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On the fourth power mean of the general k th Kloosterman sums
Advances in Difference Equations volume 2014, Article number: 103 (2014)
Abstract
Let be an integer, and let χ be a Dirichlet character modulo q. For integers m, n and k, the general k th Kloosterman sum is defined by , where ∑′ denotes the summation over all a with , , and is the inverse of a modulo q such that and . In this paper we further study the fourth power mean , and we give some identities.
MSC:11F20.
1 Introduction
Let be an integer, and let χ be a Dirichlet character modulo q. For arbitrary integers m and n, the general Kloosterman sum is defined by
where ∑′ denotes the summation over all a with , , and is the inverse of a modulo q such that and . For a prime, Chowla [1] and Malyshev [2] proved an upper bound:
where is the great common divisor of m, n and p.
For general integer , we do not know how large is. However, enjoys good value distribution properties. For fixed integer n with , Zhang [3] showed the identity
where is the divisor function, is the Euler function, and denotes the product over all prime divisors p of q with and .
For integers m, n and k, the general k th Kloosterman sum is defined by
Suppose that . Liu and Zhang [4] proved the identity:
In this paper we further consider the situation . Our result is a generalization of [4].
Theorem 1.1 Let p be an odd prime, n be any integer. Let α and k be positive integers with and . Then we have
From Theorem 1.1 we immediately get the following corollary.
Corollary 1.1 Let p be an odd prime, n be any integer. Let α and k be positive integers. Assume that one of the following conditions holds:
-
(1)
;
-
(2)
and ;
-
(3)
and .
Then we have
Theorem 1.2 Let be an odd number, n be an integer with , and let k be a positive integer. Then we have
From Theorem 1.2 we can get the following corollary.
Corollary 1.2 Let be an odd number, n be an integer with , and let k be a positive integer. Assume that one of the following conditions holds:
-
(1)
q is square-free;
-
(2)
and ;
-
(3)
and .
Then we have
2 Some lemmas
To complete the proof of theorems, we need the following lemmas.
Lemma 2.1 Let p be an odd prime, and let α and k be positive integers with . Write . Then we have
Proof For , we get
Now we assume that . It is not hard to show that
If , by (2.2) we have
If , then
On the other hand, for and , we have
Next we suppose that and with . Then
Noting that
Therefore
Now combining (2.1), (2.3)-(2.6) we immediately get
for . □
Lemma 2.2 Let n, k, , be integers with . Then for any character , there exist integers and with such that
and for these integers we have
where with and .
Proof This is Lemma 2.2 of [4]. □
3 Proof of the theorems
First we prove Theorem 1.1. Let p be an odd prime, and let α and k be positive integers. Assume that n is any integer. We have
Then from the orthogonality relation for characters and the trigonometric identity we get
Write and . By Lemma 2.1 we immediately have
This completes the proof of Theorem 1.1.
Now we prove Theorem 1.2. Let be an odd number, n be an integer with , and let k be a positive integer. Write
By Lemma 2.2 and Theorem 1.1 we have
This proves Theorem 1.2.
References
Chowla S: On Kloosterman’s sum. Norske Vid. Selsk. Forhdl. 1967, 40: 70–72.
Malyshev AV: A generalization of Kloosterman sums and their estimates. Vestn. Leningr. Univ. 1960, 15: 59–75. (Russian)
Zhang W: On the general Kloosterman sum and its fourth power mean. J. Number Theory 2004, 104: 156–161. 10.1016/S0022-314X(03)00154-9
Liu HY, Zhang WP: On the general k -th Kloosterman sums and its fourth power mean. Chin. Ann. Math., Ser. B 2004, 25: 97–102. 10.1142/S0252959904000093
Acknowledgements
This work is supported by the P.E.D. (2013JK0561) and N.S.F. (11371291) of P.R. China.
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Guo, X., Geng, G. & Pan, X. On the fourth power mean of the general k th Kloosterman sums. Adv Differ Equ 2014, 103 (2014). https://doi.org/10.1186/1687-1847-2014-103
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DOI: https://doi.org/10.1186/1687-1847-2014-103