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The hybrid power mean of the quartic Gauss sums and the two-term exponential sums
Advances in Difference Equations volume 2018, Article number: 236 (2018)
Abstract
In this paper, we use the analytic method and the properties of classical Gauss sums to study the computational problems of one kind hybrid power mean of quartic Gauss sums and two-term exponential sums, and give an interesting fourth-order linear recurrence formula for it.
1 Introduction
Let \(q\ge3\) be an integer. For any positive integer \(k\geq2\), the kth Gauss sums \(G(m, k; q)\) are defined as
where, as usual, \(e(y) = e^{2\pi i y}\).
Recently, some scholars have studied the properties of \(G(m,4;p)\) and obtained many interesting results, where p is an odd prime with \(p\equiv1\bmod4\). For example, Shimeng Shen and Wenpeng Zhang [1] proved a recurrence formula related to \(G(m,4;p)\). The author and Jiayuan Hu [2] studied the computational problem of the hybrid power mean
We proved the identity
where \(\chi_{4}\) denotes any fourth-order character modp, \(\tau(\chi)= \sum_{a=1}^{p-1} \chi(a)e (\frac{a}{p} )\) denotes the classical Gauss sums, and c̅ denotes the multiplicative inverse of \(c \bmod p\).
At the same time, the author and Jiayuan Hu [2] also pointed out how to compute the exact value of \(\tau^{2} (\overline{\chi}_{4} )+\tau^{2}(\chi_{4})\) and \(\tau^{5} (\overline{\chi}_{4} )+\tau^{5}(\chi_{4})\), these are two meaningful problems.
Zhuoyu Chen and Wenpeng Zhang [3] studied the properties of the Gauss sums
By using the analytic method and the properties of classical Gauss sums, they obtained an exact computational formula for \(G(k,p)\), which completely solved the problem proposed by the author and Jiayuan Hu in [2]. Some related works can also be found in references [4–11].
Inspired by reference [3], we will consider the following hybrid power mean:
For convenience, hereinafter, we always assume that p is a prime with \(p\equiv1\bmod4\), \((\frac{*}{p} )=\chi_{2}\) denotes the Legendre symbol modp, and
a̅ denotes the solution of the equation \(ax\equiv 1\bmod p\). The number α is closely related to prime p. In fact, we have a very important formula
where r is any integer with \((\frac{r}{p} )=-1\) (see Theorems 4–11 in [12]).
In this paper, by using the analytic method, the properties of the classical Gauss sums, and trigonometric sums, we will study the computational problem of \(M_{k}(p)\), and give an interesting fourth-order linear recurrence formula for it. That is, we will prove the following two results.
Theorem 1
If p is a prime with \(p\equiv5\bmod8\), then for any integer \(k\geq4\), we have the linear recurrence formula
where the first four items in the sequence \(\{M_{k}(p)\}\) are: \(M_{0}(p)=p(p-3)\); \(M_{1}(p)=2p\alpha\); \(M_{2}(p)=-p (p^{2}-3p-4\alpha ^{2} )\), and \(M_{3}(p)=2p^{2}\alpha (3p-14 )\).
Theorem 2
If p is a prime with \(p\equiv1\bmod8\), then for any integer \(k\geq4\), we have the linear recurrence formula
where the first four items in the sequence \(\{M_{k}(p)\}\) are: \(M_{0}(p)=p(p-3)\); \(M_{1}(p)=-6p\alpha\); \(M_{2}(p)=p (3p^{2}-17p-4\alpha ^{2} )\), and \(M_{3}(p)=6p^{2}\alpha(p-8)\).
For some special integers \(k=2\) or \(k=4\), from our theorems we may immediately deduce the following three corollaries.
Corollary 1
If p is an odd prime with \(p\equiv5\bmod 8\), then we have
Corollary 2
If p is an odd prime with \(p\equiv 1\bmod8\), then we have the identity
Corollary 3
If p is an odd prime with \(p\equiv 1\bmod8\), then we have
Notes
If \(p=4k+3\), then \((\frac{-1}{p} )=-1\). Then, in this case, for any integer m with \((m,p)=1\), we have
where \(i^{2}=-1\). Therefore, the hybrid power mean \(M_{k}(p)\) can be easily obtained.
2 Several lemmas
To prove our main results, we need several simple lemmas. Here, we will use many properties of the classical Gauss sums, all of them can be found in reference [13], so they will not be repeated here. First we have the following lemma.
Lemma 1
If p is a prime with \(p\equiv1\bmod4\), then for any fourth-order character \(\psi\bmod p\), we have the identity
Proof
First, from the trigonometric identity
the properties of character \(\psi\bmod p\) and noting that \(\psi^{4}=\chi _{0}\), the principal character modp, we have
Similarly, we also have
Since \(p\equiv1\bmod4\), so from the properties of the fourth-order character \(\psi\bmod p\), we have
where \(\psi^{2}=\chi_{2}= (\frac{*}{p} )\) denotes the Legendre symbol modp.
