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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)
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.
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  and Malyshev  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  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  proved the identity:
In this paper we further consider the situation . Our result is a generalization of .
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:
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:
q is square-free;
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
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 . □
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.
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
This work is supported by the P.E.D. (2013JK0561) and N.S.F. (11371291) of P.R. China.
The authors declare that they have no competing interests.
All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.
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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
- general k th Kloosterman sum
- fourth power mean