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On general partial Gaussian sums

Abstract

Let \(q\geq2\) be a fixed integer, \(A=A(q)\leq q\), \(B=B(q)\leq q\), and \(H=H(q)\leq q\). Define

$$\hbar(A,B,H)= \bigl\{ a\in\mathbb{Z}\mid (a,q)=1, ab\equiv1\pmod{q}, 1\leq a\leq A, 1\leq b\leq B, \vert a-b\vert \leq H \bigr\} . $$

With the aid of the estimates for the general Kloosterman sums and the properties of trigonometric sums, we obtain an upper bound of the general partial Gaussian sums over the number set \(\hbar(A,B,H)\).

Introduction

Let q, N, H, n be integers with \(q\geq2\), \(H>0\), χ be a Dirichlet character mod q and \(e(y)=e^{2\pi i y}\). The study of the following partial Gaussian sums:

$$\sum_{a= N+1}^{N+H}\chi(a)e \biggl( \frac{na}{q} \biggr) $$

is of great importance. By extending his well-known work on character sums, Burgess obtained the following.

Proposition 1

[1]

Let q be a prime and χ be a non-principal Dirichlet character mod q. Then, for any integers N, n, H, r with \(0< H< q\) and \(r\geq2\), we have

$$\sum_{a= N+1}^{N+H}\chi(a)e \biggl( \frac{na}{q} \biggr)\ll H^{1-1/r}q^{1/4(r-1)}\log^{2} q. $$

Proposition 2

[2]

Let \(q\ge2\) be an integer and χ be a primitive character mod q. Then, for any integers N, n, H with \(0< H\), we have

$$ \sum_{a= N+1}^{N+H}\chi(a)e \biggl( \frac{na}{q} \biggr)\ll H^{2/3}q^{1/8+\epsilon}. $$
(1.1)

Proposition 3

[3]

Let \(q=p^{\alpha}\) (\(\alpha>1\)) be a power of the prime \(p>3\) and χ be a non-principal Dirichlet character mod q. Then, for any integers N, n, H with \(0< H\), we have

$$ \sum_{a= N+1}^{N+H}\chi(a)e \biggl( \frac{na}{q} \biggr)\ll H^{3/4}q^{1/12}\log^{3} q. $$
(1.2)

At almost the same time, Liu [4] showed independently the following.

Proposition 4

Let q be a prime power, χ, ψ be a multiplicative and additive character mod q, respectively, with χ non-principal. Then, for any integers N, H with \(0< H\), we have

$$\sum_{a= N+1}^{N+H}\chi(a)\psi(a)\ll H^{3/4}q^{1/12+\epsilon}. $$

Now let \(q\geq2\) be a fixed integer, \(A=A(q)\leq q\), \(B=B(q)\leq q\), and \(H=H(q)\leq q\). Define

$$\hbar(A,B,H)= \bigl\{ a\in\mathbb{Z}\mid (a,q)=1, ab\equiv1\pmod{q}, 1\leq a\leq A, 1\leq b\leq B, \vert a-b\vert \leq H \bigr\} . $$

It is a direct generalization of the set of so-called H-flat numbers modq, which was studied extensively by Xi (see [5] and references therein).

This paper deals with general partial Gaussian sums of the following type:

$$G_{k}(\chi,A,B,H;q)=\sum_{a\in\hbar(A,B,H)}a^{k} \chi(a)e \biggl(\frac{na}{q} \biggr), $$

where \(k\ge0\) is an arbitrary fixed integer. For the sake of periodicity of \(e (\frac{na}{q} )\) we can also restrict n to be \(1\le n\le q\). Then with the aid of the estimates for the general Kloosterman sums and the properties of trigonometric sums, we shall obtain upper bound estimates as follows.

Theorem

Let \(q\geq2\) be an integer and χ a non-principal Dirichlet character mod q. Then

$$G_{k}(\chi,A,B,H;q)\ll A^{k}q^{1/2}d(q) \biggl( \frac{nAB d(q)}{q^{2}}+\frac{B d(q)(\log q)(\log H)}{q}+\log^{3}q \biggr), $$

which is uniformly nontrivial for any positive integer n such that \(n< q^{1/2}\).

