On Waring-Goldbach problem with Piatetski-Shapiro primes

Contents lists available atScienceDirect

Journal

of

Number

Theory

www.elsevier.com/locate/jnt

On

Waring–Goldbach

problem

with

Piatetski-Shapiro

primes

DepartmentofMathematics,BilkentUniversity,06800Bilkent,Ankara,Turkey

a r t i c l e i n f o a bs t r a c t

Article history:

Received13June2017 Receivedinrevisedform29 September2017

Accepted30September2017 Availableonline13October2017 CommunicatedbyS.J.Miller MSC: primary11P32 secondary11P05,11P55,11L03, 11L07,11L15,11L20,11B83 Keywords: Waring–Goldbach Piatetski-Shapiro Circlemethod Weylsums Exponentialsums vanderCorput

In this paper, we show that all sufficiently large natural numberssatisfyingcertainlocalconditionscanbewrittenas the sumofkth powersof Piatetski-Shapiro primes, thereby establishing a variant of Waring–Goldbach problem with primesfromasparsesequence.

©2017ElsevierInc.Allrightsreserved.

✩ _{BothauthorsweresupportedbyTÜBITAK ResearchGrantno.114F404.} * Correspondingauthor.

E-mailaddresses:[email protected](Y. Akbal),[email protected](A.M. Güloğlu).

https://doi.org/10.1016/j.jnt.2017.09.018

1. Introduction

Wedefine,foranaturalnumberk,andaprimep,ϑ= ϑ(p,k) tobethelargestnatural numbersuchthatpϑ| k,anddefine γ(p,k) by

γ = γ(p, k) =



ϑ + 2, if p = 2 and 2| k,

ϑ + 1, otherwise.

WethenputK(k)=_(p−1)|kpγ_.

LetHc(k) denotethesmallestnumberofvariabless suchthateverysufficientlylarge

integern≡ s(mod K(k)) canbe writtenintheform

n = pk₁+· · · + pk_s, with p1, . . . , ps∈ Ac, (1.1)

wherep1,. . . ,ps areprimes thatlieintheset

Ac ={mc : m ∈ N}.

Thisset is named afterI.I. Piatetski-Shapiro, sincehe wasthefirstto proveananalog ofthePrimeNumberTheorem (cf.[10])forprimesinAc forc∈ (1,12/11).

Anasymptotic formula forthe numberof representationsof n asin (1.1)was given bytheauthors in[1]. Inthis paper,we intend togive anupperbound for Hc(k) using

therecentresult ofKumchevand Wooley (cf. [8]). Beforewe stateour result,we shall givesomedefinitionsfromtheirpaper.

Put θ = 1− 1/k, σ−1_k = k(k− 1), t = ₂1k log k and u =k log(k2_/2)_{− t. Then,} define λi =  θ + σ_k−1/ki−1 (1 i  u + 1), λu+2 = k2− θt−3 k2_{+ k}− kθt−3λu+1, λu+j = k2_{− k − 1} k2_{+ k}− kθt−3θ j−3_λ u+1 (3 j  t).

Finally, set Λ = _iu+tλi, v = (k − Λ)/2σk and λ = λu+t. Note that λ is the

minimumoftheλi’s.

Theorem1.1. Forsufficientlylarge k,

Hc(k) (4k − 2) log k − (2 log 2 − 1)k − 1,

whenever

1 < c < 1 + λ

Remark1.2.It iscomputedin[8]that k− Λ 2σk =k 2+ log( √ 2/k)− 3 − γ + O(k−1/2),

where γ isthe fractionalpartofu+ t.Wecanalsoestimateλ easily to seethatk2λ∈

[1.9,2.1].Thus,thelengthoftheintervalthatc liesinisof orderk−3. 2. Preliminariesandnotation

2.1. Notation

Throughout thepaper,thelettersk, m andn arenaturalnumberswithk 4,andp

always denotesaprimenumber.Thenotation x∼ X meansthatX < x 2X forany real numberX.Furthermore,c> 1 isafixedrealnumberandweputδ = 1/c.

Givenarealnumberx,wewritee(x)= e2πix_{,{x} forthefractionalpartofx,x for} thegreatestintegernotexceedingx.WewriteL= log N .

Forany functionf ,weput

Δf (x) = f−(x + 1)δ− f(−xδ), (x > 0).

