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Upper bound estimate of character sums over Lehmer’s numbers
Journal of Inequalities and Applications volume 2013, Article number: 392 (2013)
Let p be an odd prime. For each integer a with , it is clear that there exists one and only one with such that . Let denote the set of all integers , in which a and are of opposite parity. The main purpose of this paper is using the analytic method and the properties of Kloosterman sums to study the estimate problem of the mean value , and give a sharp upper bound estimate for it, where χ denotes any non-principal even character .
Let be an odd number. For each integer a with and , it is clear that there exists one and only one b with such that . Let denote the set of cases, in which a and b are of opposite parity. For , an odd prime, professor Lehmer  asked to study or at least to say something nontrivial about it, where p is a prime, and denote the number of all elements in . We call such a number a Lehmer’s number. It is known that when . For general odd number , Zhang ( and ) studied the asymptotic properties of , and obtained a sharp asymptotic formula for it. That is, he proved the asymptotic formula
where is the Euler function, and is the Dirichlet divisor function.
Let denote the number of all integers , such that and , and . Then Zhang  also studied the mean value properties of , and proved that
In this paper, we consider the estimate problem of the character sums
and give a sharper upper bound estimate for it, where χ denotes any non-principal even character .
About character sums (1.1), it is clear that its value is zero, if χ is an odd character . If χ is a non-principal even character , then how large is the upper bound estimate of (1.1)? About this problem, it seems that none had studied it yet, at least we have not seen any related results. The problem is interesting, because it can help us to understand the deep properties of the character sums over some special sets, for example, Lehmer’s numbers.
The main purpose of this paper is using the analytic method and the properties of Kloosterman sums to study this problem, and prove the following result.
Theorem Let be an odd prime. Then for any non-principal character , we have the estimate
For general odd number , whether there exists a similar upper bound estimate for (1.1) is an interesting problem, we will further study it.
2 Several lemmas
In this section, we will give several lemmas, which are necessary in the proof of our theorem. Hereinafter, we will use many properties of character sums, all of which can be found in , so they will not be repeated here. First we have the following.
Lemma 1 Let be an odd number. Suppose that χ is an odd character , then we have the identity
Proof See reference . □
Lemma 2 Let be an odd number. Then for any even character , we have the identity
where denotes the summation over all odd characters .
Proof From the orthogonality relation for character sums and the definition of we have
where denotes the summation over all such that .
If , then
If , then
Note that if and , then . So combining (2.1), (2.2), (2.3) and Lemma 1, we have
This proves Lemma 2. □
Lemma 3 Let p be an odd prime, let χ be any character . Then for any integers m and n, we have the estimate
where , denotes the greatest common divisor of m, n and p.
Lemma 4 Let p be an odd prime, let χ be any even character . Then for any integer c with , we have the estimate
Proof For any non-principal character , applying Abel’s identity (see Theorem 4.2 of ), we have
From , we know that for any real number , we have the estimate
From (2.4), (2.5), Lemma 3, the orthogonality relation for character sums , the definition of Gauss sums, and noting that , we have
This proves Lemma 4. □
3 Proof of the theorems
In this section, we will complete the proof of our theorem. First, if , then from Theorems 12.11 and 12.20 of , we have
From (3.1), Lemma 2 and Lemma 4, we have
This completes the proof of our theorem.
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The authors would like to thank the referees for their very helpful and detailed comments, which have significantly improved the presentation of this paper. This work is supported by the N.S.F. (11071194, 11001218) of P.R. China and G.I.C.F. (YZZ12062) of NWU.
The authors declare that they have no competing interests.
DH studied the estimate problem of the mean value , and gave a sharp upper bound estimate for it. WZ participated in the research and summary of the study.
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Han, D., Zhang, W. Upper bound estimate of character sums over Lehmer’s numbers. J Inequal Appl 2013, 392 (2013). https://doi.org/10.1186/1029-242X-2013-392
- Lehmer’s numbers
- character sums
- Kloosterman sums
- upper bound estimate