1 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: is of great importance. By extending his well-known work on character sums, Burgess obtained the following.
$$\sum_{a= N+1}^{N+H}\chi(a)e \biggl( \frac{na}{q} \biggr) $$
Proposition 1
[1]
Anzeige
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]
Anzeige
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 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).
$$\hbar(A,B,H)= \bigl\{ a\in\mathbb{Z}\mid (a,q)=1, ab\equiv1\pmod{q}, 1\leq a\leq A, 1\leq b\leq B, \vert a-b\vert \leq H \bigr\} . $$
This paper deals with general partial Gaussian sums of the following type: 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.
$$G_{k}(\chi,A,B,H;q)=\sum_{a\in\hbar(A,B,H)}a^{k} \chi(a)e \biggl(\frac{na}{q} \biggr), $$
Theorem
Let
\(q\geq2\)
be an integer and
χ
a non-principal Dirichlet character mod
q. Then
which is uniformly nontrivial for any positive integer
n
such that
\(n< q^{1/2}\).
$$G_{k}(\chi,A,B,H;q)\ll A^{k}q^{1/2}d(q) \biggl( \frac{nAB d(q)}{q^{2}}+\frac{B d(q)(\log q)(\log H)}{q}+\log^{3}q \biggr), $$
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
$$G_{k}(\chi,A,B,H;q)\ll A^{k}q^{1/2}d(q) \biggl( \frac{AB d(q)}{q^{2}}+\frac{B d(q)(\log q)(\log H)}{q}+\log^{3}q \biggr). $$
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
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
This is a little stronger than the classical result of the complete Gauss sums in the case when
\((n,q)>1\).
$$G_{0}(\chi,q,q,q;q)=\sum_{a=1}^{q} \chi(a)e \biggl(\frac {na}{q} \biggr)\ll q^{1/2}d^{2}(q) \log^{3} q. $$
2 Some lemmas
To prove the theorem, we need the following lemmas.
Lemma 1
Let
q, n, ℓ, 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
where
\(s=\min(r,q-r)\)
with
\(1\leq r\leq q-1\).
$$h(r,\ell;n) \textstyle\begin{cases} =\frac{n^{\ell+1}}{\ell+1}+O(n^{\ell}),& q\mid r,\\ \ll\frac{n^{\ell}}{ \vert \sin(\pi s/q)\vert } ,& q\nmid r, \end{cases} $$
Lemma 2
Let
q
be a positive integer. Then we have
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.
$$\bigl\vert K_{\chi}(m,n;q) \bigr\vert \leq q^{1/2}(m,n,q)^{1/2}d(q), $$
Proof
See Lemma 1 of [5]. □
3 Proof of the Theorem
Now we come to prove the theorem. Note that Applying the trigonometric sums identity we obtain
$$G_{k}(\chi,A,B,H;q)=\sum_{t\leq H}\mathop{ \sum_{a\leq A,b\leq B}}_{ a-b\equiv t\pmod{q},ab\equiv1\pmod{q}}a^{k}\chi(a)e \biggl(\frac{na}{q} \biggr). $$
$$\sum^{q}_{a=1}e \biggl( \frac{ma}{q} \biggr)= \textstyle\begin{cases} q,& q\mid m,\\ 0 ,& q\nmid m, \end{cases} $$