We recall that for functions F and real nonnegative G the notations F G and

F = O(G) areequivalent tothestatementthattheinequality|F | αG holdsforsome

constantα > 0.IfF  0 also,thenF  G isequivalenttoG F .WealsowriteF  G

to indicate that F G and G F . In what follows, any implied constants in the symbols andO may dependonthe parametersc,ε,k,s,butare absolute otherwise. Weshallfrequentlyuseε withaslightabuseofnotationtomeanasmallpositivenumber, possiblyadifferentoneeachtime.

2.2. Preliminaries

Lemma 2.1 (Vaaler [3, Appendix]). Put ψ(x) = x− x − 1/2. Then, there exists a

trigonometricpolynomial

ψ∗(x) = 1|h|H

ahe(hx), (ah |h|−1)

such thatforany real x,

|ψ(x) − ψ∗_{(x)| }

|h|<H

bhe(hx), (bh H−1).

Lemma 2.2 (Vaughan’sIdentity [4,Prop.13.4]).Letu,v  1 bereal numbers. If n> v then,

Λ(n) = rs=n ru μ(r) log s− rs=n r>v s>u Λ(r) d|s du μ(d)− rst=n ru sv μ(r)Λ(s),

whereΛ is vonMangoldt’s function.

Definition2.3.Weput gk(α, x) = p∼x e(αpk), fk(α, x) = p∼x p∈Ac cp1−δe(αpk).

Lemma 2.4. Assume that c > 1, and there are coprime integers a and q with 0 a 

q P suchthat |qα − a| P x−k.Then,

fk(α, x) = gk(α, x) + O(x(44−14δ)/31P7/31)

forsufficiently largex.

Proof. Weshall assumebelowthatc∈ (1,14/13) andP  x(18−17δ)/7_{, sinceotherwise} thegivenerrorisworsethanthetrivialestimate.

Thefunction−nδ−−(n + 1)δservesasthecharacteristicfunctionofthesetAc,

anditcanberewrittenas −nδ_{−−(n + 1)}δ_{= δn}δ−1_{+ Δψ(n) + O(n}δ−2_). Thus, fk(α, x) = gk(α, x) + p∼x cp1−δe(αpk)Δψ(p) + O1/ log x. ByLemma 2.1 p∼x cp1−δe(αpk)Δ(ψ− ψ∗)(p) H−1x2−δ+ H−1 1h<H   n∼x n1−δe(hnδ).

Partialintegrationyields n∼x n1−δe(hnδ) x1−δsup t∼x   x<nt e(hnδ).

Usingtheexponentpair(1/2,1/2) (cf.[3,eqn. 3.3.4])weobtaintheestimate

x<nt

Thus, assuming1 H  x,weconcludethat

p∼x

cp1−δe(αpk)Δ(ψ− ψ∗)(p) H−1x2−δ+ H1/2x1−δ/2. (2.1)

Next,weobserve thatthesum p∼x cp1−δe(αpk)Δψ∗(p) is 1 log xsupt∼x   x<nt cn1−δe(αnk)Λ(n)Δψ∗(n) + Ox3/2−δ,

where Λ is von Mangoldt’s function. Recalling the definition of ψ∗, noting that ah

|h|−1 _andthatφ

h(t)= e



h(t + 1)δ− htδ− 1 satisfies

φh(t) |h|tδ−1, φh(t) |h|tδ−2,

we derivebypartialintegrationthat x<nt cn1−δe(αnk)Λ(n)Δψ∗(n) sup x<zt 1|h|H |Fh(z)| where Fh(z) = x<nz Λ(n)e(αnk+ hnδ).

Wehaveshownso far p∼x cp1−δe(αpk)Δψ∗(p) 1 log xsupz∼x 1|h|H |Fh(z)| + O  x3/2−δ. (2.2)

Assumethatthereexist coprimeintegersa,q with0 a q  P suchthat|qα−a|

P x−k.Then, Fh(z) = q−1 −q/2<bq/2 S(a, b; q) x<nz Λ(n)e(Gb(n)), (2.3)

where Gb(t)= βtk+ htδ− bt/q,β = α− a/q and

S(a, b; q) = q m=1 e amk+ mb q  .

