A sum analogous to the high-dimensional Kloosterman sums and its upper bound estimate
© Li and Han; licensee Springer 2013
Received: 6 December 2012
Accepted: 9 March 2013
Published: 27 March 2013
The main purpose of this paper is, using the properties of Gauss sums and the estimate for the generalized exponential sums, to study the upper bound estimate problem of one kind sums analogous to the high-dimensional Kloosterman sums and to give some interesting mean value formula and an upper bound estimate for it.
where , denotes the summation over all integers such that , and m are integers with , denotes the solution of the congruent equation ().
where χ is a Dirichlet character modq.
Now we are concerned with the upper bound estimate problem of (1.1). Regarding this contents, it seems that nobody has yet studied it, at least we have not seen any related result before. The problem is interesting because it can reflect some new properties of character sums. The main purpose of this paper is, using the analytic methods and the properties of Gauss sums, to study this problem and give a sharp upper bound estimate for (1.1). That is, we prove the following conclusions.
where denotes the principal character modp.
If , then the above formula also holds for , where denotes the Legendre symbol.
Taking and in Theorem 3, note that , we may immediately deduce the following.
where r and s are any two integers such that .
This gives another proof for a classical work in elementary number theory (i.e., see  Theorems 4-11): For any prime p with , there exist two positive integers x and y such that .
2 Several lemmas
To complete the proof of our theorems, we need the following basic lemmas.
where , denotes the principal character modp, denotes any k-order character modp and .
Now Lemma 1 follows from (2.1), (2.2) and (2.3). □
where denotes the Legendre symbol, and .
This proves formula (I).
This proves Lemma 2. □
3 Proof of the theorems
Now note that , Theorems 1 and 2 follow from (3.1), (3.2) and (3.4).
This proves Theorem 3.
where r and s are any two integers such that .
This completes the proof of our corollary.
The authors would like to thank the referee for carefully examining this paper and providing a number of important comments. This work is supported by the N.S.F. of P.R. China (11071194, 61202437), and by the Youth Science and Technology Innovation Foundation of Xi’an Shiyou University (2012QN012).
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