A polynomial with prime variables attached to cusp forms

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.

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