Sums of the higher divisor function of diagonal homogeneous forms in short intervals

1.   Introduction

For any integer $k\ge2$, let

define the $k$-th divisor function and $d(n) = d_2(n)$.

Recently, the sums of divisor function about the quadratic form $n_1^2+n_2^2+\ldots+n_l^2$ with $l\ge 3$ has drawn researchers' attention. For $k = 2$ and $l = 3$, in 2000, Calder$\rm \acute{o}$n and de Velasco ^[1] gave the following asymptotic formula,

Later, Guo and Zhai ^[4] used the classical circle method to solve the case of $k = 2$ and $l = 3$. Furthermore, the error term was eventually upgraded by Zhao ^[14] to $x\log_{}^7x$. For $k = 2$ and $l\ge 3$, Zhang ^[15] proved that

where $c_1$ and $c_2$ are constants. Moreover, Lü and Mu ^[10] considered the nonhomogeneous case, the leading term of

is $x^{1+\frac{1}{k}}(1+\log_{}x).$

In 2016, Sun and Zhang ^[12] began to take up the higher divisor function. For $k = 3$ and $l = 3$, they proved that

where $c_3$, $c_4$ and $c_5$ are constants. Later, Hu and Yang ^[8] considered the cases of $l = 4$. For $k\ge 4$ and $l\ge 3$, Hu and Lü ^[5] investigated that the main term of the sum

is $x^{\frac{l}{2}}\log_{}^{k-1}x$.

Recently, the above results were summarized by Zhou and Ding ^[17], for $r\ge 2$, $k\ge 2$ and $l > 2^{r-1}$,

where $c_6$ is a constant.

Inspired by the above research involving the divisor function, Hu and Yao ^[7] considered the sums of divisor of the ternary quadratic form in short intervals. It is stated that, for $\theta = \frac{1}{2}+2\varepsilon$ and $y = x^\theta$, there holds

where $c_7$ and $c_8$ are constants, and $L_j(x, y)$ $(j = 1, 2)$ satisfies $L_1(x, y) \asymp y^3\log_{}y$, $L_2(x, y) \asymp y^3$.

Later, Hu and Liu ^[6] explored the case of a quaternary quadratic form in short intervals. Moreover, Zhang and Li ^[16] studied the nonhomogenous case in short intervals. They proved that, $y = x^{1-\delta_k+4\varepsilon}$ with $\delta_3 = \frac{2}{15}$, $\delta_k = \frac{1}{k(2^{k-2}+1)}$ for $4\le k\le 7$ and $\delta_k = \frac{1}{k(k^2-k+1)}$ for $k\ge 8$, there holds

where $c_9$ and $c_{10}$ are constants, and $\mathfrak{L}_j(x, y)$ $(j = 1, 2)$ satisfies $\mathfrak{L}_1(x, y)\asymp y^3x^{-2+\frac{1}{k}}\log_{}x$, $\mathfrak{L}_2(x, y)\asymp y^3x^{-2+\frac{1}{k}}$.

In this paper, we will extend the result of Zhou and Ding ^[17] in short intervals. We want to consider the asymptotic formula of the sum

where $y = x^{\theta}$ with $0 < \theta\le 1$.

To give our result, let

For $0\le j\le k-1$, we define

where $a > 0$ is a positive integer. We learn from [3, p.372] that for $(a, q) = 1$, $A_j(q)$ is independent of $a$. The coefficients $c_j(b, q)$ can be written as

for some function $f$.

Meanwhile, the number of terms in $c_j (b, q)$ depends only on $k$. More accurately, the coefficients $c_j(b, q)$ are given explicitly in [2, (2.13)]. Also, from [3, (4.8)] we can get

Our result is as follows.

Theorem 1.1. Let $k\ge2$, $r\ge 3$ and $l > 2^{r-1}$ be integers. We have

where

and $S_r(q, a)$, $A_j(q)$ are defined in (1.1) and (1.2), respectively. Moreover, $\mathfrak{S}(x, y)$ is defined as

satisfying $\mathfrak{S}(x, y)\asymp y^lx^{-l+\frac{l}{r}}\log_{}^jx$.

For this problem, unlike other results, we are interested in how to find a smaller suitable value of $\theta$ to make sure that $S_k(x, y)$ has an asymptotic formula. We will establish Theorem 1.1 by the circle method and employ the estimate of the sum of the divisors over the arithmetic progression and some estimates of exponential sum in short intervals.

