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An identity involving the mean value of two-term character sums
Journal of Inequalities and Applications volume 2013, Article number: 533 (2013)
Abstract
The main purpose of this paper is, using the properties of Gauss sums and the estimate for character sums, to study the mean value problem of the two-term character sums and give an interesting identity for it.
MSC:11M20.
1 Introduction
Let be an integer, χ be a non-principal character . For any integer n with , the two-term character sums are defined as
where are two fixed positive integers. These sums play a very important role in the study of analytic number theory, so they caused many number theorists’ interest and favor. Some works related to can be found in [1–4]. If fact, the sums is a special case of the general character sums of the polynomials
where M and N are any positive integers, and is a polynomial. If is an odd prime, then Weil (see [1]) obtained the following important conclusion: Let χ be a q th-order character . If is not a perfect q th power , then we have the estimate
where ‘≪’ constant depends only on the degree of .
Now we are concerned about whether there exists a computational formula for the mean value
where are two integers. In this paper, we shall use the analytic method and the properties of Gauss sums to study this problem, and give an exact computational formula for (3). That is, we shall prove the following theorem.
Theorem 1 Let be an odd integer. Then, for any real number k and integer n with , we have the identity
where denotes the summation over all primitive characters , denotes that and , and denotes the number of all primitive characters .
Theorem 2 Let be an odd integer. Then, for any real number k and integer n with , we have the identity
From these two theorems we may immediately deduce the following corollaries.
Corollary 1 Let q be an odd square-full number (that is, for any prime p, if , then ). Then, for any real number k and integer n with , we have the identity
Corollary 2 Let q be an odd square-free number (that is, for any prime p, ). Then, for any real number k and integer n with , we have the identity
Corollary 3 Let q be an odd square-full number. Then, for any integer n with , we have the identity
For general characters , whether there exists an identity for
is an interesting open problem, where are two positive integers.
2 Several lemmas
In this section, we shall give several lemmas, which are necessary in the proof of our theorems. Hereinafter, we shall use many properties of character sums and Gauss sums, all of which can be found in references [5] and [6], so they will not be repeated here. First we have the following lemmas.
Lemma 1 Let be an integer, denotes the number of all primitive characters . Then we have the identity
where is the Möbius function, is the Euler function.
Proof This is a well-known result, here we give a simple proof. It is clear that for any non-principal character , there exists one and only one and a primitive character such that , where denotes the principal character . So, from these properties, we have
From this identity and the Möbius inversion formula, we may immediately deduce
This proves Lemma 1. □
Lemma 2 Let p be an odd prime, be an integer and . Then, for any real number k and integer n with , we have the identities
and
Proof For any primitive character , from the properties of Gauss sums, we know that
Note that and , so if χ is a primitive character , then is also a primitive character , so that . From these identities, Lemma 1 and formula (4), we may immediately deduce that
This proves the first formula of Lemma 2.
Now we prove the second formula. If χ is the Legendre symbol , then we have . So, from Lemma 1 and the method of proving (4), we have
This completes the proof of Lemma 2. □
Lemma 3 Let be an odd prime, be an integer and . Then, for any real number k and n with , we have the identities
and
Proof From the properties of primitive characters and the method of proving Lemma 2, we have
Since χ is a primitive character and , so and are also two primitive characters . So, from Lemma 1, (5) and the properties of Gauss sums, we can deduce the first identity of Lemma 3.
Note that if , is the Legendre symbol , then for any non-principal character , we have and . So, from Lemma 1 and the method of proving (5), we have
If , then there exist two 3-order characters , so from the method of proving (6), we have the identity
Combining (6) and (7), we can deduce the second identity of Lemma 3. □
Lemma 4 Let and be two integers with , and . Then, for any integer n with , we have
Proof From the properties of complete residue system , we have
This proves Lemma 4. □
3 Proof of the theorems
In this section, we shall complete the proof of our theorems. First we prove Theorem 1. For any odd number , it is clear that there exist two integers M and N such that , where M is a square-free number, and N is a square-full number. Now, for any primitive character , there exist two primitive characters and such that . Note that , so from these properties, Lemma 1, Lemma 2 and Lemma 4, we have
where denotes that and . This proves Theorem 1.
Now we prove Theorem 2. From Lemma 3, Lemma 4 and the method of proving Theorem 1, we have
This completes the proof of our theorems.
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Acknowledgements
The authors would like to thank the referee for their very helpful and detailed comments, which have significantly improved the presentation of this paper. This work is supported by the National Natural Science Foundation of P.R. China (No. 11071194).
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Guo, X., Wang, J. An identity involving the mean value of two-term character sums. J Inequal Appl 2013, 533 (2013). https://doi.org/10.1186/1029-242X-2013-533
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DOI: https://doi.org/10.1186/1029-242X-2013-533