ/2(N, h)|√N+ 1.Proof. Let n ∈ Aδ(N) and put x = n2 ∈ [N, 2N ] . From the definition of the set Aδ(N)we get|ψ((n+ 1)2)− ψ(n2)− (2n+ 1)| ≥ δ(2n+ 1)and thus|ψ(x+ h(x))− ψ(x)− h(x)| ≥ δh(x),which implies x ∈ Eδ(N, h). From Lemma 2.1 follows that there exists an effective constantc such that[x, x+ c · h(x)] ∩ [N, 2N ] ⊂ Eδ/2(N, h).Let m ∈ Aδ(N), m > n. As before we can define y = m2 ∈ [N, 2N ] and obtain, again byLemma 2.1, that[y, y + c · h(y)] ∩ [N, 2N ] ⊂ Eδ/2(N, h).If we choose c < 1, we gety − x = m2 − n2 ≥ (n+ 1)2 − n2 = 2n+ 1 = 2√x+ 1 > ch(x)and thus[x, x+ c · h(x)] ∩ [y, y + c · h(y)] = ∅.PRIMES BETWEEN CONSECUTIVE SQUARES 5Hence the lemma is proved, since for every n ∈ Aδ(N) and x = n2, with at most oneexception, we have[x, x+ c · h(x)] ⊂ [N, 2N ].The next lemma concerns the conditional estimate for the fourth power mean value ofthe function ψ(y) in short intervals.Lemma 2.3. Assume the Lindelo¨f hypothesis and let ε > 0 be arbitrarily small. Thenthere exists a function Σ(y, T ) such that for every ε > 0 we have(2.3)∫ 2NN∣∣∣ψ (y + yT)− ψ(y)− yT+ Σ(y, T )∣∣∣4 dy  N4+εT−3and(2.4) Σ(y, T ) yT log y,uniformly for N ≥ 2, 1 ≤ T ≤ N and N ≤ y ≤ 2N .Lemma 2.3 is due to G. Yu [7, Lemma B]. The interesting consequence of this lemma isthat it allows to obtain the following conditional bound for the exceptional set Eδ(N, h),with h(x) = 2√x+ 1.Lemma 2.4. Assume the Lindelo¨f hypothesis, let h(x) = 2√x + 1 and let ε > 0 bearbitrarily small. Then(2.5) |Eδ(N, h)|  N1/2+ε.Proof. In order to prove the lemma, we subdivide [N, 2N ] into O(log2N) intervals of typeIj = [Nj, Nj + Y ] withN ≤ Nj < 2N and Y  Nlog2N.For every y ∈ Eδ(N, h) we have|ψ(y + h(y))− ψ(y)− h(y)|  N1/2and then|Eδ(N, h)|N2 ∫|ψ(y + h(y))− ψ(y)− h(y)|4 dyEδ(N,h)(2.6)∑j∫|ψ(y + h(y))− ψ(y)− h(y)|4 dy,Ejδ(N,h)6 D. BAZZANELLAwhere Ejδ(N, h) = Eδ(N, h) ∩ [Nj, Nj + Y ].If we choose Tj = N1/2j /2 we have, by Lemma 2.3, that there exist functions Σj(y, Tj)satisfy the conditions (2.3) and (2.4), for all values of j. From the Brun–Titchmarshtheorem, see H. L. Montgomery and R. C. Vaughan [6], we can deduce(ψ(y + h(y))− ψ(y)− h(y))−(ψ(y +yTj)− ψ(y)− yTj+ Σj(y, Tj)) yTj log y,for every j and every y ∈ Ejδ(Xn, θ). Then from (2.6) it follows that|Eδ(N, h)|N2 ∑j∫ ∣∣∣∣ψ(y + yTj )− ψ(y)− yTj + Σj(y, Tj)∣∣∣∣4 dyEjδ(N,h)+∑j∫ ∣∣∣∣ yTj log y∣∣∣∣4 dyEjδ(N,h)∑j∫ 2NN∣∣∣∣ψ(y + yTj )− ψ(y)− yTj + Σj(y, Tj)∣∣∣∣4 dy+ |Eδ(N, h)| N2log4Nwhich implies(2.7) |Eδ(N, h)|N2 ∑j∫ 2NN∣∣∣∣ψ(y + yTj )− ψ(y)− yTj + Σj(y, Tj)∣∣∣∣4 dy.At this point we use Lemma 2.3 to get|Eδ(N, h)|N2 ∑j∫ 2NN∣∣∣∣ψ(y + yTj )− ψ(y)− yTj + Σj(y, Tj)∣∣∣∣4 dy∑jN4+εT−3j N5/2+εand this leads to (2.5). 