$$\begin{aligned}& G_{k}(\chi,A,B,H;q) \\ & \quad = \sum_{a\in\hbar(A,B,H)}a^{k}\chi(a)e \biggl( \frac{na}{q} \biggr) \\ & \quad = \sum_{t\leq H} \mathop{\mathop{\sum \nolimits'}_{a\leq A,b\leq B}}_{ a-b\equiv t\pmod{q},ab\equiv1\pmod{q}} a^{k}\chi(a)e \biggl(\frac{na}{q} \biggr) \\ & \quad = \frac{1}{q}\sum_{m\leq q}\sum _{t\leq H}e \biggl(-\frac {mt}{q} \biggr) \mathop{\mathop{\sum \nolimits'}_{a\leq A,b\leq B }}_{ ab\equiv 1\pmod{q}} a^{k} \chi(a)e \biggl(\frac{(m+n)a-mb}{q} \biggr) \\ & \quad = \frac{1}{q^{3}}\sum_{m\leq q}\sum _{t\leq H}e \biggl(-\frac {mt}{q} \biggr) \mathop{\mathop{\sum \nolimits'}_{a,b\leq q }}_{ ab\equiv1\pmod{q}} \chi(a)e \biggl( \frac{(m+n)a-mb}{q} \biggr) \\ & \quad\quad{} \times\sum_{c\leq A}c^{k}\sum _{r\leq q}e \biggl(\frac {r(a-c)}{q} \biggr)\sum _{d\leq B}\sum_{s\leq q}e \biggl( \frac {s(b-d)}{q} \biggr) \\ & \quad = \frac{1}{q^{3}}\sum_{r,s\leq q}\sum _{m\leq q}\sum_{t\leq H}e \biggl(- \frac{mt}{q} \biggr) \mathop{\mathop{\sum\nolimits'}_{a,b\leq q }}_{ ab\equiv1\pmod{q}} \chi(a) e \biggl(\frac{(m+r+n)a-(m-s)b}{q} \biggr) \\ & \quad\quad{} \times\sum_{c\leq A}c^{k}e \biggl(- \frac{rc}{q} \biggr)\sum_{d\leq B}e \biggl(- \frac{sd}{q} \biggr) \\ & \quad = \frac{1}{q^{3}}\sum_{r,s\leq q}\sum _{m\leq q}\sum_{t\leq H}e \biggl(- \frac{mt}{q} \biggr)K_{\chi }(m+r+n,s-m;q)h(-r,k;A)h(-s,0;B) \\ & \quad = \frac{1}{q^{3}}\sum_{m\leq q}\sum _{t\leq H}e \biggl(-\frac {mt}{q} \biggr)K_{\chi}(m+n,-m;q)h(-q,k;A)h(-q,0;B) \\ & \quad\quad{} +\frac{1}{q^{3}}\sum_{r\leq q-1}\sum _{m\leq q}\sum_{t\leq H}e \biggl(- \frac{mt}{q} \biggr)K_{\chi }(m+r+n,-m;q)h(-r,k;A)h(-q,0;B) \\ & \quad\quad{} +\frac{1}{q^{3}}\sum_{s\leq q-1}\sum _{m\leq q}\sum_{t\leq H}e \biggl(- \frac{mt}{q} \biggr)K_{\chi}(m+n,s-m;q)h(-q,k;A)h(-s,0;B) \\ & \quad\quad{} +\frac{1}{q^{3}}\sum_{r,s\leq q-1}\sum _{m\leq q}\sum_{t\leq H}e \biggl(- \frac{mt}{q} \biggr)K_{\chi}(m+r+n,s-m;q)h(-r,k;A)h(-s,0;B). \end{aligned}$$
Then from Lemma 1 and Lemma 2, we have where \(\Vert x\Vert =\min_{a\in Z}\vert x-a\vert \).
$$\begin{aligned}& \frac{1}{q^{3}}\sum_{m\leq q}\sum _{t\leq H}e \biggl(-\frac {mt}{q} \biggr)K_{\chi}(m+n,-m;q)h(-q,k;A)h(-q,0;B) \\& \quad \ll\frac{1}{q^{3}}\sum_{m\leq q}\min \biggl(H, \biggl\Vert \frac {m}{q} \biggr\Vert ^{-1} \biggr) \bigl\vert K_{\chi}(m+n,-m;q) \bigr\vert \cdot \bigl\vert h(-q,k;A) \bigr\vert \cdot \bigl\vert h(-q,0;B) \bigr\vert \\& \quad \ll HA^{k+1}Bq^{-5/2}d(q)\sum _{m\leq q/H}(m+n,q)^{1/2} \\& \quad\quad{} +A^{k+1}Bq^{-3/2}d(q) \sum_{q/H< m\leq q-1}\frac{(m+n,q)^{1/2}}{m}, \end{aligned}$$
Combining the estimates and we have
$$\begin{aligned} \sum_{m\leq q/H}(m+n,q)^{1/2}&=\sum _{d\mid q}d^{1/2}\mathop{\sum _{m\leq q/H }}_{ d\mid (m+n)}1 \\ & =\sum_{d\mid q}d^{1/2}\sum _{m\leq q/(dH)+n/d}1 \\ & \ll H^{-1}qd(q)+nd(q) \end{aligned}$$
$$\begin{aligned} \sum_{q/H< m\leq q-1}\frac{(m+n,q)^{1/2}}{m}&= \sum _{d\mid q}d^{1/2}\mathop{\sum _{q/H< m\leq q-1}}_{ d\mid (m+n)}\frac{1}{m} \\ & = \sum_{d\mid q}d^{1/2}\sum _{\frac{q}{Hd}+\frac{n}{d}< m\leq\frac {q-1+n}{d}}\frac{1}{dm-n} \\ &\ll\sum_{d\mid q}d^{-1/2}\sum _{\frac{q}{Hd}+\frac{n}{d}< m\leq\frac {q-1+n}{d}}\frac{1}{m} \biggl(1+\frac{n}{dm} \biggr) \\ &\ll d(q)\log H+nd(q), \end{aligned}$$
