Character sums over generalized Lehmer numbers

Abstract

Let $q>2$ be an integer, $n\geqslant2$ be a fixed integer with $(n,q)=1$, ψ be a non-principal Dirichlet character modq. An upper bound estimate for character sums of the form

$$\sum_{a\in\textit{C}(1,q)}\psi(a) $$

is given, where $\mathcal{C}(1,q)=\{a \mid 1\leqslant a\leqslant q-1, a\overline{a}\equiv1 (\bmod q), n\nmid(a+\overline{a})\}$.

1 Introduction

Let q be an odd integer, c be a fixed positive integer with $(c,q)=1$. For each integer a with $1\leqslant a\leqslant q-1$ and $(a,q)=1$, it is clear that there exists one and only one integer b with $1\leqslant b\leqslant q-1$ such that $ab\equiv c (\bmod q)$. If a and b are of opposite parity, then a is called a Lehmer number. Let $\mathcal{A}(c,q)$ denote the set of all Lehmer numbers, and $r(c,q)$ the number of $\mathcal{A}(c,q)$. Lehmer [1] posed the problem of finding $r(1,q)$.

Before proceeding we need to recall that the notations $U = O(V)$ and $U \ll V$ are equivalent to $\vert U\vert \le c V$ for some constant $c>0$. We write $\ll_{\rho}$ and $O_{\rho}$ to indicate that this constant may depend on the parameter ρ. $\sum^{'}$ means summing over reduced residue classes, a̅ denotes the multiplicative inverse of a modulo q and for a real x we denote $e(x)=e^{2\pi i x}$, $\{x\}$ the fractal part of x, and $\langle x\rangle=\min\{\{x\},1-\{x\}\}$.

In 1993, Zhang [2] proved that

$$\begin{aligned}& r \bigl(1,p^{\alpha} \bigr)=\frac{\phi(p^{\alpha})}{2}+O \bigl(p^{\alpha/2}\ln ^{3} \bigl(p^{\alpha} \bigr) \bigr), \\ & r(1,pl)=\frac{\phi(pl)}{2}+O \bigl((pl)^{1/2}\ln^{2}(pl) \bigr), \end{aligned}$$

where p, l are two distinct odd primes, α is a positive integer, and $\phi(q)$ is the Euler function. For arbitrary odd integer $q\ge3$, he [3] soon obtained

$$r(1,q)=\frac{\phi(q)}{2}+O \bigl(q^{1/2}d^{2}(q) \ln^{2}q \bigr), $$

where $d(q)$ is the classical divisor function.

Later, Lu and Yi [4] generalized this problem to incomplete intervals. In fact, let $q\ge3$ be an integer, $n\geqslant2$ and c be two fixed integers with $(n,q)=(c,q)=1$, $0<\delta_{1},\delta _{2}\leqslant1$, they defined

$$r_{n}(\delta_{1},\delta_{2},c;q)=\mathop{ \mathop{\mathop{{\sum }'}_{a\leqslant\delta_{1}q}\ \mathop{{\sum }'}_{b\leqslant\delta_{2}q}}_{ab\equiv c (\bmod q)}}_{n\nmid(a+b)}1, $$

and got an asymptotic formula as follows:

$$r_{n}(\delta_{1},\delta_{2},c;q)= \biggl(1- \frac{1}{n} \biggr)\delta _{1}\delta_{2}\phi(q) +O_{n} \bigl(q^{1/2}d^{6}(q)\log^{2}q \bigr). $$

Recently, interesting connections between Lehmer numbers and character sums were investigated by some scholars. For example, for an odd prime p, and a fixed prime w less than p, let

$$\mathcal{B}(w,p)= \bigl\{ a\mid 1\leqslant a\leqslant p-1, a\overline{a}\equiv1 ( \bmod p), a\equiv\overline{a} (\bmod w) \bigr\} . $$