3. Proof of the theoremsLet h(x) = 2√x+ 1 and use the classical explicit formula, see H. Davenport [3, chapter17], to write(3.1) ψ(x+ h(x))− ψ(x)− h(x) = −∑|γ|≤Txρcρ(x) +O(N log2NT),PRIMES BETWEEN CONSECUTIVE SQUARES 7uniformly for N ≤ x ≤ 2N , where 10 ≤ T ≤ N , ρ = β + iγ runs over the non-trivialzeros of ζ(s) andcρ(x) =(1 + h(x)/x)ρ − 1ρ.Let(3.2) T =√N log3Nand note that(3.3) cρ(x) min(1√N,1|γ|).If we follow the method of D. R. Heath-Brown we can find a constant 0 < u < 1 suchthat∑|γ|≤T, β>uxρcρ(x)√NlogN,see [5, pag. 319].Remark. More precisely from (3.2) it follows that we can choose 5/6 ≤ u < 1. Note thatunder the assumption of the Density Hypothesis we have the weaker condition 11/14 ≤u < 1.Thus we haveψ(x+ h(x))− ψ(x)− h(x) = −∑|γ|≤T, β≤uxρcρ(x) +O( √NlogN)and then(3.4) |Eδ(N, h)|N2 ∫ 2NN∣∣∣∣∣∣∑|γ|≤T, β≤uxρcρ(x)∣∣∣∣∣∣4dx.To estimate the fourth power integral we divide the interval [0, u] into O(logN)subintervals Ik of the formIk =[klogN,k + 1logN].By the Ho¨lder inequality we have∣∣∣∣∣∣∑|γ|≤T, β≤uxρcρ(x)∣∣∣∣∣∣4 log3N∑k∣∣∣∣∣∣∑|γ|≤T, β∈Ikxρcρ(x)∣∣∣∣∣∣4.8 D. BAZZANELLAFollowing again the method of D. R. Heath-Brown, we can get(3.5)∫ 2NN∣∣∣∣∣∣∑|γ|≤T, β≤uxρcρ(x)∣∣∣∣∣∣4dx N−1+ε maxσ≤uN4σM(σ, T ),whereM(σ, T ) =∑β1,...,β4≥σ|γ1|≤T,...,|γ4|≤T11 + |γ1 + γ2 − γ3 − γ4|and(3.6) M(σ, T ) N∗(σ, T ) logN,see [5, p. 336]. From (3.4), (3.5) and (3.6) it follows that(3.7) |Eδ(N, h)|  N−3+ε maxσ≤uN4σN∗(σ, T ).To prove Theorem 1.1 we assume (1.3). Then we write|Eδ(N, h)| N−3+ε maxσ≤uN4σT (10−11σ)/(2−σ)+ε12≤ σ ≤ 34N−3+ε maxσ≤uN4σT (13−15σ)/(3σ−1)+ε34≤ σ ≤ 1315.The above upper bound for the exceptional set attains its maximum at σ = 3/4, so weget|Eδ(N, h)|  N7/10+ε,for every δ > 0. From Lemma 2.2 we conclude|Aδ(N)| δ |Eδ/2(N, h)|√N+ 1 N1/5+ε,for every δ > 0. Then the proof of Theorem 1.1 is complete.Remark. In Theorem 1.1 we need to assume the upper bound (1.3) only for (139 −√761)/160 ≤ σ ≤ 5/6. For smaller values of σ the estimate of N∗(σ, T ) proved byD. R. Heath-Brown, see [5, Lemma 12.7], is enough and the bigger values are not involvedin (3.7).To prove Theorem 1.2 we assume the Lindelo¨f hypothesis. By Lemma 2.4 we have (2.5)and then, again by Lemma 2.2, we obtain that|Aδ(N)| δ |Eδ/2(N, h)|√N+ 1 N ε,for every δ > 0. Then Theorem 1.2 is proved.PRIMES BETWEEN CONSECUTIVE SQUARES 9In order to prove Theorem 1.3 we follow the proof of Theorem 1.1 until the equation(3.7). Then we use (1.1) and the Density Hypothesis to have|Eδ(N, h)|  N−7/2+ε(maxσ≤uNσN(σ, T ))4 N−7/2+ε(maxσ≤uNσT 2(1−σ))4 N1/2+ε.By Lemma 2.2, we can conclude that|Aδ(N)| δ |Eδ/2(N, h)|√N+ 1 N ε,for every δ > 0, and this complete the proof of Theorem 1.3.References[1] D. Bazzanella, Primes between consecutive squares, Arch. Math. 75 (2000), 29–34.[2] D. Bazzanella and A. Perelli, The exceptional set for the number of primes in short intervals, J.Number Theory 80 (2000), 109–124 .[3] H. Davenport, Multiplicative Number Theory, Second edition, Graduate Texts in Mathematics 74.Springer–Verlag, New York (1980).[4] D. R. Heath-Brown, The difference between consecutive primes II, J. London Math. Soc. (2), 19(1979), 207–220.[5] A. Ivic´, The Riemann Zeta-Function , John Wiley & Sons, New York (1985).[6] H. L. Montgomery and R. C. Vaughan, The large sieve , Mathematika 20 (1973), 119–134.[7] G. Yu, The differences between consecutive primes, Bull. London Math. Soc. 28 (1996), no. 3,242–248.Dipartimento di Matematica, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129Torino - ItalyE-mail address: danilo.bazzanella@polito.it