WeshallapplyLemma 2.2withu,v 1 and1 uv  x towrite x<nz Λ(n)e(Gb(n)) = Σ1+ Σ2+ Σ3, where Σ1= ru μ(r) x/r<sz/r e(Gb(rs)) log s − ru r=wt wu tv μ(w)Λ(t)  x/r<sz/r e(Gb(rs)), and Σ2=− x<rsz s>v r>u Λ(s) d|r du μ(d)e(Gb(rs)), Σ3=− x<rsz u<ruv  r=wt wu tv μ(w)Λ(t)e(Gb(rs)). Bypartial summation Σ1 log x ru sup z∼x   x/r<sz/r e(Gb(rs))  . Wehave ∂2_G b(rs) ∂s2 = k(k− 1)βr k_sk−2_{+ hδ(δ− 1)s}δ−2_rδ_.

Note thatsince P  x(18−17δ)/7 _{and c < 4/3, it follows thatP = o(x}δ_{). Furthermore,}

|β| P x−k_{.Thus,thesecondtermabovedominatesthefirstforsufficientlylargex when}

s ∼ x/r;that is, f(s)  r2_xδ−2_{|h|, where f (s)= G}

b(rs). Applying van der Corput’s

estimatein[4,Cor. 8.13]to f (s) on(x/r,z/r],weconcludethat Σ1 log2x



u|h|1/2xδ/2+ x1−δ/2|h|−1/2. (2.4) Next,usingdyadicdivisionwecanwrite

Σ2 xε

R,S

where T (R, S) = r∼R γr s∼S x<rsz λse(Gb(rs))

withR > u,S > v,RS x,and|λs|,|γr| 1.Wenote thatΣ3 log xR,S|T (R,S)|,

where T (R,S) is asimilar bilinearsum with differentcoefficients and R,S satisfy u< R  uv,RS  x.To estimate T (R,S), we apply Weyl–van der Corput inequality(cf.

[3, Lemma 2.5])toget T (R, S)2 (RS) 2 L + RS2 L 1||L max S<s,s+2S|Γ(, s, R)|, (2.5)

where 1 L S istobechosenoptimally,and Γ(, s, R) =

r∈I

e (Gb(r(s + ))− Gb(rs)) .

Here,I⊆ (R,2R] isanintervaldeterminedbytheconditionsr∼ R,x< sr,(s+ )r z. Wehave

Gb(r(s + ))− Gb(rs) = βrk



(s + )k− sk+ hrδ(s + ))δ− sδ− br/q. Thus, weconcludethatwhenr∼ R andforsufficientlylargex,

∂2(Gb(r(s + ))− Gb(rs))

∂s2

  xδ−1|h|R−1_.

Applying [4,Cor. 8.13]onceagain,weobtain

Γ(, s, R) R1/2(|h|xδ−1)1/2+ (x1−δ|h|−1)1/2.

Insertingthisboundin(2.5)andusing[3,Lemma 2.4]tochooseL∈ [1,S] optimallywe obtain

T (R, S) R−1/6x(δ+5)/6|h|1/6+ x1−δ/4|h|−1/4

+ R1/2x1/2+ R−1/4x(δ+3)/4|h|1/4+ xR−1/4.

This leadsto theestimate

x−ε(Σ2+ Σ3) xv−1/2+ (uv)1/2x1/2+ x1−δ/4|h|−1/4+ xu−1/4 + u−1/6x(δ+5)/6|h|1/6+ u−1/4x(δ+3)/4|h|1/4.

Combiningestimatesin(2.4)and (2.6)and choosingv = (x/u)1/2yields thebound x−ε x<nz Λ(n)e(Gb(n)) x1−δ/4|h|−1/4+ u1/4x3/4 + u−1/4x + u−1/6x(δ+5)/6|h|1/6 + u−1/4x(δ+3)/4|h|1/4+ uxδ/2|h|1/2.

Inserting thisestimate in(2.3) and thenapplying the bounds (cf. [11, Lemma 4.1 and Theorem 4.2])

S(a, b; q) q1/2+εgcd(b, q) for b= 0, S(a, 0; q) q1−1/k,

whichholdfor(a,q)= 1,wededucethat

Fh(z) xεq1/2+2ε



x1−δ/4|h|−1/4+ u1/4x3/4+ uxδ/2|h|1/2 + u−1/6x(δ+5)/6|h|1/6+ u−1/4x(δ+3)/4|h|1/4+ u−1/4x.