Taking n a constant, we can immediately obtain the following.

Corollary 1

Let \(q\geq2\) be an integer and χ a non-principal Dirichlet character mod q. Then we have

$$G_{k}(\chi,A,B,H;q)\ll A^{k}q^{1/2}d(q) \biggl( \frac{AB d(q)}{q^{2}}+\frac{B d(q)(\log q)(\log H)}{q}+\log^{3}q \biggr). $$

Taking \(k=0\), \(B=q\) in Corollary 1, we obtain the following.

Corollary 2

Let \(q\geq2\) be an integer and χ a non-principal Dirichlet character mod q. Then, for any positive integers A, H such that \(A, H\le q\), we have

$$ G_{0}(\chi,A,q,H;q)\ll q^{1/2}d^{2}(q) \log^{3} q. $$
(1.3)

Remark

It is easy to see that (1.3) is stronger than (1.1) for any integer H such that \(q^{9/16+\epsilon}< H\le q\). It is also stronger than (1.2) for any integer H such that \(q^{5/9}d^{8/3}(q)< H\le q\). These results reveal that more cancelations occurred in the number set \(\hbar(A,B,H)\).

Taking \(A=H=q\) in Corollary 2, we obtain the following.

Corollary 3

Let \(q\geq2\) be an integer and χ a non-principal Dirichlet character mod q. Then we have

$$G_{0}(\chi,q,q,q;q)=\sum_{a=1}^{q} \chi(a)e \biggl(\frac {na}{q} \biggr)\ll q^{1/2}d^{2}(q) \log^{3} q. $$

This is a little stronger than the classical result of the complete Gauss sums in the case when \((n,q)>1\).

Some lemmas

To prove the theorem, we need the following lemmas.

Lemma 1

Let q, n, , r be integers with \(q>2\), \(q>n\), and \(\ell\ge0\). Define \(h(r,\ell;n)= \sum^{n}_{a=1}a^{\ell} e (\frac {ra}{q} )\). Then we have

$$h(r,\ell;n) \textstyle\begin{cases} =\frac{n^{\ell+1}}{\ell+1}+O(n^{\ell}),& q\mid r,\\ \ll\frac{n^{\ell}}{ \vert \sin(\pi s/q)\vert } ,& q\nmid r, \end{cases} $$

where \(s=\min(r,q-r)\) with \(1\leq r\leq q-1\).

Proof

See Lemma 3 of [6] or Lemma 2.4 of [7]. □

Lemma 2

Let q be a positive integer. Then we have

$$\bigl\vert K_{\chi}(m,n;q) \bigr\vert \leq q^{1/2}(m,n,q)^{1/2}d(q), $$

where \(K_{\chi}(m,n;q)= \sum_{a\pmod{q}}\chi(a)e (\frac{ma+n\bar{a}}{q} )\) is the general Kloosterman sum, with \(a\bar{a}\equiv1\pmod{q}\) and \((m,n,q)\) the greatest common divisor of m, n, q.

Proof

See Lemma 1 of [5]. □

Proof of the Theorem

Now we come to prove the theorem. Note that

$$G_{k}(\chi,A,B,H;q)=\sum_{t\leq H}\mathop{ \sum_{a\leq A,b\leq B}}_{ a-b\equiv t\pmod{q},ab\equiv1\pmod{q}}a^{k}\chi(a)e \biggl(\frac{na}{q} \biggr). $$