Some conditional results on primes between consecutive squares

CAMERLO, RICCARDO

Archivio istituzionale della ricerca - Università di Genova

SOME CONDITIONAL RESULTS ON PRIMES
BETWEEN CONSECUTIVE SQUARES
Danilo Bazzanella
Dipartimento di Scienze Matematiche, Politecnico di Torino, Italy
danilo.bazzanella@polito.it
Autor’s version
Published in Funct. Approx. Comment. Math. 45 (2011), n. 2, 255-263
The original publication is available at http://projecteuclid.org/
1
SOME CONDITIONAL RESULTS ON PRIMES BETWEEN
CONSECUTIVE SQUARES
DANILO BAZZANELLA
Abstract. A well-known conjecture about the distribution of primes asserts that be-
tween two consecutive squares there is always at least one prime number. The proof of
this conjecture is quite out of reach at present, even under the assumption of the Riemann
Hypothesis. The aim of this paper is to provide the upper bounds for the exceptional set
for this conjecture under the assumption of some heuristic hypotheses.
1. Introduction
A well known conjecture about the distribution of primes asserts that for every positive
integer n, the interval [n2, (n+1)2] contains at least one prime. The proof of this conjecture
is quite out of reach at present, even under the assumption of the Riemann Hypothesis.
However it is not difficult to prove unconditionally that the conjecture holds for almost all
positive integers n. Indeed, we can prove that almost all intervals of the type [n2, (n+1)2]
contain the expected number of primes.
This paper is concerned with the exceptional set for the distribution of primes between
two consecutive squares, under the assumption of some unproved heuristic hypotheses.
The basic idea was to connect the exceptional set for the distribution of primes in in-
tervals of the type [n2, (n + 1)2] to the exceptional set of the asymptotic formula for the
distribution of primes in short intervals. The properties of the latter set, see D. Bazzanella
and A. Perelli [2], were thus used to obtain the desired results.
In a previous paper, see D. Bazzanella [1], the author proved that each of the intervals
[n2, (n+1)2] ⊂ [1, N ], with at most O(N1/4+ε) exceptions, contained the expected number
of primes, for every constant ε > 0. Under the assumption of the Riemann Hypothesis,
the author also proved that each of the intervals [n2, (n + 1)2] ⊂ [1, N ], with at most
O(f(N) log2N) exceptions, contained the expected number of primes, for every real valued
function f(x)→∞ arbitrarily slowly.
Under the assumption of a stronger hypotheses than the Riemann Hypothesis, we
would expect to obtain smaller bounds for the exceptional set. Infact, assuming a strong
form of the Montgomery Conjecture, the author proved that, with n sufficiently large, each
of the intervals [n2, (n+1)2] contained the expected number of primes without exceptions,
1991 Mathematics Subject Classification. 11NO5.
Key words and phrases. distribution of prime numbers, primes between squares.
2
PRIMES BETWEEN CONSECUTIVE SQUARES 3
see [1]. On the other hand, under the assumption of a weaker hypotheses than the Riemann
Hypothesis, we would expect to achieve upper bounds of sizes between O(f(N) log2N)
and O(N1/4+ε).
In order to estimate some sums which arose in our argument we employed the counting
functions N(σ, T ) and N∗(σ, T ). The former is defined as the number of zeros ρ = β + iγ
of the Riemann zeta function which satisfy σ ≤ β ≤ 1 and |γ| ≤ T , while N∗(σ, T ) is
defined as the number of ordered sets of zeros ρj = βj + iγj (1 ≤ j ≤ 4), each counted
by N(σ, T ), for which |γ1 + γ2 − γ3 − γ4| ≤ 1. As D. Bazzanella and A. Perelli [2] we
considered the heuristic assumption that there exists a constant T0 such that
(1.1) N∗(σ, T ) N(σ, T )
4
T
T ε
for every T ≥ T0 and arbitrarily small ε > 0, which is close to being the best possible, in
view of the trivial estimate
N∗(σ, T ) N(σ, T )
4
T
.
Recall that the Ingham–Huxley density estimate [5, Theorem 11.1] implies
(1.2) N(σ, T )