Going back to (2.2) and applying [3, Lemma 2.4] to choose 1  u  x optimally, we obtain p∼x cp1−δe(αpk)Δψ∗(p) x3/2−δ+ xεq1/2+2εx1−δ/4H3/4 + x(δ+6)/8H9/8+ x(3δ+10)/14H17/14+ x7/8H + xδ/2H3/2 + x(3δ+6)/10H13/10+ x(δ+8)/10H11/10+ x3/4H + x(δ+4)/6H7/6+ x(δ+2)/4H5/4+ x(δ+8)/10H11/10. (2.7)

Combining(2.1)and(2.7)weseethatx−ε(fk(α,x)− gk(α,x)) is

H−1_x2−δ_{+ H}1/2_x1−δ/2_{+ x}3/2−δ_{+ P}1/2_x1−δ/4_H3/4_{+ x}δ/2_H3/2 + x(δ+6)/8H9/8+ x7/8H + x(3δ+10)/14H17/14+ x(3δ+6)/10H13/10

+ x(δ+8)/10H11/10+ x(δ+4)/6H7/6+ x(δ+2)/4H5/4.

Using[3,Lemma 2.4]tochooseH∈ [1,x] optimallyyieldsthebound

x(4−2δ)/3+ x(10−4δ)/7P2/7+ x(30−10δ)/21P5/21+ x(24−8δ)/17P4/17

+ x(23−8δ)/16P1/4+ x(44−14δ)/31P7/31+ x(32−10δ)/23P5/23

+ x(6−2δ)/5P1/5+ x(18−6δ)/13P3/13+ x(12−4δ)/9P2/9

TheresultfollowsbynotingthatforP  x(18−17δ)/7andc∈ (1,4/3),whichweassumed above,thetermx(44−14δ)/31P7/31dominatestheotherterms. 2

3. ProofofTheorem 1.1

Given n∈ N,weputN =1₂n1/k,anddefinetheintegral

Ik,c(n) = 1  0 fk(α, N )F(α)2e(−αn)dα, where F(α) = fk(α, N )v u+t  j=1 fk(α, Nλj).

Definition 3.1 (Major and minor arcs). For 1 P  Nk/2_{, wedefine theset of major}

arcsM= M(P ) as theunionoftheintervals

M(a, q; P ) ={α ∈ [0, 1) : |qα − a|  P N−k}

with 0 a  q  P and (a,q) = 1.We define the corresponding set of minor arcs by putting m= m(P )= [0,1)\ M.

Lemma 3.2.Fors= 2(u+ t+ v)+ 1,P  N(14δ−13)λ/38−sε_{,and anyA> 0,}  M fk(α, N )F(α)2e(−αn)dα = Ss,k(n)Jk,s(n) + OA  XN−kL−s−A, where X = N2Λ+2v+1_,S

s,k(n) isthesingularseries

Ss,k(n) = q1 1aq (a,q)=1 ϕ(q)−1 1xq (x,q)=1 eaxk/q s e(−na/q),

and Jk,s(n) isthesingularintegral

Jk,s(n) = ∞  −∞ V (β; N ) u+t+v_ i=1 V (β; Nλi₎2_e(−βn)dβ, in which V (β; Z) = 2Z  Z e(βγk) log γ dγ.

Remark 3.3. As noted in [7, eqn. (4.3)], for all sufficiently large integers n with n ≡ s(mod K(k)),

Ss,k(n)Jk,s(n) XL−s,

and theconditionss  3k + 1 andn ≡ s(mod K(k)) ensure thatthesingular series is positive. Proof. Let G(α) = gk(α, N )v u+t  j=1 gk(α, Nλj).

For α ∈ M(a,q;P ) with coprime a,q satisfying 0 a  q  P , we have |qα − a| 

P N−k P N−λik_{.Thus,Lemma 2.4yields}

fk(α, Nλi) = gk(α, Nλi) + O(Nλi(44−14δ)/31P7/31) (3.1)

forsufficientlylargeN .Furthermore, by[5,Theorem 2]wehave

gk(α, Nλi) P1/2N11λi/20+ε+

qε_Nλi_{(log N )}C

(q + Nkλi|qα − a|)1/2

for some absolute constant C > 0. Here, the second term dominates for each i with

1 i u+ t,providedthat

P  N9λ/20, (3.2)

inwhichcase,wehave

gk(α, Nλi) q−1/2Nλi+ε.