Applying the trigonometric sums identity

$$\sum^{q}_{a=1}e \biggl( \frac{ma}{q} \biggr)= \textstyle\begin{cases} q,& q\mid m,\\ 0 ,& q\nmid m, \end{cases} $$

we obtain

$$\begin{aligned}& G_{k}(\chi,A,B,H;q) \\ & \quad = \sum_{a\in\hbar(A,B,H)}a^{k}\chi(a)e \biggl( \frac{na}{q} \biggr) \\ & \quad = \sum_{t\leq H} \mathop{\mathop{\sum \nolimits'}_{a\leq A,b\leq B}}_{ a-b\equiv t\pmod{q},ab\equiv1\pmod{q}} a^{k}\chi(a)e \biggl(\frac{na}{q} \biggr) \\ & \quad = \frac{1}{q}\sum_{m\leq q}\sum _{t\leq H}e \biggl(-\frac {mt}{q} \biggr) \mathop{\mathop{\sum \nolimits'}_{a\leq A,b\leq B }}_{ ab\equiv 1\pmod{q}} a^{k} \chi(a)e \biggl(\frac{(m+n)a-mb}{q} \biggr) \\ & \quad = \frac{1}{q^{3}}\sum_{m\leq q}\sum _{t\leq H}e \biggl(-\frac {mt}{q} \biggr) \mathop{\mathop{\sum \nolimits'}_{a,b\leq q }}_{ ab\equiv1\pmod{q}} \chi(a)e \biggl( \frac{(m+n)a-mb}{q} \biggr) \\ & \quad\quad{} \times\sum_{c\leq A}c^{k}\sum _{r\leq q}e \biggl(\frac {r(a-c)}{q} \biggr)\sum _{d\leq B}\sum_{s\leq q}e \biggl( \frac {s(b-d)}{q} \biggr) \\ & \quad = \frac{1}{q^{3}}\sum_{r,s\leq q}\sum _{m\leq q}\sum_{t\leq H}e \biggl(- \frac{mt}{q} \biggr) \mathop{\mathop{\sum\nolimits'}_{a,b\leq q }}_{ ab\equiv1\pmod{q}} \chi(a) e \biggl(\frac{(m+r+n)a-(m-s)b}{q} \biggr) \\ & \quad\quad{} \times\sum_{c\leq A}c^{k}e \biggl(- \frac{rc}{q} \biggr)\sum_{d\leq B}e \biggl(- \frac{sd}{q} \biggr) \\ & \quad = \frac{1}{q^{3}}\sum_{r,s\leq q}\sum _{m\leq q}\sum_{t\leq H}e \biggl(- \frac{mt}{q} \biggr)K_{\chi }(m+r+n,s-m;q)h(-r,k;A)h(-s,0;B) \\ & \quad = \frac{1}{q^{3}}\sum_{m\leq q}\sum _{t\leq H}e \biggl(-\frac {mt}{q} \biggr)K_{\chi}(m+n,-m;q)h(-q,k;A)h(-q,0;B) \\ & \quad\quad{} +\frac{1}{q^{3}}\sum_{r\leq q-1}\sum _{m\leq q}\sum_{t\leq H}e \biggl(- \frac{mt}{q} \biggr)K_{\chi }(m+r+n,-m;q)h(-r,k;A)h(-q,0;B) \\ & \quad\quad{} +\frac{1}{q^{3}}\sum_{s\leq q-1}\sum _{m\leq q}\sum_{t\leq H}e \biggl(- \frac{mt}{q} \biggr)K_{\chi}(m+n,s-m;q)h(-q,k;A)h(-s,0;B) \\ & \quad\quad{} +\frac{1}{q^{3}}\sum_{r,s\leq q-1}\sum _{m\leq q}\sum_{t\leq H}e \biggl(- \frac{mt}{q} \biggr)K_{\chi}(m+r+n,s-m;q)h(-r,k;A)h(-s,0;B). \end{aligned}$$

Then from Lemma 1 and Lemma 2, we have

$$\begin{aligned}& \frac{1}{q^{3}}\sum_{m\leq q}\sum _{t\leq H}e \biggl(-\frac {mt}{q} \biggr)K_{\chi}(m+n,-m;q)h(-q,k;A)h(-q,0;B) \\& \quad \ll\frac{1}{q^{3}}\sum_{m\leq q}\min \biggl(H, \biggl\Vert \frac {m}{q} \biggr\Vert ^{-1} \biggr) \bigl\vert K_{\chi}(m+n,-m;q) \bigr\vert \cdot \bigl\vert h(-q,k;A) \bigr\vert \cdot \bigl\vert h(-q,0;B) \bigr\vert \\& \quad \ll HA^{k+1}Bq^{-5/2}d(q)\sum _{m\leq q/H}(m+n,q)^{1/2} \\& \quad\quad{} +A^{k+1}Bq^{-3/2}d(q) \sum_{q/H< m\leq q-1}\frac{(m+n,q)^{1/2}}{m}, \end{aligned}$$

where \(\Vert x\Vert =\min_{a\in Z}\vert x-a\vert \).

