Exponential sums involving the divisor function over arithmetic progressions

1.   Introduction

The divisor functions

are the basic arithmetic functions in number theory, and it generate the Dirichlet series $\zeta^{k}(s)$ which are the simplest $GL_{k} \; L$-functions. Hence the behavior of the divisor functions are very important in the theory of automorphic $L$-funcions. In this article, we will study the sum of the type

where $\phi(x)$ is a $C^{\infty}$-function supported on $[1, 2]$ with derivatives bounded by $\phi^{(j)}(x)\ll 1$ and $k\in \mathbb{Z}^{+}$.

Studying the asymptotic distribution of this type of sums that involving the Fourier coefficients and the nonlinear exponential functions are very classical in analytic number theory. The oscillation behavior of Fourier coefficients of $GL_{2}$ automorphic forms is studied by Ren and Ye ^[13] and they proved an asymptotic formula for the sum

where $\lambda_f(n)$ is the $n$-th Fourier coefficient of a holomorphic cusp form for $GL_{2}$. The analytic properties of $\lambda_f(n)$ were studied by many authors, see ^{[7,8,9,10,19,20,21,22,23]}. For the Maass forms on $GL_{2}$, Sun and Wu ^[16] proved the similar asymptotic formula. Acharya and Singh ^[1] gave the upper bound of the sum

where $\alpha$, $\theta$ are real numbers with $0 < \theta < 1$, and $\nu(n)$ is either $\mu(n)$ or $\Lambda(n)$. If $f(x)$ is a Hecke-Maass cusp form for $SL(3, \mathbb{Z})$, Ren and Ye ^[14] proved an asymptotic formula for the sum

where $A_f(1;n)$ is the $(1, n)$-th Fourier coefficients of $f$. Let $f$ be a full-level cusp form for $GL_{m}(\mathbb{Z})$ with Fourier coefficients $A_{f}\left(n_{1}, \cdots, n_{m-1}\right)$, Ren and Ye ^[15] considered the following exponential sums:

They obtained the asymptotic formulas and upper bounds for these sums.

When the summation is restricted in arithmetic progressions, Yan ^[18] has proved an asymptotic formula for

where $\lambda_f(n)$ is the $n$-th Fourier coefficient of a holomorphic cusp form for $SL(2, \mathbb{Z})$. Ma and Yan ^[12] also focused on the oscillation behavior of the exponential sum twisted by $r(n)$ over the arithmetic progressions, where $r(n)$ denote{s} the number of representations of a positive integer $n$ as a sum of two squares. He ^[3] studied the asymptotic formula for the corresponding $GL_{3}$ exponential sum

The divisor functions are involving the theory of $GL_{k} \; L$-functions. Sun and Zhang ^[17] studied the average behavior of the divisor functions over values of quadratic forms and got its asymptotic formula. And the general divisor problems involving Hecke eigenvalues also have attracted many authors, see ^[4,5]. In this paper, we consider the oscillation behavior of the divisor functions $d_{k}(n)$ in arithmetic progressions when $k = 3$. More precisely, the aim of this paper is to prove the following result.

Theorem 1. Let $k, l, q \in\mathbb{N}$. Then for any $\epsilon > 0$ and $q\ll X^{\frac{1}{3}-\epsilon}$, we have

where $A\left(k, \frac{a}{q}\right)$ and $B\left(k, \frac{a}{q}\right)$ are defined in Lemma 2.1 and $c(\phi) = \int_{0}^{\infty}u\phi(u^{3})du$.

An interesting generalization of this problem is to replace the exponential function $e\left(\frac{3\sqrt[3]{kn}}{q}\right)$ in (1.1) with the $q$-exponential function $e_{q}(z)$ or the degenerate exponential function $e_{\lambda}^{t}$ for their definitions and properties see Chung-Kim-Kwon ^[2] and Kim-Kim ^[11] respectively.

2.   Some preliminary lemmas

To prove our theorem, we need the following lemmas.

Lemma 2.1. Let $f(x)$ be a smooth function of compact support in $(0, \infty)$ and $d_3(n)$ be the number of ways to write $n$ as a product of three factors. Then we have

where

and

Proof. See the Section 5 of Ivić ^[6].

Lemma 2.2. If $A\left(n, \frac{h}{k}\right)$ is defined by (2.2), then for $(h, k) = 1$ we have

Proof. See the Section 8 of Ivić ^[6].

