Resonance between the Representation Function and Exponential Functions over Arithemetic Progression

Abstract

Let denote the number of representations of a positive integer as a sum of two squares, i.e., , where and are integers. We study the behavior of the exponential sum twisted by over the arithmetic progressions where , , , and means . Here, is a large parameter, are integers, and . We obtain the upper bounds in different situations.

1. Introduction

In analytic number theory, the problems concerning nonlinear exponential twisting arithmetic functions arise naturally in investigating equidistribution theory, zero-distribution of L-functions, and so on. Let be some arithmetic number-theoretic function. We usually consider the general nonlinear exponential sum of the form

Here, means , and . When and is the von Mangoldt function, the sum was studied by Vinogradov [1]. For and ( is the Möbius function), the sums were studied by Iwaniec, Luo, and Sarnak, and they showed these sums are intimately related to -functions of . If is a holomorphic cusp form of even weight on the upper half plane, they also proved that a good upper bound of implies a quasi-Riemann hypothesis for [2]. In addition, studying the behavior of the Fourier coefficients of automorphic forms has great significance in modern number theory. Analytic number theorists always estimate the mean value or the twisted sums such as mentioned above to obtain some information about the Fourier coefficients (for examples, see [3–10]).

If is a variable and are the Fourier coefficients of automorphic forms, these sums were studied by Ren and Ye [7] and Sun and Wu [11]. They proved that the resonance phenomenon occurs only when and is close to . Let denote the number of representations of a positive integer as a sum of two squares, i.e., , where and are integers. Sun and Wu [11] also studied the case that and obtained the resonance phenomenon. Yan [12] studied the nonlinear exponential sum twisting the Fourier coefficients of Maass forms over the arithmetic progress and obtained an asymptotic formula for the sumwhere is a Maass cusp form for and is the n-th Fourier coefficient of . These analogues to the arithmetic progression are the main motivation of this paper.

In this paper, we study the nonlinear exponential sumwhere , . Here, is a large parameter. are integers, and . We consider the case that tends to infinity as and obtain analogues of the result of Sun and Wu [11].

The principal aim of this paper is to prove the following result.

Theorem 1. Let , , and . Let and . Let .(i)If , then we have(ii)If and , then we have(iii)If and , then for or , we have For , we have where  and or 0 according to if there exists a positive integer for satisfying  or not.(iv)In particular, if with , then we haveTo prove Theorem 1, we shall follow the steps in [7, 11, 12] first. Then, we will use a new Voronoi-type summation formula generalized by Hu et al. [13] to get the asymptotic formula, and this is the key to success. Thus, we can get the Kloosterman sum, use Weil’s bound to get the saving in the -aspect, and then obtain a similar main term as that in [12].

2. Some Lemmas

To prove Theorem 1, we need to quote some lemmas. First, we consider the Kloosterman sum, which is defined aswhere denotes the inverse of modulo . The famous Weil’s bound of the Kloosterman sum iswhere denotes the divisor function.

Let denote the standard J-Bessel function. Let denote the number of representations of by the quadratic form , namely,

If , we denote . Let be a symmetric positive definite integral matrix associated to , and let denote the discriminant of . Let , , , and be a positive definite integral quadratic form, which is defined in terms of the Smith normal form of (see [14]), and the adjoint form of .

Then, we have the following Voronoi summation formula [13].

Lemma 1. Let and be a smooth compact function. We havewhereand is the Mellin transform of , which is given by

Remark 1. In our situation, , but we still want to compute the dependence of in our proof. If one can obtain the asymptotic formula for general , then our result can be applied directly to get the analogues for .
For asymptotic expansions of the Bessel functions, we quote the following lemma.

Lemma 2. For large, we haveFor the mean value of , we have the classical result [16].

Lemma 3. We also need the following result [17].

Lemma 4. Let and be real functions in with being monotonic. Suppose that .(a)If or , then(b)If or , then

3. Proof of Theorem 1

In this section, we will finish the proof of Theorem 1. By the formula of the Ramanujan sumwe getwhere means the summation is restricted by .

Let , and let be a function supported on [1, 2], which is identically 1 on and satisfies for . Using the bound in Lemma 3, we getwhere

Applying Lemma 1 with and , we havewhere

For the first term, changing variables and applying Lemma 4(a), we get

By [16], we have . Thus, the contribution of the first term to (22) is

Next, we turn to estimate the contribution from the term involving . Using Lemma 2, we have

Taking , we obtain

Putting this in , we get

Changing variable , we obtainwherewith

The O-term contributes

The integral defined in (36) was studied by Ren and Ye [7], Sun and Wu [11], and Yan [12]. Here, we follow their steps and choose the parameters with a few differences to get the -aspect saving.

Apply the method given in [7], we obtain

We will take here. Then, implies , where

Thus, for , one obtains by (34) and (38) thatwhere

Thus, the contrition from with is

Let

Applying (41) withwe get

By the bound , we obtain

Similarly, using the bound , we get

Finally, we obtain the contribution of (45) as

If , then , by (25), (29), (37), and (48), and choosingwe get

Next, we will handle the situation that . By the trivial estimate and putting it into (22), we get the second term in the right-hand side of (34) which contributes

Applying (38) to bound the terms with , we havewhere

Since the contribution from the -term is absorbed by (51), we only need to estimate

We distinguish two cases according to or not.

If , we have

By integration by parts and [17], we have

Hence, (55) is

If , letand write

Then,

Moreover, for , we have

Then, we obtain

Hence, in this situation, (55) contributes

It remains to estimatewhere

Note that

If , then , and we have (65) which vanishes. And if , then by trivial estimate, (65) contributes

If , then there is at most one integer satisfyingfor every , say . Hence, (55) becomeswhereand or 0 according to if there exists such or not. Recall

When and and applying (64) and (68), we derive that (55) equals to

This proves Theorem 1(iii).

In particular, if with , then for , , we have

Thus, for in this situation.

Hence, (73) becomes

This proves Theorem 1(iv).

Data Availability

The data used to support the findings of this study are included within the article.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

Xiaofei Yan is supported by National Natural Science Foundation of China (Grant No. 11801327) and Natural Science Foundation of Shandong Province (Grant No. ZR201709280100).