Combining the estimates

$$\begin{aligned} \sum_{m\leq q/H}(m+n,q)^{1/2}&=\sum _{d\mid q}d^{1/2}\mathop{\sum _{m\leq q/H }}_{ d\mid (m+n)}1 \\ & =\sum_{d\mid q}d^{1/2}\sum _{m\leq q/(dH)+n/d}1 \\ & \ll H^{-1}qd(q)+nd(q) \end{aligned}$$

and

$$\begin{aligned} \sum_{q/H< m\leq q-1}\frac{(m+n,q)^{1/2}}{m}&= \sum _{d\mid q}d^{1/2}\mathop{\sum _{q/H< m\leq q-1}}_{ d\mid (m+n)}\frac{1}{m} \\ & = \sum_{d\mid q}d^{1/2}\sum _{\frac{q}{Hd}+\frac{n}{d}< m\leq\frac {q-1+n}{d}}\frac{1}{dm-n} \\ &\ll\sum_{d\mid q}d^{-1/2}\sum _{\frac{q}{Hd}+\frac{n}{d}< m\leq\frac {q-1+n}{d}}\frac{1}{m} \biggl(1+\frac{n}{dm} \biggr) \\ &\ll d(q)\log H+nd(q), \end{aligned}$$

we have

$$\begin{aligned}& \frac{1}{q^{3}}\sum_{m\leq q}\sum _{t\leq H}e \biggl(-\frac {mt}{q} \biggr)K_{\chi}(m+n,-m;q)h(-q,k;A)h(-q,0;B) \\& \quad \ll nA^{k+1}Bq^{-3/2}d^{2}(q)+A^{k+1}Bq^{-3/2}d^{2}(q) \log H. \end{aligned}$$
(3.1)

Applying Lemma 1 and Lemma 2 again, we obtain

$$\begin{aligned}& \frac{1}{q^{3}}\sum_{r\leq q-1}\sum _{m\leq q}\sum_{t\leq H}e \biggl(- \frac{mt}{q} \biggr)K_{\chi}(m+r+n,-m;q)h(-r,k;A)h(-q,0;B) \\& \quad \ll\frac{1}{q^{3}}\sum_{r\leq q-1}\sum _{m\leq q}\min \biggl(H, \biggl\Vert \frac{m}{q} \biggr\Vert ^{-1} \biggr) \bigl\vert K_{\chi }(m+r+n,-m;q) \bigr\vert \cdot \bigl\vert h(-r,k;A) \bigr\vert \cdot \bigl\vert h(-q,0;B) \bigr\vert \\& \quad \ll HBq^{-5/2}d(q)\sum_{r\leq q-1}\sum _{m\leq q/H}(m+r+n,-m,q)^{1/2}\cdot \frac{A^{k}}{\vert \sin(\frac{\pi r}{q})\vert } \\& \quad \quad{}+Bq^{-3/2}d(q)\sum_{r\leq q-1}\sum _{q/H< m\leq q-1}\frac {(m+r+n,-m,q)^{1/2}}{m}\cdot\frac{A^{k}}{\vert \sin(\frac{\pi r}{q})\vert } \\& \quad \ll HA^{k}Bq^{-3/2}d(q)\sum _{r\leq q-1} \frac{1}{r}\sum_{m\leq q/H}(m+r+n,-m,q)^{1/2} \\& \quad\quad{} +A^{k}Bq^{-1/2}d(q)\sum _{r\leq q-1} \frac{1}{r}\sum_{q/H< m\leq q-1} \frac{(m+r+n,-m,q)^{1/2}}{m} . \end{aligned}$$