WerewriteIk,c(n) intheform

1  0 s  i=1 fk(α, xi)e(−αn)dα, (3.3)

where eachxi stands forNλji for someappropriateindex 1 ji  u+ t. Noteinthis

notation,s_i=1xi = X.Itfollows from(3.1)and(3.3)that

 M fk(α, N )F(α)2− gk(α, N )G(α)2  e(−αn)dα (3.4) is

1qP 0aq (a,q)=1 P q−1N−k X(44−14δ)/31P7s/31 + s−1 =1 1i1<...<is (xi1· · · xi) (13−14δ)/31 P7/31q−(s−1−)/2X1+ε  P N−k _X(44−14δ)/31_P1+7s/31 + X1+ε 1qP q−(s−1)/2 s−1 =1  Nλ(13−14δ)/31P7/31q1/2  .

In thelast sumover  above,we needNλ(13−14δ)/31P7/31q1/2 < 1, sinceotherwise the estimate above is worse thanthe trivialestimate. Thus,with this assumption and the factthats> 5,theaboveestimateis

P N−k_X(44−14δ)/31_P1+7s/31_{+ X}1+ε_Nλ(13−14δ)/31_P7/31 

.

It followsthat(3.4)isAXN−kL−s−A ifwefurtherimpose theconditionthat

P  N(14δ−13)λ/38−sε. (3.5)

Notethatifweassume(3.5),then(3.2)alsoholds.Finally,itfollowsfrom[9,Theorem 3]

thatforanypositiveA,

 M s  i=1 gk(α, xi)e(−nα)dα = Ss,k(n)Jk,s(n) + OA  XN−kL−s−A.

Therefore, theresultfollows. 2

Next, wedeal withminor arc contribution.We chooseP = N(14δ−13)λ/38−sε. Given

α∈ m, useDiophantine approximation to find coprimeintegersa,q with1 a q 

Nk_{/P suchthat |qα − a| P N}−k_{. Since α∈ m, q > P . Using [6, Theorem 1.2]with}

(θ = 1,k 4 andρ= ρ(k))togetherwith[8,Lemma 2.1]wederive thatforanyε> 0,

gk(α, N ) N1−σk/3+ε+ N1+εP−1/2

N1+sε_N−σk/3_{+ N}−(14δ−13)λ/76_.

NotethatTheorem 1.2in[6]canbeusedwiththeimprovedexponentσkasismentioned

intheproofof[8,Lemma 2.2]duetotherecentdevelopmentsinVinogradov’smeanvalue theorem.

Itfollowsfrom [2,Lemma 2.11]thatforc∈ (1,2) andanyε> 0, fk(α, N )− gk(α, N ) N1−(δ(1−A)−A)+ε, whereA= (ν− 1)/(2ν − 1) and ν = ⎧ ⎪ ⎨ ⎪ ⎩ k(k + 1)2_{, if 4 k  11,} 23k/2 (3k/22− 1) 3k/2 − k , if k 12. Notethat δ(1− A) − A < 1 − 2A = 1 2ν− 1 < 1 3k(k− 1) = σk/3. Therefore,combiningaboveestimates,wecanwrite

sup

α∈m|fk(α, N )| N

1+ε_N−η1_{+ N}−η2_,

whereη1= (14δ− 13)λ/76,andη2= δ(1− A)− A.Thus,  m fk(α, N )F(α)2dα N1+ε  N−η1_{+ N}−η2 1  0 |F(α)|2_dα N1+ε_N−η1_{+ N}−η2_{(X/N )}1−δ 1  0 |G(α)|2_dα,

where the passage from the integrand |F|2 to |G|2 is justified by interpreting the inte-gral as aweightedsumover thesolution set of asystemof Diophantine equations. By

[8, Lemma 2.3], 1  0 G(α)2 dα XN−1−k+ε. Hence,weconclude  m fk(α, N )F(α)2dα  N−η1_{+ N}−η2_N2(1−δ)(Λ+v)+2ε_XN−k = o(XL−sN−k), providedthat

c− 1 < min 1 (4ν− 2)(Λ + v) + ν − 1, λ 152(Λ + v) + 13λ  . (3.6) ItfollowsfromRemark 1.2thatforsufficientlylargek,Λ+ v >7k₆ .Wecanalsoseethat

ν < 27k2_{/2 andλ< 2.1/k}2_{. Then,} λ  4ν− 2 +ν− 14 Λ + v  <2.1 k2 54k2+81k 2 7k  < 152.

This shows that the second term on the right side in (3.6) is the minimum. Finally, the proof of Theorem 1.1 follows by combining the results in this section and using

[8, Theorem 1]. Acknowledgment

Wethanktherefereeforcarefullyreadingthemanuscript. References

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