Then, for any non-principal Dirichlet character $\chi \bmod w$, Ma, Zhang and Zhang [5] got an upper bound estimate of character sums over $\mathcal{B}(w,p)$ as

$$\mathop{\mathop{\sum}_{a=1}}_{a\in\mathcal{B}(w,p)}^{p-1}\chi (a)\ll_{w} p^{1/2+\epsilon}. $$

At almost the same time, Han and Zhang [6] obtained an upper bound estimate of the character sums over Lehmer numbers as

$$\begin{aligned} \sum_{a\in\mathcal{A}(1,p)}\chi (a)= \mathop{\mathop{ \sum }_{a=1}}_{2 \nmid(a+\overline {a})}^{p-1}\chi(a)\ll p^{1/2} \ln^{2}p, \end{aligned}$$

(1.1)

where χ is an arbitrary non-principal character modulo an odd prime p.

The results of character sums over other special numbers or polynomials can also be found in [7] and [8]. For more properties of character sums and their various applications, see [9, 10] and the references therein.

It seems that (1.1) cannot be extended to arbitrary integer q by their methods in [6]. However, relying on the methods in [4], we can overcome the obstacles.

Let $q\ge3$ be an integer, $n\geqslant2$ be a fixed integer with $(n,q)=1$, ψ be a non-principal Dirichlet character modulo q. If $n\nmid (a+\overline{a})$, then a is called a generalized Lehmer number. Denote the set of all generalized Lehmer numbers by

$$\mathcal{C}(1,q)= \bigl\{ a \mid 1\leqslant a\leqslant q-1, a\overline{a}\equiv 1 (\bmod q), n\nmid(a+\overline{a}) \bigr\} . $$

Following the same technique as in [4], we obtain the following.

Theorem

Let $q\ge3$ be an integer, $n\geqslant2$ be a fixed integer with $(n,q)=1$, ψ be a non-principal Dirichlet character $\bmod\ q$. Then we have the upper bound estimate

$$\sum_{a\in\mathcal{C}(1,q)}\psi(a)=\mathop{\mathop {{\sum }'}_{a=1}}_{n\nmid(a+\overline{a})}^{q}\psi(a) \ll_{n} q^{1/2}d^{5}(q)\log^{2}q. $$

Let $q\ge3$ be an odd integer, $n=2$ in the theorem, we may immediately obtain the following.

Corollary 1

Let ψ be a non-principal Dirichlet character modulo q. Then we have

$$\sum_{a\in\mathcal{A}(1, q)}\psi(a)=\mathop{\mathop {{\sum }'}_{a=1}}_{2\nmid(a+\overline{a})}^{q}\psi(a)\ll q^{1/2}d^{5}(q)\log^{2}q. $$

Let q be an odd prime p, $n=2$ in Corollary 1, then (1.1) can be deduced directly as follows.

Corollary 2

Let ψ be a non-principal Dirichlet character modulo p. Then we have

$$\sum_{a\in\mathcal{A}(1, p)}\psi(a)\ll p^{1/2} \log^{2}p. $$

2 Some lemmas

To prove the theorem, we need the following several lemmas. First we need an upper bound estimate of the general Kloosterman sum $S(m,n,\chi;q)$ as follows.

Lemma 1

Let q be a positive integer and χ a Dirichlet character $\bmod\ q$. Then for any integers m and n, we have

$$S(m,n,\chi;q)\ll q^{1/2}(m,n,q)^{1/2}d(q), $$

where $S(m,n,\chi;q)$ is defined by

$$S(m,n,\chi;q)=\sum_{a \bmod q}\chi(a)e \biggl( \frac {ma+n\overline{a}}{q} \biggr). $$

Proof

See Lemma 1 of [7]. □

Lemma 2

Let q be a positive integer, $\chi_{0}$ be the principal Dirichlet character $\bmod\ q$, ψ be a non-principal character modq, $r_{1}$, $r_{2}$ be integers with $1\leqslant r_{1}, r_{2}\leqslant q-1$. Then we have

$$\bigl\vert G(r_{1},\psi)G(r_{2},\chi_{0}) \bigr\vert \leqslant q^{1/2}(r_{1},q) (r_{2},q). $$

Proof

By Lemma 2 of Chapter 1.2 in [11], we have

$$G(r_{2},\chi_{0})=\mu \biggl(\frac{q}{(r_{2},q)} \biggr) \phi(q)\phi ^{-1} \biggl(\frac{q}{(r_{2},q)} \biggr)\leqslant (r_{2},q), $$

where we have used the fact $\phi(q)/\phi(t)\leqslant q/t$ if $t \mid q$.