T 3(1−σ)/(2−σ)+ν
1
2
≤ σ ≤ 3
4
T 3(1−σ)/(3σ−1)+ν
3
4
≤ σ ≤ 1
,
for arbitrarily ν > 0. Thus from (1.1) it follows that
(1.3) N∗(σ, T )

T (10−11σ)/(2−σ)+ε
1
2
≤ σ ≤ 3
4
T (13−15σ)/(3σ−1)+ε
3
4
≤ σ ≤ 13
15
.
Remark. The estimate of N∗(σ, T ) proved by D. R. Heath-Brown, see [5, Lemma 12.7],
implies that the upper bound (1.3) holds for 1/2 ≤ σ ≤ 2/3.
Assuming that (1.3) holds also for σ > 2/3 we obtain the first result.
Theorem 1.1. Let ε > 0 be arbitrarily small and assume (1.3). Then each of the intervals
[n2, (n + 1)2] ⊂ [1, N ], with at most O(N1/5+ε) exceptions, contains the expected number
of primes.
If we assume the Lindelo¨f hypothesis, which states that, for every η > 0, the Riemann
Zeta-function satisfies
ζ(σ + it) tη (1/2 ≤ σ ≤ 1, t ≥ 2),
we can get the following stronger result.
4 D. BAZZANELLA
Theorem 1.2. Let ε > 0 be arbitrarily small and assume the Lindelo¨f hypothesis. Then
each of the intervals [n2, (n + 1)2] ⊂ [1, N ], with at most O(N ε) exceptions, contains the
expected number of primes.
Finally we assume the Density Hypothesis, which states that for every η > 0 the
counting function N(σ, T ) satisfies
N(σ, T ) T 2(1−σ)+η (1/2 ≤ σ ≤ 1),
and we thus obtain our last result.
Theorem 1.3. Let ε > 0 be arbitrarily small, assume the Density Hypothesis and (1.1).
Then each of the intervals [n2, (n+1)2] ⊂ [1, N ], with at most O(N ε) exceptions, contains
the expected number of primes.
2. Definitions and preliminary lemmas
We will always assume that n, x and N are sufficiently large as prescribed by the
various statements, and ε > 0 is arbitrarily small and not necessarily the same at each
occurrence. As in [2] we define a set related to the asymptotic formula
(2.1) ψ(x+ h(x))− ψ(x) ∼ h(x) (x→∞)
as
Eδ(N, h) = {N ≤ x ≤ 2N : |ψ(x+ h(x))− ψ(x)− h(x)| ≥ δh(x)},
where h(x) is an increasing function such that xε ≤ h(x) ≤ x for some ε > 0. It is clear that
(2.1) holds if and only if for every δ > 0 there exists N0(δ) such that Eδ(N, h) = ∅ for every
N ≥ N0(δ). Hence for small δ > 0, N tending to∞ and h(x) which is suitably small with
respect to x, the set Eδ(N, h) contains the exceptions, if any, to the expected asymptotic
formula for the number of primes in short intervals. Our first lemma is concerned with
the structure of the exceptional set above.
Lemma 2.1. Let δ > 0, xε ≤ h(x) ≤ x, for some ε > 0 and N be sufficiently large
depending on the function h(x). If x0 ∈ Eδ(N, h) then there exists an effective constant c,
depending on δ and h(x), such that
