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        A polynomial with prime variables attached to cusp forms

        2018-06-23 12:22:56LiuDanLiuHuafengZhangDeyu

        Liu Dan,Liu Huafeng,Zhang Deyu

        (School of Mathematics and Statistics,Shandong Normal University,Ji′nan 250014,China)

        1 Introduction

        Letfbe a holomorphic cusp form for the group Γ=SL2(Z)of even integral weightk,with Fourier coefficientsa(n):

        We normalizefwith the first coefficient being 1,and setλ(n)=a(n)/n(k?1)/2.From the Ramanujan conjecture it is easy to know that|λ(n)|≤d(n),whered(n)is the Dirichlet divisor function(this result is due to Deligne).

        Many scholars are interested in researching the properties of quadratic forms.In 1963,Vinogradov[1]and Chen[2]independently studied the number of lattice points in the 3-dimensional balland showed the asymptotic formula

        Subsequently,the exponent 2/3 in the above error term was improved to 29/44 by Chamizo and Iwaniec[3],and to 21/32 by Heath-Brown[4].Friedlander and Iwaniec[5]studied the number of prime vectors among integer lattice points in 3-dimensional ball.Letπ3(x)denote the number of integer points(m1,m2,m3)∈Z3withThey proved that

        which can be viewed as a generalization of the prime number theorem.

        Let Λ(n)stands for von Mangoldt function.Guo and Zhai[6]studied the asymptotic behavior of sum

        and obtained for any fixed constantA>0,

        whereC,Iare computable constants.In 2015,Hu[7]studied the sum

        and obtained its upper boundx3/2logc1x,wherec1is a suitable constant.Later,G.Zaghloul[8]improved this result towherec2is arbitrary.Zhang and Wang[9]studied the sum

        and proved

        In 1938,Hua[10]established that almost alln≤Nsubject to the natural congruence conditions can be represented as the sum of two prime squares and akth power of prime,namely

        Later many results for the casek=1 were proved(see[8-9],[11-12]etc.).

        In this paper,motivated by above results we firstly study hybrid problems of the Fourier coefficientsλ(n)and a polynomialand get the following results.

        Theorem 1.1Let

        Then we have

        Theorem 1.2Let

        Then we have

        wherec>0 is arbitrary.

        To prove our Theorems,we follow the classical line of the circle method,the di ff erence is that here we will deal with the sum

        on the major arcs,which makes it difficult to establish the Voronoi′s summation formula ofλ(n)over the arithmetic progression.

        Notation 1.1Throughout this paper,the lettercwith or without subscript denotes a constant,not necessarily the same in all occurrences.εdenotes an arbitrary positive real number.R,Z,N,P denotes the sets of all real numbers,integers,natural numbers and primes,respectively.As usual,we writee(z)for exp(2πiz).The letterp,with or without subscript,denotes a prime number.

        2 Outline of circle method

        Throughout this paper,xis a large positive integer andL=logx.For anyα∈Randy>1,de fine

        By the de finitions ofπλ(x),πλ,Λ(x)and the well-known identity

        we have

        In order to apply the circle method,letP,Qare positive parameters,which will be decided later.By Dirichlet′s lemma on rational approximation,eachcan be written in the formfor some integersa,qwith 1≤a≤q≤Qand(a,q)=1.We de fine the major arcs M and the minor arcs m as follows:

        where

        It follows from 2P≤Qthat M(a,q)are mutually disjoint.Then we can rewrite the integral from 0 to 1 as a sum of two integrals on the major arcs and the minor arcs,respectively.For example,we can rewriteπλ(x)as

        Therefore,the problems are reduced to handle the integrals on the major arcs and the minor arcs.

        3 Some lemmas

        In this section,we give some lemmas which will be used in the proof of our theorems.

        Lemma 3.1Supposeα∈M.Then,

        whereB>0 is a suitable positive constant.

        ProofThis is lemma 3.4 in[8].

        Lemma 3.2 Supposeα∈m.Let

        Then

        ProofThis is well-known result of Vinogradov,which can be found in[12].

        Lemma 3.3Supposeα∈m.Fork≥2,and setK=2k?1.Let

        Then

        ProofThis is Harman′s result which can be found in[13].

        Lemma 3.4Let

        For anyf∈Hkand anyε>0,we have

        uniformly for 2≤M≤x.

        Moreover,for any positive integerr,

        ProofThis is Theorem 3.1 in[9].

        4 Proof of Theorem 1.1

        In this section,we prove Theorem 1.1.To do this,we need the estimate ofλ(n)over the arithematic progressions.

