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On general partial Gaussian sums
Journal of Inequalities and Applications volume 2016, Article number: 295 (2016)
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
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 \(\hbar(A,B,H)\).
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.
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
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
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
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
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).
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.
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 \(n< q^{1/2}\).
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
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
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
This is a little stronger than the classical result of the complete Gauss sums in the case when \((n,q)>1\).
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 \(1\leq r\leq q-1\).
Proof
See Lemma 3 of [6] or Lemma 2.4 of [7]. □
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.
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
Then from Lemma 1 and Lemma 2, we have
where \(\Vert x\Vert =\min_{a\in Z}\vert x-a\vert \).
Combining the estimates
and
we have
Applying Lemma 1 and Lemma 2 again, we obtain
Combining
and
we have
Similarly, we get the estimate
Noting that
Using the estimates
and
we have
Now combining (3.1)-(3.4), we have
This completes the proof of the theorem.
References
Burgess, DA: Partial Gaussian sums. Bull. Lond. Math. Soc. 20(6), 589-592 (1988)
Burgess, DA: Partial Gaussian sums. II. Bull. Lond. Math. Soc. 21(2), 153-158 (1989)
Burgess, DA: Partial Gaussian sums. III. Glasg. Math. J. 34(2), 253-261 (1992)
Liu, CL: On incomplete Gaussian sums. Acta Math. Sin. New Ser. 12(2), 141-150 (1996)
Xi, P, Yi, Y: On character sums over flat numbers. J. Number Theory 130(5), 1234-1240 (2010)
Xu, ZF: Distribution of the difference of an integer and its m-th power mod n over incomplete intervals. J. Number Theory 133(12), 4200-4223 (2013)
Xu, ZF, Zhang, TP: High-dimensional D.H. Lehmer problem over short intervals. Acta Math. Sin. Engl. Ser. 30(2), 213-228 (2014)
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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DOI: https://doi.org/10.1186/s13660-016-1238-2