Lemma 2.3. If $U(x)$ and $V(x)$ are defined by (2.4), then for any fixed integer $K\geq1$ and $x\geq x_0 > 0$

with suitable constants $c_j, \cdots, l_j$, and in particular

Proof. See the Lemma 3 of Ivić ^[6].

3.   The proof of Theorem 1

Denote $\alpha = \frac{3\sqrt[3]{k}}{q}$. We consider the sum

Note that

Then

Applying Lemma 2.1 with $f(x) = \phi(x/X)e(\alpha \sqrt[3]{x})$, we have

where

Applying Lemma 2.3 with $K = 3$, we obtain

By Lemma 2.2, the $O$-term in (3.5) contributes to $\mathcal{S}_{1}(a, c)$

By Lemma 2.2, the $O$-term in (3.6) contributes to $\mathcal{S}_{2}(a, c)$

For the integral in (3.5) and (3.6), we make a change of variable $x = Xu^{3}$ to get

where

By (3.4)–(3.7), we have

Let

Note that

for $q < X^{\frac{1}{3}-\epsilon}$. Then

and by integration by parts many times, we show that $I_{j}^{+}\left(\frac{n}{c^{3}}\right)$ is negligible. For $n\geq \frac{2c^{3}\alpha^{3}}{27}$ or $n\leq \frac{c^{3}\alpha^{3}}{100}$, we also have

thus, $I_{j}^{-}\left(\frac{n}{c^{3}}\right)$ is negligible for $n\geq \frac{2c^{3}\alpha^{3}}{27}$ or $n\leq \frac{c^{3}\alpha^{3}}{100}$. Denote

Then we can get

For $j\geq2$, we use the trivial bound $I_{j}^{-}\left(\frac{n}{c^{3}}\right)\ll1$. Then the contribution from $I_{j}^{-}\left(\frac{n}{c^{3}}\right)$, $j = 2, 3$ to $\mathcal{S}_{1}(a, c)$ in (3.9) and $\mathcal{S}_{2}(a, c)$ in (3.10) is at most

Then, by Lemma 2.3, (3.9) and (3.10), we have

Thus, by (3.11) and (3.12), we can get

Note that the first term in (3.13) disappears when $\alpha\leq \frac{3}{\sqrt[3]{2}c}$. For $\alpha > \frac{3}{\sqrt[3]{2}c}$, let $n_{\alpha}\geq1$ be the integer such that

Then for any positive integer $n$, we have

For $n\neq n_{\alpha}$, the right-hand side of above equation

and then

By integrating by parts, we can obtain

Thus, by Lemma 2.2, the contribution of the term $n\neq n_{\alpha}$ to (3.13) is at most

When $n = n_{\alpha}$, we can get

For $|\lambda|\geq\frac{1}{10}$, we have $I_{1}^{-}\left(\frac{n_{\alpha}}{c^{3}}\right)\ll c^{3}\alpha^{2}X^{-\frac{1}{3}}$ by integrating by parts. Hence, the first term in (3.13) is bounded by $c^{\frac{9}{4}+\epsilon}\alpha^{\frac{7}{4}+\epsilon}X^{\frac{1}{3}}$. If $c^{-3}\alpha^{-2}X^{\frac{1}{3}}|\lambda|\gg X^{\epsilon}$, then $I_{1}^{-}\left(\frac{n_{\alpha}}{c^{3}}\right)$ is negligible. This proves that

where

By (3.2), (3.3) and (3.15), we have

Since $\alpha = \frac{3\sqrt[3]{k}}{q}$ with $k\in\mathbb{Z}$, we have $c\neq q$. Then

Thus, $\frac{kc^{3}}{q^{3}} = 0$ for $c\neq q$. Otherwise, we take

Therefore,

where

and

This proves our theorem.

4.   Conclusions

This paper focused on the study of the nonlinear exponential sum twisting the divisor functions $d_{3}(n)$ over the arithmetic progression and established its asymptotic formula. The main techniques used to prove the theorem were the estimation of exponential sum, Voronoi summation formula, and the weighted stationary phase. Moreover, it is also an interesting question to generalize the resonance theory to the divisor functions $d_{k}(n)$ if there is a better Voronoi summation formula for $k > 3$, and we can try to obtain a better bound for the error term respect to $q$ which is a prime number.

Acknowledgments

Zhang, Li and Yan are supported by the National Natural Science Foundation of China [Grant No. 12171286].

Conflict of interest

The authors declare no conflict of interest.