Resonance between the Representation Function and Exponential Functions over Arithemetic Progression

Here, n ∼ X means X≤ n≤ 2X, and e(z) � e2πiz. When β � 1/2 and an � Λ(n) is the von Mangoldt function, the sum S(X, α) was studied by Vinogradov [1]. For an � Λ(n) and an � μ(n) (μ is the Möbius function), the sums S(X, α) were studied by Iwaniec, Luo, and Sarnak, and they showed these sums are intimately related to L-functions of GL2. If f is a holomorphic cusp form of even weight on the upper half plane, they also proved that a good upper bound of S(X, α) implies a quasi-Riemann hypothesis for L(s, f) [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 S(X, α) mentioned above to obtain some information about the Fourier coefficients (for examples, see [3–10]). If β is a variable and an are the Fourier coefficients of automorphic forms, these sums were studied by Ren and Ye [7] and Sun and Wu [11]. ,ey proved that the resonance phenomenon occurs only when β � 1/2 and |α| is close to 2 � k √ , k ∈ Z. Let r(n) denote the number of representations of a positive integer n as a sum of two squares, i.e.,


Introduction
In analytic number theory, the problems concerning nonlinear exponential twisting arithmetic functions arise naturally in investigating equidistribution theory, zerodistribution of L-functions, and so on. Let a n be some arithmetic number-theoretic function. We usually consider the general nonlinear exponential sum of the form S(X, α) � n∼X a n e αn β , 0 ≠ α ∈ R, 0 ≤ β ≤ 1. (1) Here, n ∼ X means X ≤ n ≤ 2X, and e(z) � e 2πiz . When β � 1/2 and a n � Λ(n) is the von Mangoldt function, the sum S(X, α) was studied by Vinogradov [1]. For a n � Λ(n) and a n � μ(n) (μ is the Möbius function), the sums S(X, α) were studied by Iwaniec, Luo, and Sarnak, and they showed these sums are intimately related to L-functions of GL 2 . If f is a holomorphic cusp form of even weight on the upper half plane, they also proved that a good upper bound of S(X, α) implies a quasi-Riemann hypothesis for L(s, f) [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 S(X, α) mentioned above to obtain some information about the Fourier coefficients (for examples, see [3][4][5][6][7][8][9][10]).
If β is a variable and a n are the Fourier coefficients of automorphic forms, these sums were studied by Ren and Ye [7] and Sun and Wu [11]. ey proved that the resonance phenomenon occurs only when β � 1/2 and |α| is close to 2 � k √ , k ∈ Z + . Let r(n) denote the number of representations of a positive integer n as a sum of two squares, i.e., n � x 2 1 + x 2 2 , where x 1 and x 2 are integers. Sun and Wu [11] also studied the case that a n � r(n) 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 sum n∼X n≡lmodq where g is a Maass cusp form for SL(2, Z) and λ g (n) is the nth Fourier coefficient of g. ese analogues to the arithmetic progression are the main motivation of this paper.
In this paper, we study the nonlinear exponential sum n∼X n≡lmodq where 0 ≠ α ∈ R, 0 < β < 1. Here, X > 1 is a large parameter. 1 ≤ l ≤ q are integers, and (l, q) � 1. We consider the case that q tends to infinity as X ⟶ ∞ and obtain analogues of the result of Sun and Wu [11]. e principal aim of this paper is to prove the following result. Theorem 1. Let X > 1, 0 < β < 1, and 0 ≠ α ∈ R. Let l, q ∈ N and l ≤ q ≤ X 1/2 . Let δ � (q, 4). For where ε a, n c � δ c ε α a 1 2 and δ c � 1 or 0 according to if there exists a positive integer n c for c|q satisfying or not.
To prove Theorem 1, we shall follow the steps in [7,11,12] first. en, 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. us, we can get the Kloosterman sum, use Weil's bound to get the saving in the q-aspect, and then obtain a similar main term as that in [12].

Some Lemmas
To prove eorem 1, we need to quote some lemmas. First, we consider the Kloosterman sum, which is defined as where d denotes the inverse of d modulo c.
e famous Weil's bound of the Kloosterman sum is where d(c) denotes the divisor function. Let J v denote the standard J-Bessel function. Let r(n, Q) denote the number of representations of n by the quadratic form Q, namely, If Q(x) � x 2 1 + x 2 2 + · · · + x 2 l , we denote r(n, Q): � r l (n). Let A be a symmetric positive definite integral matrix associated to Q, and let D � detA denote the discriminant of A. Let G Q (c, a) � xmodc e(aQ(x)/c), δ � (c, D), D � δD c , and Q * be a positive definite integral quadratic form, which is defined in terms of the Smith normal form of Q † (see [14]), and Q † the adjoint form of Q. en, we have the following Voronoi summation formula [13]. where and Φ(s) is the Mellin transform of F(x), which is given by Remark 1. In our situation, D � 4, but we still want to compute the dependence of D in our proof. If one can obtain the asymptotic formula for general r(n, Q), then our result can be applied directly to get the analogues for r(n, Q). For asymptotic expansions of the Bessel functions, we quote the following lemma.

Lemma 2. For z > 0 large, we have
For the mean value of r(n), we have the classical result [16].
We also need the following result [17].

Proof of Theorem 1
In this section, we will finish the proof of eorem where * means the summation is restricted by (a, c) � 1. Let Δ > 1, and let 0 ≤ ϕ(x) ≤ 1 be a C ∞ function supported on [1,2], which is identically 1 on and satisfies ϕ (r) (x) ≪ Δ r for r ≥ 0. Using the bound in Lemma 3, we get where (25) where For the first term, changing variables x � Xt and applying Lemma 4(a), we get By [16], we have G Q (c, a) ≪ c. us, the contribution of the first term to (22) is Next, we turn to estimate the contribution from the term involving G(n). Using Lemma 2, we have Taking z � (4π Putting this in G(y), we get 4 Journal of Mathematics Changing variable x � Xt 2 , we obtain where with e integral P ± (w) 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 q-aspect saving.