Note that ψ is a non-principal character $\bmod\ q$, we only need to consider the following cases.

If $(r_{1},q)=1$, we have

$$\bigl\vert G(r_{1},\psi) \bigr\vert = \bigl\vert \overline{ \psi}(r_{1})G(1,\psi ) \bigr\vert = \bigl\vert G(1,\psi) \bigr\vert =q^{1/2}. $$

If $(r_{1},q)>1$, and ψ is a primitive character $\bmod\ q$, we have

$$\bigl\vert G(r_{1},\psi) \bigr\vert = \bigl\vert \overline{ \psi}(r_{1})G(1,\psi ) \bigr\vert \le q^{1/2}. $$

If $(r_{1},q)>1$, and ψ is a non-primitive character $\bmod\ q$, then Lemma 5 of Chapter 1.2 in [11] indicates that there exists one and only one $q^{\ast}$ such that $q^{\ast} \mid q$, with $\chi^{\ast}$ the primitive character $\bmod\ q^{\ast}$ corresponding χ. Thus

$$\begin{aligned} \bigl\vert G(r_{1},\psi) \bigr\vert \leqslant& \biggl\vert \overline{\chi }^{\ast} \biggl(\frac{r_{1}}{(r_{1},q)} \biggr) \chi^{\ast} \biggl(\frac{q}{q^{\ast}(r_{1},q)} \biggr)\mu \biggl(\frac{q}{q^{\ast}(r_{1},q)} \biggr) \phi(q)\phi^{-1} \biggl(\frac{q}{(r_{1},q)} \biggr)\tau \bigl( \chi^{\ast } \bigr) \biggr\vert \\ \leqslant &q^{1/2}(r_{1},q). \end{aligned}$$

Combining the above, we have

$$\bigl\vert G(r_{1},\psi)G(r_{2},\chi_{0}) \bigr\vert \leqslant q^{1/2}(r_{1},q) (r_{2},q). $$

 □

Lemma 3

Let $q\ge3$ be an integer, χ, ψ be Dirichlet characters modq such that $\psi\neq\chi_{0}$ and $\psi\overline{\psi}=\chi_{0}$. Then we have the estimate

$$\mathop{\mathop{\sum_{\chi\bmod q}}_{\chi\neq\chi_{0}}}_{\chi \neq\overline{\psi}} G(r_{1},\chi\psi)G(r_{2},\chi)\ll\phi (q)q^{1/2}(r_{1},q)^{1/2}(r_{2},q)^{1/2}d(q). $$

Proof

Combining Lemmas 1 and 2, we have

$$\begin{aligned}& \mathop{\mathop{\sum_{\chi\bmod q}}_{\chi\neq \chi_{0}}}_{\chi\neq\overline{\psi}} G(r_{1},\chi\psi)G(r_{2},\chi) \\& \quad =\sum_{\chi\bmod q}G(r_{1},\chi \psi)G(r_{2},\chi) -G(r_{1},\psi)G(r_{2}, \chi_{0})-G(r_{1},\chi_{0})G(r_{2}, \overline {\psi}) \\& \quad =\sum_{\chi\bmod q}\sum_{a=1}^{q} \chi\psi (a)e \biggl(\frac{ar_{1}}{q} \biggr) \sum_{b=1}^{q} \chi(b)e \biggl( \frac{br_{2}}{q} \biggr) \\& \quad\quad{} -G(r_{1},\psi)G(r_{2},\chi_{0})-G(r_{1}, \chi _{0})G(r_{2},\overline{\psi}) \\& \quad =\phi(q) \mathop{{\sum}'}_{a=1}^{q} \psi(a) \mathop{\mathop{{\sum}'}_{b=1}}_{ab\equiv1 (\bmod q)}^{q}e \biggl(\frac{ar_{1}+br_{2}}{q} \biggr) \\& \quad =\phi(q)S(r_{1},r_{2},\psi;q) -G(r_{1}, \psi)G(r_{2},\chi _{0})-G(r_{1}, \chi_{0})G(r_{2},\overline{\psi}) \\& \quad \ll\phi (q)q^{1/2}(r_{1},r_{2},q)^{1/2}d(q)+q^{1/2}(r_{1},q) (r_{2},q) \\& \quad \ll\phi(q)q^{1/2}(r_{1},q)^{1/2}(r_{2},q)^{1/2}d(q). \end{aligned}$$