        Recall the de finition of Kloosterman sum,we have

        Noting that(a,q)=1,we can know that the inner sum isqifd=qandu?a≡0 modq.Otherwise the inner sum is 0.Therefore,

        Let

        By Lemma 3.4 and(1),we have that forx≥q,

        where we takeM=q2/3x1/3.Ifx

        uniformly forx>0.De fine

        IfT?q,thenH(T;q,a)?q1+εT.Ifq?T,consider the integralApplying(1)and Lemma 3.3 withM=U,we obtain

        So,by a splitting argument,we have

        Supposeα=a/q+βwithWe can get

        We use the Abel′s summation and Lemma 3.4 to get

        By partial integration twice and then using(2)and(3)we have

        noting thatx=QP,andqsatis fiesq≤Pon the major arcs.

        Next,we prove the Theorem 1.1.We first handle the casek≥2.It is enough to take

        By the trivial estimates ofand(4),we have

        where we use

        Next,we estimate the contribution from the integral on the minor arcs.By Cauchy′s inequality,we obtain

        We also have

        whereK=2k?1.

        By partial summation formula,we have

        whereK=2k?1.

        Inserting this into(6),we can get that

        Combing(5)and(7),we can get

        Thus,we can get the desired bound of Theorem 1.1.Fork=1,we use Lemma 3.2 instead of Lemma 3.3 and takeP=x4/19,Q=x5/19.Then following a similar argument ofk≥2,we can get

        It is easy to verify

        5 Proof of Theorem 1.2

        In this section,we give the proof of Theorem 1.2.To do this,we can set

        whereCis a positive constant.Similarly,we can writeπλ,Λ(x)as a sum of two integrals i.e.

        Now we deal with the casek≥2.We first treat the integral on the major arcs.Using the trivial estimates ofand lemma 3.1,we have

        RecallingPandQare de fined by(8),we get

        Next,we estimate the contribution from the integral on the minor arcs.Using Cauchy′s inequality,we have

        where we have used the well-known estimate

        For the estimate ofwe use the following result.

        Lemma 5.1Supposeα∈m.Fork≥2,let

        Then

        ProofWe can easily get

        From Lemma 3.3,we have

        Noting thatP

        Combing these three formulas above,we have

        By partial summation formula and Lemma 3.1,we can easily get

        Inserting this into(11),we can get that

        From(9)and(10),we have

        Fork=1,using Lemma 3.2 and following Lemma 3.1,we can get

        Then,following the ideal ofk≥2,we can also get

        Combing(12)and(13),we complete the proof of Theorem 2.2.

        Reference

        [1]Vinogradov I M.On the number of interger points in a sphere[J].Izv.Akad.Nauk SSSR.Ser.Mat.,1963,27:957-968.

        [2]Chen J.Improvement of asymptotic formulas for the number of lattice points in a region of three dimensions(II)[J].Sci.Sinica,1963,12:751-764.

        [3]Chamizo F,Iwaniec H.On the sphere problem[J].Rev.Mat.Iberoamer,1995,11:417-429.

        [4]Heath-Brown D R.Lattice points in the sphere[J].Number Theory in Progress,1999,2:883-892.

        [5]Friedlander J B,Iwaniec H.Hyperbolic prime number theorem[J].Acta Math.,2009,202:1-19.

        [6]Guo R T,Zhai W G.Some problem about the ternary quadratic form[J].Acta Arith.,2012,156:101-121.

        [7]Hu L Q.Quadratic forms connected with Fourier coefficients of Maass cusp forms[J].Front.Math.China,2015,10:1101-1112.

        [8]Zaghloul G.Quadratic forms connected with Fourier coefficients of holomorphic and Maass cusp forms[J].J.Number Theory,2016,167:118-127.

        [9]Zhang D Y,Wang Y N.Ternary quadratic form with prime variables attached to Fourier coefficients of holomorphic cusp form[J].Journal of Number Theory,2017,176:211-225.

        [10]Hua L K.Some results in the additive prime number theory[J].Quart.J.Math.(Oxford),1938,9:68-80.

        [11]Bauer C,Liu M C,Liu J Y.On a sum of three prime squares[J].J.Number Theory,2000,85:336-359.

        [12]Liu J Y,Zhan T.New developments in the Addictive Theory of Prime Numbers[M].Singapore,World Scienti fic:2012.

        [13]Harman G.Trigonometric sums over primes(I)[J].Mathematika,1981,28:249-254.

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