[x0, x0 + c · h(N)] ∩ [N, 2N ] ⊂ Eδ/2(N, h).
Lemma 2.1 may be proved along the same lines as Theorem 1 of [2], and essentially
says that if we have a single exception in Eδ(N, h), with a fixed δ, then we necessarily have
an interval of exceptions in Eδ/2(N, h). Moreover we define a set related to the asymptotic
formula
(2.2) ψ((n+ 1)2)− ψ(n2) ∼ 2n (n→∞)
PRIMES BETWEEN CONSECUTIVE SQUARES 5
as
Aδ(N) = {
√
N ≤ n ≤
√
2N : |ψ((n+ 1)2)− ψ(n2)− (2n+ 1)| ≥ δ(2n+ 1)},
that contains the exceptions, if any, to the expected asymptotic formula for the number
of primes in intervals of the type [n2, (n + 1)2] ⊂ [N, 2N ]. The main tool of the proofs is
the following lemma.
Lemma 2.2. For h(x) = 2
√
x+ 1 and every δ > 0 we have
|Aδ(N)| δ |Eδ/2(N, h)|√
N
+ 1.
Proof. Let n ∈ Aδ(N) and put x = n2 ∈ [N, 2N ] . From the definition of the set Aδ(N)
we get
|ψ((n+ 1)2)− ψ(n2)− (2n+ 1)| ≥ δ(2n+ 1)
and thus
|ψ(x+ h(x))− ψ(x)− h(x)| ≥ δh(x),
which implies x ∈ Eδ(N, h). From Lemma 2.1 follows that there exists an effective constant
c such that
[x, x+ c · h(x)] ∩ [N, 2N ] ⊂ Eδ/2(N, h).
Let m ∈ Aδ(N), m > n. As before we can define y = m2 ∈ [N, 2N ] and obtain, again by
Lemma 2.1, that
[y, y + c · h(y)] ∩ [N, 2N ] ⊂ Eδ/2(N, h).
If we choose c < 1, we get
y − x = m2 − n2 ≥ (n+ 1)2 − n2 = 2n+ 1 = 2√x+ 1 > ch(x)
and thus
[x, x+ c · h(x)] ∩ [y, y + c · h(y)] = ∅.
Hence the lemma is proved, since for every n ∈ Aδ(N) and x = n2, with at most one
exception, we have
[x, x+ c · h(x)] ⊂ [N, 2N ].

The next lemma concerns the conditional estimate for the fourth power mean value of
the function ψ(y) in short intervals.
Lemma 2.3. Assume the Lindelo¨f hypothesis and let ε > 0 be arbitrarily small. Then
there exists a function Σ(y, T ) such that for every ε > 0 we have
(2.3)
∫ 2N
N
∣∣∣ψ (y + y
T
)
− ψ(y)− y
T
+ Σ(y, T )
∣∣∣4 dy  N4+εT−3
6 D. BAZZANELLA
and
(2.4) Σ(y, T ) y
T log y
,
uniformly for N ≥ 2, 1 ≤ T ≤ N and N ≤ y ≤ 2N .
Lemma 2.3 is due to G. Yu [7, Lemma B]. The interesting consequence of this lemma is
that it allows to obtain the following conditional bound for the exceptional set Eδ(N, h),
with h(x) = 2
√
x+ 1.
Lemma 2.4. Assume the Lindelo¨f hypothesis, let h(x) = 2
√
x + 1 and let ε > 0 be
arbitrarily small. Then
(2.5) |Eδ(N, h)|  N1/2+ε.
Proof. In order to prove the lemma, we subdivide [N, 2N ] into O(log2N) intervals of type
Ij = [Nj, Nj + Y ] with
N ≤ Nj < 2N and Y  N
log2N
.
For every y ∈ Eδ(N, h) we have
|ψ(y + h(y))− ψ(y)− h(y)|  N1/2
and then
|Eδ(N, h)|N2 
∫
|ψ(y + h(y))− ψ(y)− h(y)|4 dy
Eδ(N,h)
(2.6)