 □

Lemma 4

Let $0<\rho\leqslant\frac{1}{2}$, $x_{0},x_{1},\ldots, x_{k}$ be a sequence of real numbers such that

$$\langle x_{k}-x_{k'}\rangle \geqslant\rho,\quad\quad x_{k}\neq x_{k'}, $$

and $\langle x_{0}\rangle =\min \{\langle x_{1}\rangle , \ldots, \langle x_{k}\rangle \}$. Then we have

$$\sum_{k=1}^{K}\frac{1}{\langle x_{k}\rangle }\ll \rho^{-1}\log(K+1). $$

Proof

See Lemma 2 of Chapter 5.1 in [11]. □

Lemma 5

Let $q\ge3$ be an integer, ψ be a character $\bmod \ q$, $n\geqslant2$ be a fixed integer with $(n,q)=1$, l be an integer with $1\leqslant l\leqslant n$. Then we have

$$\mathop{{\sum}'}_{a=1}^{q}\mathop{{ \sum}'}_{b=1}^{q}\psi(a) e \biggl( \frac{(a+b)l}{n} \biggr)\ll q^{1/2}\phi(q)d^{2}(q)\log q. $$

Proof

The relations

$$1\leqslant l\leqslant n, \quad\quad 1\leqslant r\leqslant q-1,\quad\quad (n,q)=1 $$

imply that

$$\frac{l}{n}-\frac{r}{q}\neq0. $$

And also

$$\psi(a)=\frac{1}{q}\sum_{r=1}^{q}G(r, \psi)e \biggl(-\frac{ar}{q} \biggr)= \frac{1}{q}\sum _{r=1}^{q-1}G(r,\psi)e \biggl(-\frac{ar}{q} \biggr). $$

Thus

$$\begin{aligned}& \mathop{{\sum}'}_{a=1}^{q}\mathop{{ \sum }'}_{b=1}^{q}\psi(a) e \biggl( \frac{(a+b)l}{n} \biggr) \\& \quad =\sum_{a=1}^{q}\psi(a)e \biggl( \frac{al}{n} \biggr) \mathop{{\sum}'}_{b=1}^{q}e \biggl(\frac{bl}{n} \biggr) \\& \quad =\sum_{a=1}^{q}\frac{1}{q} \sum _{r=1}^{q-1}G(r,\psi)e \biggl(- \frac{ar}{q} \biggr)e \biggl(\frac{al}{n} \biggr) \mathop{{\sum }'}_{b=1}^{q}e \biggl( \frac{bl}{n} \biggr) \\& \quad =\frac{1}{q}\sum_{r=1}^{q-1}G(r, \psi)\mathop{{\sum}'}_{b=1}^{q}e \biggl( \frac{bl}{n} \biggr) \sum_{a=1}^{q}e \biggl( \biggl(\frac{l}{n}-\frac{r}{q} \biggr)a \biggr) \\& \quad =\frac{1}{q}\mathop{{\sum}'}_{b=1}^{q}e \biggl(\frac{bl}{n} \biggr) \Biggl(\sum_{r=1}^{q-1} G(r,\psi)\frac{f(l,r,n,q)}{e (\frac{r}{q}-\frac{l}{n} )-1} \Biggr), \end{aligned}$$

where $f(l,r,n,q)=1-e ( (\frac{l}{n}-\frac{r}{q} )q )$.