$$\begin{aligned}& \frac{1}{q^{3}}\sum_{m\leq q}\sum _{t\leq H}e \biggl(-\frac {mt}{q} \biggr)K_{\chi}(m+n,-m;q)h(-q,k;A)h(-q,0;B) \\& \quad \ll nA^{k+1}Bq^{-3/2}d^{2}(q)+A^{k+1}Bq^{-3/2}d^{2}(q) \log H. \end{aligned}$$
(3.1)
Applying Lemma 1 and Lemma 2 again, we obtain Combining and we have Similarly, we get the estimate
$$\begin{aligned}& \frac{1}{q^{3}}\sum_{r\leq q-1}\sum _{m\leq q}\sum_{t\leq H}e \biggl(- \frac{mt}{q} \biggr)K_{\chi}(m+r+n,-m;q)h(-r,k;A)h(-q,0;B) \\& \quad \ll\frac{1}{q^{3}}\sum_{r\leq q-1}\sum _{m\leq q}\min \biggl(H, \biggl\Vert \frac{m}{q} \biggr\Vert ^{-1} \biggr) \bigl\vert K_{\chi }(m+r+n,-m;q) \bigr\vert \cdot \bigl\vert h(-r,k;A) \bigr\vert \cdot \bigl\vert h(-q,0;B) \bigr\vert \\& \quad \ll HBq^{-5/2}d(q)\sum_{r\leq q-1}\sum _{m\leq q/H}(m+r+n,-m,q)^{1/2}\cdot \frac{A^{k}}{\vert \sin(\frac{\pi r}{q})\vert } \\& \quad \quad{}+Bq^{-3/2}d(q)\sum_{r\leq q-1}\sum _{q/H< m\leq q-1}\frac {(m+r+n,-m,q)^{1/2}}{m}\cdot\frac{A^{k}}{\vert \sin(\frac{\pi r}{q})\vert } \\& \quad \ll HA^{k}Bq^{-3/2}d(q)\sum _{r\leq q-1} \frac{1}{r}\sum_{m\leq q/H}(m+r+n,-m,q)^{1/2} \\& \quad\quad{} +A^{k}Bq^{-1/2}d(q)\sum _{r\leq q-1} \frac{1}{r}\sum_{q/H< m\leq q-1} \frac{(m+r+n,-m,q)^{1/2}}{m} . \end{aligned}$$
$$\begin{aligned}& \sum_{r\leq q-1}\frac{1}{r}\sum _{m\leq q/H}(m+r+n,-m,q)^{1/2} \\ & \quad =\sum_{d\mid q}d^{1/2}\mathop{\sum _{r\leq q}}_{ d\mid (r+n)}\frac{1}{r}\mathop{\sum _{m\leq q/H }}_{ d\mid m} 1 \\ & \quad =\sum_{d\mid q}d^{1/2}\sum _{(n+1)/d \le r\leq(q+n-1)/d}\frac {1}{dr-n}\sum_{m\leq q/(Hd)} 1 \\ & \quad \ll q/H\sum_{d\mid q}d^{-3/2}\sum _{(n+1)/d \le r\leq(q+n-1)/d}\frac {1}{r} \biggl(1+\frac{n}{dr} \biggr) \\ & \quad \ll H^{-1}qd(q)\log \biggl(\frac{q+n-1}{n+1} \biggr) \end{aligned}$$
$$\begin{aligned}& \sum_{r\leq q-1}\frac{1}{r}\sum _{q/H< m\leq q-1}\frac {(m+r+n,-m,q)^{1/2}}{m} \\ & \quad =\sum_{d\mid q}d^{1/2}\mathop{\sum _{r\leq q-1}}_{ d\mid (r+n)}\frac{1}{r}\mathop{\sum _{q/H< m\leq q-1}}_{ d\mid m}\frac{1}{m} \\ & \quad =\sum_{d\mid q}d^{-1/2}\sum _{(n+1)/d \le r\leq(q+n-1)/d}\frac {1}{dr-n}\sum_{q/(Hd)< m\leq q/d} \frac{1}{m} \\ & \quad \ll(\log H)\sum_{d\mid q}d^{-3/2}\sum _{r\leq q/d}\frac{1}{r(1-\frac {n}{dr})} \\ & \quad \ll d(q) (\log H) \log \biggl(\frac{q+n-1}{n+1} \biggr), \end{aligned}$$
$$\begin{aligned}& \frac{1}{q^{3}}\sum_{r\leq q-1}\sum _{m\leq q}\sum_{t\leq H}e \biggl(- \frac{mt}{q} \biggr)K_{\chi}(m+r+1,-m;q)h(-r,k;A)h(-q,0;B) \\ & \quad \ll A^{k}Bq^{-1/2}d^{2}(q) (\log H) \log \biggl( \frac{q+n-1}{n+1} \biggr). \end{aligned}$$
(3.2)
$$\begin{aligned}& \frac{1}{q^{3}}\sum_{s\leq q}\sum _{m\leq q}\sum_{t\leq H}e \biggl(- \frac{mt}{q} \biggr)K_{\chi}(m+1,s-m;q)h(-q,k;A)h(-s,0;B) \\ & \quad \ll A^{k+1}q^{-3/2}d^{2}(q)\log \biggl( \frac{q-1}{n+1} \biggr). \end{aligned}$$
(3.3)
Noting that Using the estimates and we have Now combining (3.1)-(3.4), we have This completes the proof of the theorem.