∑
j
∫
|ψ(y + h(y))− ψ(y)− h(y)|4 dy,
Ejδ(N,h)
where Ejδ(N, h) = Eδ(N, h) ∩ [Nj, Nj + Y ].
If we choose Tj = N
1/2
j /2 we have, by Lemma 2.3, that there exist functions Σj(y, Tj)
satisfy the conditions (2.3) and (2.4), for all values of j. From the Brun–Titchmarsh
theorem, see H. L. Montgomery and R. C. Vaughan [6], we can deduce(
ψ(y + h(y))− ψ(y)− h(y)
)
−
(
ψ(y +
y
Tj
)− ψ(y)− y
Tj
+ Σj(y, Tj)
)
 y
Tj log y
,
PRIMES BETWEEN CONSECUTIVE SQUARES 7
for every j and every y ∈ Ejδ(Xn, θ). Then from (2.6) it follows that
|Eδ(N, h)|N2 
∑
j
∫ ∣∣∣∣ψ(y + yTj )− ψ(y)− yTj + Σj(y, Tj)
∣∣∣∣4 dy
Ejδ(N,h)
+
∑
j
∫ ∣∣∣∣ yTj log y
∣∣∣∣4 dy
Ejδ(N,h)

∑
j
∫ 2N
N
∣∣∣∣ψ(y + yTj )− ψ(y)− yTj + Σj(y, Tj)
∣∣∣∣4 dy
+ |Eδ(N, h)| N
2
log4N
which implies
(2.7) |Eδ(N, h)|N2 
∑
j
∫ 2N
N
∣∣∣∣ψ(y + yTj )− ψ(y)− yTj + Σj(y, Tj)
∣∣∣∣4 dy.
At this point we use Lemma 2.3 to get
|Eδ(N, h)|N2 
∑
j
∫ 2N
N
∣∣∣∣ψ(y + yTj )− ψ(y)− yTj + Σj(y, Tj)
∣∣∣∣4 dy

∑
j
N4+εT−3j
 N5/2+ε
and this leads to (2.5). 
3. Proof of the theorems
Let h(x) = 2
√
x+1 and use the classical explicit formula, see H. Davenport [3, chapter
17], to write
(3.1) ψ(x+ h(x))− ψ(x)− h(x) = −
∑
|γ|≤T
xρcρ(x) +O
(
N log2N
T
)
,
uniformly for N ≤ x ≤ 2N , where 10 ≤ T ≤ N , ρ = β+ iγ runs over the non-trivial zeros
of ζ(s) and
cρ(x) =
(1 + h(x)/x)ρ − 1
ρ
.
Let
(3.2) T =
√
N log3N
8 D. BAZZANELLA
and note that
(3.3) cρ(x) min
(
1√
N
,
1
|γ|
)
.
If we follow the method of D. R. Heath-Brown we can find a constant 0 < u < 1 such
that ∑
|γ|≤T, β>u
xρcρ(x)
√
N
logN
,
see [5, pag. 319].
Remark. More precisely from (3.2) it follows that we can choose 5/6 ≤ u < 1. Note that
under the assumption of the Density Hypothesis we have the weaker condition 11/14 ≤
u < 1.
Thus we have
ψ(x+ h(x))− ψ(x)− h(x) = −
∑
|γ|≤T, β≤u
xρcρ(x) +O
( √
N
logN
)
and then
(3.4) |Eδ(N, h)|N2 
∫ 2N
N
∣∣∣∣∣∣
∑
|γ|≤T, β≤u
xρcρ(x)
∣∣∣∣∣∣
4
dx.
To estimate the fourth power integral we divide the interval [0, u] into O(logN) subinter-
vals Ik of the form
Ik =
[
k
logN
,
k + 1
logN
]
.
By the Ho¨lder inequality we have∣∣∣∣∣∣
∑
|γ|≤T, β≤u
xρcρ(x)
∣∣∣∣∣∣
4
 log3N
∑
k
∣∣∣∣∣∣
∑
|γ|≤T, β∈Ik
xρcρ(x)
∣∣∣∣∣∣
4
.
Following again the method of D. R. Heath-Brown, we can get
(3.5)
∫ 2N
N
∣∣∣∣∣∣
∑
|γ|≤T, β≤u
xρcρ(x)
∣∣∣∣∣∣
4
dx N−1+ε max
σ≤u
N4σM(σ, T ),
where
M(σ, T ) =
∑
β1,...,β4≥σ
|γ1|≤T,...,|γ4|≤T
1
1 + |γ1 + γ2 − γ3 − γ4|
and
(3.6) M(σ, T ) N∗(σ, T ) logN,
PRIMES BETWEEN CONSECUTIVE SQUARES 9
see [5, p. 336]. From (3.4), (3.5) and (3.6) it follows that
(3.7) |Eδ(N, h)|  N−3+ε max
σ≤u
N4σN∗(σ, T ).
To prove Theorem 1.1 we assume (1.3). Then we write
|Eδ(N, h)| 