Combining

$$\begin{aligned}& \sum_{r\leq q-1}\frac{1}{r}\sum _{m\leq q/H}(m+r+n,-m,q)^{1/2} \\ & \quad =\sum_{d\mid q}d^{1/2}\mathop{\sum _{r\leq q}}_{ d\mid (r+n)}\frac{1}{r}\mathop{\sum _{m\leq q/H }}_{ d\mid m} 1 \\ & \quad =\sum_{d\mid q}d^{1/2}\sum _{(n+1)/d \le r\leq(q+n-1)/d}\frac {1}{dr-n}\sum_{m\leq q/(Hd)} 1 \\ & \quad \ll q/H\sum_{d\mid q}d^{-3/2}\sum _{(n+1)/d \le r\leq(q+n-1)/d}\frac {1}{r} \biggl(1+\frac{n}{dr} \biggr) \\ & \quad \ll H^{-1}qd(q)\log \biggl(\frac{q+n-1}{n+1} \biggr) \end{aligned}$$

and

$$\begin{aligned}& \sum_{r\leq q-1}\frac{1}{r}\sum _{q/H< m\leq q-1}\frac {(m+r+n,-m,q)^{1/2}}{m} \\ & \quad =\sum_{d\mid q}d^{1/2}\mathop{\sum _{r\leq q-1}}_{ d\mid (r+n)}\frac{1}{r}\mathop{\sum _{q/H< m\leq q-1}}_{ d\mid m}\frac{1}{m} \\ & \quad =\sum_{d\mid q}d^{-1/2}\sum _{(n+1)/d \le r\leq(q+n-1)/d}\frac {1}{dr-n}\sum_{q/(Hd)< m\leq q/d} \frac{1}{m} \\ & \quad \ll(\log H)\sum_{d\mid q}d^{-3/2}\sum _{r\leq q/d}\frac{1}{r(1-\frac {n}{dr})} \\ & \quad \ll d(q) (\log H) \log \biggl(\frac{q+n-1}{n+1} \biggr), \end{aligned}$$

we have

$$\begin{aligned}& \frac{1}{q^{3}}\sum_{r\leq q-1}\sum _{m\leq q}\sum_{t\leq H}e \biggl(- \frac{mt}{q} \biggr)K_{\chi}(m+r+1,-m;q)h(-r,k;A)h(-q,0;B) \\ & \quad \ll A^{k}Bq^{-1/2}d^{2}(q) (\log H) \log \biggl( \frac{q+n-1}{n+1} \biggr). \end{aligned}$$
(3.2)

Similarly, we get the estimate

$$\begin{aligned}& \frac{1}{q^{3}}\sum_{s\leq q}\sum _{m\leq q}\sum_{t\leq H}e \biggl(- \frac{mt}{q} \biggr)K_{\chi}(m+1,s-m;q)h(-q,k;A)h(-s,0;B) \\ & \quad \ll A^{k+1}q^{-3/2}d^{2}(q)\log \biggl( \frac{q-1}{n+1} \biggr). \end{aligned}$$
(3.3)

Noting that

$$\begin{aligned}& \frac{1}{q^{3}}\sum_{r,s\leq q-1}\sum _{m\leq q}\sum_{t\leq H}e \biggl(- \frac{mt}{q} \biggr)K_{\chi }(m+r+n,s-m;q)h(-r,k;A)h(-s,0;B) \\ & \quad \ll q^{-5/2}\tau(q)\sum_{r,s\leq q-1}\sum _{m\leq q}\min \biggl(H, \biggl\Vert \frac{m}{q} \biggr\Vert ^{-1} \biggr)\cdot (m+r+n,s-m,q)^{1/2} \frac{A^{k}}{\vert \sin(\frac{\pi r}{q})\vert }\cdot\frac{1}{\vert \sin(\frac{\pi s}{q})\vert } \\ & \quad \ll HA^{k}q^{-1/2}d(q)\sum _{r,s\leq q-1} \frac{1}{rs}\sum_{m\leq q/H}(m+r+n,s-m,q)^{1/2} \\ & \quad\quad{} +A^{k}q^{1/2}d(q)\sum _{r,s\leq q-1} \frac{1}{rs}\sum_{q/H< m\leq q-1} \frac{(m+r+n,s-m,q)^{1/2}}{m} . \end{aligned}$$