Notation: Throughout this paper, $x$ always denotes a sufficiently large positive integer and $y = x^{\theta}$ with $0 < \theta\le 1$, we take $N_1 = x-y$ and $N_2 = x+y$, $f(x)\ll g(x)$ means that $f(x) = O\big(g(x)\big)$, $f(x)\asymp g(x)$ means that $f(x)\ll g(x) \ll f(x)$. As usual, $\vert\vert \alpha \vert\vert$ denotes the distance from $\alpha$ to the nearest integer, $e(x) = e^{2\pi ix}$, $\varepsilon$ always denotes an arbitrary small positive constant, which may not be the same at different occurrences. For $\lambda \in \mathbb{R}$, we set

We define

2.   Lemmas

We list the following lemmas that will be used in subsequent sections.

Lemma 2.1. For any real numbers $\alpha$ and $\tau \ge 1$, and integers a and q satisfying $(a, q) = 1, 1\le q\le \tau$, $\alpha$ can be written as

Proof. See Pan and Pan [11, Lemma 5.19]. □

Lemma 2.2. For any $a, q\in \mathbb{Z}$ with $1\le a\le q$, $(a, q) = 1$.

Proof. See Vaughan [13, Theorem 4.2]. □

Lemma 2.3. Let $r \ge 3$ and $N_1, N_2$ be defined as above. Then we get

Proof. For (2.1), (2.2) and (2.3), see [16, Lemmas 2.5 and 2.9]. □

Lemma 2.4. For r $\ge$ 3, We have

Proof. See [16, Lemma 2.10]. □

Lemma 2.5. Let $g(\alpha)$ be defined in (1.5). Then there holds

Proof. By the definition of $g(\alpha)$, we have

It follows from [9, Theorem 1] that,

where $k = o(\sqrt{\log_{}\log_{}x})$ and $c_{11}, c_{12} > 0$ are constants. Then we have

Noting the fact that $\log_{}(1+u)\le u$ for $u\ge 0$, we can deduce that

Therefore, by similar arguments, it's not hard to derive that

□

Lemma 2.6. For $r\ge 2$ and $f_r(\alpha)$ defined in (1.5), we have

Proof. The proof is similar to that of [13, Lemma 2.5]. We notice that the key of proof of [13, Lemma 2.5] depends on the summation interval. It is showed that the only difference between these two proofs is the length of interval and easily to get $N_2^{\frac{1}{r}}-N_1^{\frac{1}{r}} = (x+y)^{\frac{1}{r}}-(x-y)^{\frac{1}{r}}\asymp yx^{-1+\frac{1}{r}}$. This completes the proof of Lemma 2.6. □

Lemma 2.7. Suppose that $(a, q) = 1$, $q\le Q \le y^{\frac{1}{k}}$. We have

where $\eta = {\frac{k-1}{k+1}}$ for $2\le k \le 3$, $\eta = {\frac{k-1}{k+2}}$ for $k\ge 4$, and $A_j(q)$ is defined as in (1.2).

Proof. When $2\le k\le 3$, by the definition of (1.5), we write

where

According to the divisor problem for arithmetic progressions, we utilize it to isolate the main term and estimate the remainder. For the sake of simplicity, we denote

We learn from [2, Theorem 1] that main term has the form

where $L_j(u)$ is an antiderivative of $\log_{}^j u/ j!$. Thus, the contribution of main term to $g(\alpha)$ can be written as

where $A_j(q)$ is defined as in (1.2), and $I_j(\lambda)$ is defined as in (2.4). After integrating by parts, one has

It follows from [3, (4.4)] that the contribution from error term is bounded by

when $k \ge 4$, the proof is similar, so we omit it.

□

3.   Proof of Theorem 1.1

In order to apply the circle method, we set

By Lemma 2.1, each $\alpha\in{I: = \left[\frac{1}{\tau}, 1+\frac{1}{\tau} \right)}$ can be written as

for two positive integers $a, q$ with $1\le a \le q \le \tau$ and $(a, q) = 1$.

We define the major arcs $\mathfrak{M}$ and $\mathfrak{m}$ as follows:

where ${\mathfrak{M}}(q, a) = \left\{\alpha:\alpha = \frac{a}{q}+\lambda, |\lambda|\le\frac{1}{q\tau} \right\}.$

It is obvious that the major arcs $\mathfrak{M}$ are actually disjoint when $1 < Q < \frac{\tau}{2}$. Noting that $f_r(\alpha)$ and $g(\alpha)$ are defined in (1.5), we obtain

First, we deal with the integrals over the minor arcs.

Lemma 3.1. Suppose $\alpha \in \mathfrak{m}$ with $Q < q\le \tau$.