N−3+ε max
σ≤u
N4σT (10−11σ)/(2−σ)+ε
1
2
≤ σ ≤ 3
4
N−3+ε max
σ≤u
N4σT (13−15σ)/(3σ−1)+ε
3
4
≤ σ ≤ 13
15
.
The above upper bound for the exceptional set attains its maximum at σ = 3/4, so we
get
|Eδ(N, h)|  N7/10+ε,
for every δ > 0. From Lemma 2.2 we conclude
|Aδ(N)| δ |Eδ/2(N, h)|√
N
+ 1 N1/5+ε,
for every δ > 0. Then the proof of Theorem 1.1 is complete.
Remark. In Theorem 1.1 we need to assume the upper bound (1.3) only for (139 −√
761)/160 ≤ σ ≤ 5/6. For smaller values of σ the estimate of N∗(σ, T ) proved by
D. R. Heath-Brown, see [5, Lemma 12.7], is enough and the bigger values are not involved
in (3.7).
To prove Theorem 1.2 we assume the Lindelo¨f hypothesis. By Lemma 2.4 we have (2.5)
and then, again by Lemma 2.2, we obtain that
|Aδ(N)| δ |Eδ/2(N, h)|√
N
+ 1 N ε,
for every δ > 0. Then Theorem 1.2 is proved.
In order to prove Theorem 1.3 we follow the proof of Theorem 1.1 until the equation
(3.7). Then we use (1.1) and the Density Hypothesis to have
|Eδ(N, h)|  N−7/2+ε
(
max
σ≤u
NσN(σ, T )
)4
 N−7/2+ε
(
max
σ≤u
NσT 2(1−σ)
)4
 N1/2+ε.
By Lemma 2.2, we can conclude that
|Aδ(N)| δ |Eδ/2(N, h)|√
N
+ 1 N ε,
for every δ > 0, and this complete the proof of Theorem 1.3.
10 D. BAZZANELLA
References
[1] D. Bazzanella, Primes between consecutive squares, Arch. Math. 75 (2000), 29–34.
[2] D. Bazzanella and A. Perelli, The exceptional set for the number of primes in short intervals, J.
Number Theory 80 (2000), 109–124 .
[3] H. Davenport, Multiplicative Number Theory, Second edition, Graduate Texts in Mathematics 74.
Springer–Verlag, New York (1980).
[4] D. R. Heath-Brown, The difference between consecutive primes II, J. London Math. Soc. (2), 19
(1979), 207–220.
[5] A. Ivic´, The Riemann Zeta-Function , John Wiley & Sons, New York (1985).
[6] H. L. Montgomery and R. C. Vaughan, The large sieve , Mathematika 20 (1973), 119–134.
[7] G. Yu, The differences between consecutive primes, Bull. London Math. Soc. 28 (1996), no. 3,
242–248.
Dipartimento di Matematica, Politecnico di Torino, Corso Duca degli Abruzzi 24,
10129 Torino - Italy
E-mail address: danilo.bazzanella@polito.it


Some conditional results on primes between consecutive squares

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