Using the estimates

$$\begin{aligned}& \sum_{r,s\leq q-1}\frac{1}{rs}\sum _{m\leq q/H}(m+r+n,s-m,q)^{1/2} \\ & \quad =\sum_{d\mid q}d^{1/2}\sum _{m\leq q/H}\mathop{\sum_{r\leq q-1}}_{ d\mid (m+r+n)} \frac{1}{r}\mathop{\sum_{s\leq q-1}}_{ d\mid (s-m)} \frac{1}{s} \\ & \quad =\sum_{d\mid q}d^{1/2}\sum _{m\leq q/H}\sum_{(m+n+1)/d \le r\leq (q+m+n-1)/d}\frac{1}{dr-m-n} \sum_{(1-m)/d\le s\leq(q-m-1)/d}\frac {1}{ds+m} \\ & \quad \ll H^{-1}q d(q) (\log q) \log(2q+n-1) \end{aligned}$$

and

$$\begin{aligned}& \sum_{r,s\leq q-1}\frac{1}{rs}\sum _{q/H< m\leq q-1}\frac {(m+r+n,s-m,q)^{1/2}}{m} \\ & \quad =\sum_{d\mid q}d^{1/2}\sum _{q/H < m\leq q-1}\frac{1}{m}\mathop{\sum _{r\leq q-1}}_{ d\mid (m+r+n)}\frac{1}{r}\mathop{\sum _{s\leq q-1}}_{ d\mid (s-m)}\frac {1}{s} \\ & \quad =\sum_{d\mid q}d^{1/2}\sum _{q/H < m\leq q-1}\frac{1}{m}\sum_{(m+n+1)/d \le r\leq(q+m+n-1)/d} \frac{1}{dr-m-n}\sum_{(1-m)/d\le s\leq (q-m-1)/d}\frac{1}{ds+m} \\ & \quad \ll d(q) \bigl(\log^{2}q \bigr) \log(2q+n-1), \end{aligned}$$

we have

$$\begin{aligned}& \frac{1}{q^{3}}\sum_{r,s\leq q-1}\sum _{m\leq q}\sum_{t\leq H}e \biggl(- \frac{mt}{q} \biggr)K_{\chi }(m+r+n,s-m;q)h(-r,k;A)h(-s,0;B) \\ & \quad \ll A^{k}q^{1/2}d(q) \bigl(\log^{2} q \bigr) \log(2q+n-1). \end{aligned}$$
(3.4)

Now combining (3.1)-(3.4), we have

$$G_{k}(\chi,A,B,H;q)\ll A^{k}q^{1/2}d(q) \biggl( \frac{nAB d(q)}{q^{2}}+\frac{B d(q)(\log q)(\log H)}{q}+\log^{3}q \biggr). $$

This completes the proof of the theorem.

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Acknowledgements

The article was supported by the National Natural Science Foundation of China (Grant Nos. 11201275, 11471258), the Research Fund for the Doctoral Program of Higher Education of China (Grant No. 20106101120001), the Natural Science Foundation of Shaanxi Province of China (Grant Nos. 2011JQ1010, 2016JM1017), the Scientific Research Program Funded by Shaanxi Provincial Education Department (Grant No. 15JK1794) and the Fundamental Research Funds for the Central Universities (Grant No. GK201503014). The authors want to show their great thanks to the anonymous referee for his/her helpful comments and suggestions.

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DH drafted the manuscript. GR and TZ participated in its design and coordination and helped to draft the manuscript. All authors read and approved the final manuscript.

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Ren, G., He, D. & Zhang, T. On general partial Gaussian sums. J Inequal Appl 2016, 295 (2016). https://doi.org/10.1186/s13660-016-1238-2

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MSC

  • 11L05
  • 11L40

Keywords

  • partial Gaussian sums
  • Kloosterman sums