$\bullet$ For $3\le r \le 7$, we have

$\bullet$ For $r \ge 8$, we have

Proof. The Cauchy's inequality gives us that

We now use the result of ^[16] and get

From that Lemma 2.5, Lemma 2.6 and (3.2), we get result of Lemma 3.1. □

3.1.   The major arcs

By the definition of major arcs, we clearly have

By (2.1), Lemmas 2.2 and 2.4, we can conclude that

This combined with Lemma 2.7 gives that

where

and

Taking these into (3.3), we obtain

where the error terms $\mathscr{U}_i(i = 1, 2)$ satisfy

3.1.1.   Estimate of ${\mathscr{U}}_1$

The estimate of ${\mathscr{U}}_1$ will be separated into two cases.

If $2^{r-1} < l < 2^r$, it can be obtained by Hölder's inequality and Lemma 2.6 that

If $l\ge 2^r$, by Lemma 2.6, we have

3.1.2.   Estimate of ${\mathscr{U}}_2$

By (1.3) and (2.4), we have

3.1.3.   Major arcs for ${\mathscr{U}}(q, \lambda)$

By means of (2.3) and (2.4), we have

First, according to (1.3), (2.1) and Lemma 2.2, the contribution of the O-term to the integrals over the major arcs is

Hence, combining (3.4) and integrating over the major arcs, we obtain

In the integral of above identity, we extend the interval to $[-\frac{1}{2}, \frac{1}{2}]$. By (2.1) and (2.2)

Noting the condition (1.3), (3.1) and Lemma 2.2, the contribution of the above upper bound to the integrals over the major arcs is

From (3.4) – (3.7), we have

where

Therefore, we have established the following identity

By § 3.1.1, § 3.1.2 and (3.8), we get the following lemma.

Lemma 3.2. For $\alpha \in \mathfrak{M}$, let $k\ge 2$, $r\ge 3$, $l > 2^{r-1}$ be integers.

$\bullet$ For $2^{r-1} < l < 2^r$, we have

$\bullet$ For $l\ge2^r$, we have

where $S_r(q, a)$, $A_j(q)$ and $\mathfrak{S}(x, y)$ are defined in (1.1), (1.2) and (1.4), respectively.

4.   Completion of the proof

Finally, combining Lemma 3.1, Lemma 3.2 and noting that $\eta = {\frac{k-1}{k+1}}$ for $2\le k \le 3$ and $\eta = {\frac{k-1}{k+2}}$ for $k\ge 4$ by condition of Lemma 2.7, we can get the following that.

Case 1. For $2^{r-1} < l < 2^r$, $2\le k\le 3$, $3\le r\le 7$,

Case 2. For $2^{r-1} < l < 2^r$, $k\ge 4$, $3\le r \le 7$,

Case 3. For $l\ge2^r$, $2\le k\le 3$, $3\le r\le 7$,

Case 4. For $l\ge2^r$, $k\ge 4$, $3\le r\le 7$,

Case 5. For $2^{r-1} < l < 2^r$, $2\le k\ge 3$, $r\ge 8$,

Case 6. For $2^{r-1} < l < 2^r$, $k\ge 4$, $r \ge 8$,

Case 7. For $l\ge2^r$, $2\le k\le 3$, $r\ge 8$,

Case 8. For $l\ge2^r$, $k\ge 4$, $r\ge 8$,

For Cases 1–4, we take $y = x^{\theta}, \theta = \frac{(1-\frac{1}{r})(\frac{r}{2}+\frac{l-2^{r-1}}{k\cdot2^{r-1}}-\varepsilon)}{\frac{r}{2}+\frac{l-2^{r-1}}{k\cdot2^{r-1}}-\frac{1}{2}-\varepsilon}$ by condition of Theorem 1.1 and replace $Q$ and $\tau$ in (3.1) by

It is easy to see that

In fact, we can check that the argument of Theorem 1.1 is valid if the parameters $Q$, $\tau$ and y satisfy the conditions

and

Therefore,

For Cases 4–8, we take $y = x^{\theta}, \theta = \frac{(1-\frac{1}{r})(\frac{r}{2}+\frac{l-2r(r-1)}{k\cdot2r(r-1)}-\varepsilon)}{\frac{r}{2}+\frac{l-2r(r-1)}{k\cdot2r(r-1)}-\frac{1}{2}-\varepsilon}$ by condition of Theorem 1.1. According to (4.1), we find

Actually, we can also deduce that Theorem 1.1 is available if the parameters $Q$, $\tau$ and y satisfying (4.2) and

Similarly,

This completes the proof of Theorem 1.1.

Use of AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

This work is supported by Natural Science Foundation of China (Grant Nos. 11761048), Natural Science Foundation of Jiangxi Province for Distinguished Young Scholars (Grant Nos. 20212ACB211007) and Natural Science Foundation of Jiangxi Province (Grant Nos. 20224BAB201001). The authors would like to express their thanks to the referee for many useful suggestions and comments on the manuscript.

Conflict of interest

The authors declare there is no conflict of interests.