Apply the upper bound

$$\bigl\vert G(r,\psi) \bigr\vert \leqslant q^{1/2}(r,q), $$

we have

$$\begin{aligned} \sum_{r=1}^{q-1} G(r,\psi) \frac{f(l,r,n,q)}{e (\frac{r}{q}-\frac{l}{n} )-1} \ll& q^{1/2}\sum_{r=1}^{q-1} \frac{(r,q)}{\vert e (\frac{r}{q}-\frac{l}{n} )-1\vert } \\ \ll& q^{1/2}\sum_{r=1}^{q-1} \frac{(r,q)}{\vert \sin\pi (\frac{r}{q}-\frac{l}{n} )\vert } \ll q^{1/2}\sum_{r=1}^{q-1} \frac{(r,q)}{\langle \frac {r}{q}-\frac{l}{n}\rangle } \\ =&q^{1/2}\mathop{\mathop{\sum}_{d\mid q}}_{d< q} \mathop{\mathop{\sum }_{r\leqslant q-1}}_{(r,q)=d} \frac{d}{\langle \frac{r}{q}-\frac{l}{n}\rangle }=q^{1/2}\mathop {\mathop{\sum}_{d\mid q}}_{d< q}d \mathop{\mathop{\sum}_{m\leqslant\frac{q-1}{d}}}_{(m,q)=1}\frac {1}{\langle \frac{md}{q}-\frac{l}{n}\rangle } \\ =&q^{1/2}\mathop{\mathop{\sum}_{d\mid q}}_{d< q}d \sum_{k\mid q}\mu(k)\sum_{m\leqslant\frac{q-1}{kd}} \frac{1}{\langle \frac{mkd}{q}-\frac{l}{n}\rangle }. \end{aligned}$$

Now write $\frac{k}{q/d}=\frac{h_{0}}{q_{0}}$, where $q_{0}\geqslant 1$, $(h_{0},q_{0})=1$, we have $\frac{q}{kd}=\frac{q_{0}}{h_{0}}\leqslant q_{0}\leqslant \frac{q}{d}$. Then Lemma 4 implies

$$\biggl\langle \frac{m_{i}kd}{q}-\frac{m_{j}kd}{q} \biggr\rangle = \biggl\langle \frac {(m_{i}-m_{j})h_{0}}{ q_{0}} \biggr\rangle \geqslant\frac{1}{q_{0}} \quad \text{if } i\neq j, 1\leqslant i, j\leqslant\frac{q-1}{kd}. $$

So we get

$$\begin{aligned} \sum_{r=1}^{q-1} G(r,\psi) \frac{f(l,r,n,q)}{e (\frac{r}{q}-\frac{l}{n} )-1} \ll &q^{1/2}\mathop{\mathop{\sum}_{d\mid q}}_{d< q}d \sum_{k\mid q}q_{0}\log \biggl( \frac{q-1}{kd}+1 \biggr) \\ \ll& q^{1/2}\mathop{\mathop{\sum}_{d\mid q}}_{d< q}d \sum_{k\mid q}\frac {q}{d}\log q \ll q^{3/2}d^{2}(q)\log q. \end{aligned}$$

(2.1)

Thus

$$\mathop{{\sum}'}_{a=1}^{q}\mathop{{ \sum}'}_{b=1}^{q}\chi_{1}(a) e \biggl(\frac{(a+b)l}{n} \biggr) \ll q^{1/2}\phi(q)d^{2}(q) \log q. $$

 □

3 Proof of the theorem

In this section, we shall complete the proof of the theorem.