From the properties of the classical Gauss sums, we have
Note that \(\tau(\chi_{2})=\sqrt{p}\), from (5), (6), and (7) we have
Combining (3), (4), and (8) and noting the orthogonality properties of characters modp, we have the identity
This proves Lemma 1. □
Lemma 2
Let p be an odd prime with \(p\equiv1\bmod4\). Then, for the Legendre symbol \({\chi_{2}\bmod p}\), we have the identity
Proof
First, from (2) and the method of proving Lemma 1, we have
Combining (9) and (10), we can deduce the identity
This proves Lemma 2. □
Lemma 3
Let p be an odd prime with \(p\equiv1\bmod4\), ψ be any fourth-order character modp. Then we have the identity
Proof
See Lemma 2.2 in [3]. □
Lemma 4
Let p be an odd prime with \(p\equiv1\bmod4\). Then we have the identity
Proof
Since the congruence equation \(x^{4}\equiv1\bmod p\) has four different solutions in a reduced residue system modp, so from (2) we have
This proves Lemma 4. □
3 Proofs of the theorems
Now we complete the proofs of our theorems. First we prove Theorem 1. For convenience, we let
Then, for any integer m with \((m, p)=1\), from (2) and the properties of the fourth-order character \(\psi\bmod p\), we have
If \(p=8r+5\), then \(\psi(-1)=-1\). In this case, from Lemma 1 we have the identity
It is clear that from Lemma 4 we have
From (12), Lemma 2, and Lemma 3, we have
Similarly, noting that \(\tau(\psi)\tau (\overline{\psi} )=-p\), from (12) and Lemma 4 we also have
From [1] (see Lemma 3) we have
So, if \(k\geq4\), then we have
Combining (13)–(16) and (18), we immediately complete the proof of Theorem 1.
Now we prove Theorem 2. If \(p=8k+1\), then note that \(\psi(-1)=1\), from Lemma 4 we have
From (11), Lemma 1, Lemma 2, and Lemma 3, we have
Applying Lemma 1, Lemma 2, and Lemma 3, we also have
From [1] (see Lemma 3) we have
So if \(k\geq4\), then we have the fourth-order linear recurrence formula
Now Theorem 2 follows from (19)–(22) and (24).
If \(p\equiv5\bmod8\), then note that \(\overline{\tau(\psi)}=-\tau (\overline{\psi} )\), from (11) we have
Thus, from (25) and \(\tau(\psi)\tau (\overline{\psi} )=-p\), we have
Applying (26) and Lemma 2, we may immediately deduce Corollary 1.
If \(p\equiv1\bmod8\), then note that \(B(m)\) is a real number. So Corollary 2 and Corollary 3 follow from Theorem 2.
This completes the proofs of all our results.
References
Shen, S.M., Zhang, W.P.: On the quartic Gauss sums and their recurrence property. Adv. Differ. Equ. 2017, Article ID 43 (2017)
Li, X.X., Hu, J.Y.: The hybrid power mean quartic Gauss sums and Kloosterman sums. Open Math. 15, 151–156 (2017)
Chen, Z.Y., Zhang, W.P.: On the fourth-order linear recurrence formula related to classical Gauss sums. Open Math. 15, 1251–1255 (2017)
Zhang, W.P., Liu, H.N.: On the general Gauss sums and their fourth power mean. Osaka J. Math. 42, 189–199 (2005)
Han, D.: A hybrid mean value involving two-term exponential sums and polynomial character sums. Czechoslov. Math. J. 64, 53–62 (2014)
Zhang, W.P., Han, D.: On the sixth power mean of the two-term exponential sums. J. Number Theory 136, 403–413 (2014)
Cochrane, T.: Exponential sums modulo prime powers. Acta Arith. 101, 131–149 (2002)
Cochrane, T., Pinner, C.: Using Stepanov’s method for exponential sums involving rational functions. J. Number Theory 116, 270–292 (2006)
Ren, G.L.: On the fourth power mean of one kind special quadratic Gauss sums. J. Northwest Univ. Nat. Sci. 46(3), 321–324 (2016)
Zhang, H., Zhang, W.P.: The fourth power mean of two-term exponential sums and its application. Math. Rep. 19, 75–83 (2017)
Zhang, W.P., Hu, J.Y.: The number of solutions of the diagonal cubic congruence equation modp. Math. Rep. 20, 73–80 (2018)
Zhang, W.P., Li, H.L.: Elementary Number Theory. Shaanxi Normal University Press, Xi’an (2008)
Apostol, T.M.: Introduction to Analytic Number Theory. Springer, New York (1976)
Acknowledgements
The author would like to thank the referees for their very helpful and detailed comments, which have significantly improved the presentation of this paper.
Funding
The research was supported by the National Natural Science Foundation of China (Grant No. 11771351; 11501452), the Natural Science Basic Research Plan in Shaanxi Province (Grant No. 2018JQ1093), University’s Scientific Research Project (Grant No. 2018KY0208) and the Doctoral Scientific Research Project of Xi’an Aeronautical University.
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Li, X. The hybrid power mean of the quartic Gauss sums and the two-term exponential sums. Adv Differ Equ 2018, 236 (2018). https://doi.org/10.1186/s13662-018-1658-z
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DOI: https://doi.org/10.1186/s13662-018-1658-z
MSC
- 11L05
- 11L07
Keywords
- Quartic Gauss sums
- Two-term exponential sums
- Hybrid power mean
- Analytic method