$$\begin{aligned}& \frac{1}{q^{3}}\sum_{r,s\leq q-1}\sum _{m\leq q}\sum_{t\leq H}e \biggl(- \frac{mt}{q} \biggr)K_{\chi }(m+r+n,s-m;q)h(-r,k;A)h(-s,0;B) \\ & \quad \ll q^{-5/2}\tau(q)\sum_{r,s\leq q-1}\sum _{m\leq q}\min \biggl(H, \biggl\Vert \frac{m}{q} \biggr\Vert ^{-1} \biggr)\cdot (m+r+n,s-m,q)^{1/2} \frac{A^{k}}{\vert \sin(\frac{\pi r}{q})\vert }\cdot\frac{1}{\vert \sin(\frac{\pi s}{q})\vert } \\ & \quad \ll HA^{k}q^{-1/2}d(q)\sum _{r,s\leq q-1} \frac{1}{rs}\sum_{m\leq q/H}(m+r+n,s-m,q)^{1/2} \\ & \quad\quad{} +A^{k}q^{1/2}d(q)\sum _{r,s\leq q-1} \frac{1}{rs}\sum_{q/H< m\leq q-1} \frac{(m+r+n,s-m,q)^{1/2}}{m} . \end{aligned}$$
$$\begin{aligned}& \sum_{r,s\leq q-1}\frac{1}{rs}\sum _{m\leq q/H}(m+r+n,s-m,q)^{1/2} \\ & \quad =\sum_{d\mid q}d^{1/2}\sum _{m\leq q/H}\mathop{\sum_{r\leq q-1}}_{ d\mid (m+r+n)} \frac{1}{r}\mathop{\sum_{s\leq q-1}}_{ d\mid (s-m)} \frac{1}{s} \\ & \quad =\sum_{d\mid q}d^{1/2}\sum _{m\leq q/H}\sum_{(m+n+1)/d \le r\leq (q+m+n-1)/d}\frac{1}{dr-m-n} \sum_{(1-m)/d\le s\leq(q-m-1)/d}\frac {1}{ds+m} \\ & \quad \ll H^{-1}q d(q) (\log q) \log(2q+n-1) \end{aligned}$$
$$\begin{aligned}& \sum_{r,s\leq q-1}\frac{1}{rs}\sum _{q/H< m\leq q-1}\frac {(m+r+n,s-m,q)^{1/2}}{m} \\ & \quad =\sum_{d\mid q}d^{1/2}\sum _{q/H < m\leq q-1}\frac{1}{m}\mathop{\sum _{r\leq q-1}}_{ d\mid (m+r+n)}\frac{1}{r}\mathop{\sum _{s\leq q-1}}_{ d\mid (s-m)}\frac {1}{s} \\ & \quad =\sum_{d\mid q}d^{1/2}\sum _{q/H < m\leq q-1}\frac{1}{m}\sum_{(m+n+1)/d \le r\leq(q+m+n-1)/d} \frac{1}{dr-m-n}\sum_{(1-m)/d\le s\leq (q-m-1)/d}\frac{1}{ds+m} \\ & \quad \ll d(q) \bigl(\log^{2}q \bigr) \log(2q+n-1), \end{aligned}$$
$$\begin{aligned}& \frac{1}{q^{3}}\sum_{r,s\leq q-1}\sum _{m\leq q}\sum_{t\leq H}e \biggl(- \frac{mt}{q} \biggr)K_{\chi }(m+r+n,s-m;q)h(-r,k;A)h(-s,0;B) \\ & \quad \ll A^{k}q^{1/2}d(q) \bigl(\log^{2} q \bigr) \log(2q+n-1). \end{aligned}$$
(3.4)
$$G_{k}(\chi,A,B,H;q)\ll A^{k}q^{1/2}d(q) \biggl( \frac{nAB d(q)}{q^{2}}+\frac{B d(q)(\log q)(\log H)}{q}+\log^{3}q \biggr). $$
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.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
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.