Proof of the theorem

From the orthogonality relation for Dirichlet characters $\bmod\ q$ and the trigonometric sum identity, we can get

$$\begin{aligned} \sum_{a\in\mathcal{C}(1,q)}\psi (a) =&\sum _{a=1}^{q}\psi(a)- \mathop{\mathop{\mathop{{\sum }}_{a=1}}_{n\mid(a+\overline {a})}}^{q}\psi(a) \\ =&\sum_{a=1}^{q}\psi(a)- \mathop{ \mathop{\mathop{{\sum}'}_{a=1}^{q} \mathop{{\sum }'}_{b=1}^{q}}_{n\mid(a+b)}}_{ab\equiv1(\bmod q)} \psi(a) \\ =&-\frac{1}{\phi(q)}\sum_{\chi\bmod q}\mathop{\mathop{{ \sum }'}_{a=1}^{q}\mathop{{\sum }'}_{b=1}^{q}}_{n\mid(a+b)}\psi(a)\chi (ab) \\ =&-\frac{1}{n\phi(q)}\sum_{\chi\bmod q}\mathop{{\sum }'}_{a=1}^{q}\mathop{{\sum }'}_{b=1}^{q} \psi(a)\chi(ab)\sum _{l=1}^{n}e \biggl(\frac{(a+b)l}{n} \biggr) \\ =&-\frac{1}{n\phi(q)}\mathop{\mathop{\sum_{\chi\bmod q}}_{\chi \neq\chi_{0}}}_{\chi\neq\overline{\psi}} \mathop{{\sum }'}_{a=1}^{q}\mathop{{ \sum }'}_{b=1}^{q} \psi(a)\chi(ab)\sum _{l=1}^{n}e \biggl(\frac{(a+b)l}{n} \biggr) \\ &{}-\frac{1}{n\phi(q)}\sum_{l=1}^{n} \mathop{{\sum }'}_{a=1}^{q}\mathop{{\sum }'}_{b=1}^{q} \psi(a)e \biggl( \frac{(a+b)l}{n} \biggr) \\ &{} -\frac{1}{n\phi(q)}\sum_{l=1}^{n} \mathop{{\sum}'}_{a=1}^{q}\mathop {{ \sum }'}_{b=1}^{q} \overline{\psi}(b)e \biggl(\frac{(a+b)l}{n} \biggr) \\ :=&-E_{1}-E_{2}-E_{3}. \end{aligned}$$

