An identity involving the mean value of two-term character sums
© Guo and Wang; licensee Springer. 2013
Received: 6 July 2013
Accepted: 17 October 2013
Published: 12 November 2013
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.
where ‘≪’ constant depends only on the degree of .
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.
where denotes the summation over all primitive characters , denotes that and , and denotes the number of all primitive characters .
From these two theorems we may immediately deduce the following corollaries.
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  and , so they will not be repeated here. First we have the following lemmas.
where is the Möbius function, is the Euler function.
This proves Lemma 1. □
This proves the first formula of Lemma 2.
This completes the proof of Lemma 2. □
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.
Combining (6) and (7), we can deduce the second identity of Lemma 3. □
This proves Lemma 4. □
3 Proof of the theorems
where denotes that and . This proves Theorem 1.
This completes the proof of our theorems.
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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