First of all, we shall estimate $E_{1}$. Making use of Lemma 3, we get

$$\begin{aligned} E_{1} =&\frac{1}{n\phi(q)}\mathop{\mathop{\sum _{\chi\bmod q}}_{\chi\neq\chi_{0}}}_{\chi\neq\overline{\psi}}\mathop{{\sum }'}_{a=1}^{q}\mathop{{\sum}'}_{b=1}^{q} \psi(a)\chi(ab)\sum_{l=1}^{n}e \biggl( \frac{(a+b)l}{n} \biggr) \\ =&\frac{1}{n\phi(q)}\mathop{\mathop{\sum_{\chi\bmod q}}_{\chi\neq\chi_{0}}}_{\chi\neq\overline{\psi}} \sum_{l=1}^{n}\sum _{a=1}^{q}\chi\psi(a)e \biggl(\frac {al}{n} \biggr) \sum_{b=1}^{q}\chi(b)e \biggl( \frac{bl}{n} \biggr) \\ =&\frac{1}{n\phi(q)}\mathop{\mathop{\sum_{\chi\bmod q}}_{\chi\neq\chi_{0}}}_{\chi\neq\overline{\psi}} \sum_{l=1}^{n}\sum _{a=1}^{q}\frac{1}{q}\sum _{r_{1}=1}^{q-1}G(r_{1},\chi\psi) e \biggl(- \frac{ar_{1}}{q} \biggr)e \biggl(\frac{al}{n} \biggr) \\ &{} \times\sum_{b=1}^{q} \frac{1}{q} \sum_{r_{2}=1}^{q-1}G(r_{2}, \chi) e \biggl(-\frac{br_{2}}{q} \biggr)e \biggl(\frac{bl}{n} \biggr) \\ =&\frac{1}{n\phi(q)q^{2}}\mathop{\mathop{\sum_{\chi \bmod q}}_{\chi\neq\chi_{0}}}_{\chi\neq\overline{\psi}} \sum_{l=1}^{n}\sum _{r_{1}=1}^{q-1}G(r_{1},\chi\psi) \sum _{r_{2}=1}^{q-1}G(r_{2},\chi) \\ & {} \times\sum_{a=1}^{q}e \biggl( \biggl( \frac{l}{n}- \frac{r_{1}}{q} \biggr)a \biggr)\sum _{b=1}^{q}e \biggl( \biggl(\frac{l}{n}- \frac{r_{2}}{q} \biggr)b \biggr) \\ =&\frac{1}{n\phi(q)q^{2}}\sum_{l=1}^{n} \sum _{r_{1}=1}^{q-1}\sum_{r_{2}=1}^{q-1} \frac {f_{1}(l,r_{1},n,q)f_{2}(l,r_{2},n,q)}{ (e (\frac{l}{n}-\frac{r_{1}}{q} )-1 ) (e (\frac{l}{n}-\frac{r_{2}}{q} )-1 )} \\ &{} \times\mathop{\mathop{\sum_{\chi\bmod q}}_{\chi\neq\chi_{0}}}_{\chi\neq\overline{\psi}}G(r_{1}, \chi\psi)G(r_{2},\chi) \\ \ll&\frac{1}{\phi(q)q^{2}}\sum_{l=1}^{n} \sum_{r_{1}=1}^{q-1}\sum _{r_{2}=1}^{q-1}\frac {\phi(q)q^{1/2}(r_{1},q)^{1/2}(r_{2},q)^{1/2}d(q)}{ \vert e (\frac{l}{n}-\frac{r_{1}}{q} )-1\vert \vert e (\frac{l}{n}-\frac{r_{2}}{q} )-1\vert } \\ =&\frac{d(q)}{q^{3/2}}\sum_{l=1}^{n} \sum _{r_{1}=1}^{q-1}\sum_{r_{2}=1}^{q-1} \frac {(r_{1},q)^{1/2}(r_{2},q)^{1/2}}{ \vert e (\frac{l}{n}-\frac{r_{1}}{q} )-1\vert \vert e (\frac{l}{n}-\frac{r_{2}}{q} )-1\vert } \\ \ll&\frac{d(q)}{q^{3/2}}\sum_{l=1}^{n} \Biggl(\sum_{r=1}^{q-1}\frac{(r,q)^{1/2}}{ \vert e (\frac{l}{n}-\frac{r}{q} )-1\vert } \Biggr)^{2}. \end{aligned}$$

Similar to (2.1), we have

$$\sum_{r=1}^{q-1}\frac{(r,q)^{1/2}}{ \vert e (\frac{l}{n}-\frac{r}{q} )-1\vert } \ll \mathop{\mathop{\sum}_{d\mid q}}_{d< q}d^{1/2} \sum _{k\mid q}\frac {q}{d}\log q =q\log q\mathop{ \mathop{\sum}_{d\mid q}}_{d< q}d^{-1/2}\sum _{k\mid q}1 \ll qd^{2}(q)\log q. $$

Then

$$\begin{aligned} E_{1}\ll\frac{d(q)}{q^{3/2}}q^{2}d^{4}(q) \log ^{2}q=q^{1/2}d^{5}(q)\log^{2}q. \end{aligned}$$

(3.1)

Second, we estimate $E_{2}$. By Lemma 5, we have

$$\begin{aligned} E_{2}\ll\frac{1}{\phi(q)}q^{1/2} \phi(q)d^{2}(q)\log q=q^{1/2}d^{2}(q)\log q. \end{aligned}$$

(3.2)

In the same way we can get the estimate

$$\begin{aligned} E_{3}\ll q^{1/2}d^{2}(q)\log q. \end{aligned}$$

(3.3)

Combining (3.1), (3.2), and (3.3), we obtain the result. □