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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 750964, 8 pages
http://dx.doi.org/10.1155/2013/750964
Research Article

On a Kind of Dirichlet Character Sums

1School of Science, Northwestern Polytechnical University, Xi'an, Shaanxi 710072, China
2The School of Electronic and Information Engineering, Xi'an Jiaotong University, Xi'an, Shaanxi 710049, China

Received 16 March 2013; Accepted 2 August 2013

Academic Editor: Hassan Eltayeb

Copyright © 2013 Rong Ma and Yulong Zhang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Let be a prime and let denote the Dirichlet character modulo . For any prime with , define the set . In this paper, we study a kind of mean value of Dirichlet character sums , and use the properties of the Dirichlet -functions and generalized Kloosterman sums to obtain an interesting estimate.

1. Introduction

Let be an integer and let denote the Dirichlet character modulo , for any real number , many scholars have studied the following sums: where are positive integers.

Perhaps one of the most famous results is Pólya’s inequality [1]. That is, when is the primitive character modulo , we have

In fact, the result can be extended to the nonprincipal character modulo [2]. Further details about the estimates of character sums can be found in the literature, for example, [3, 4].

For any fixed integer and any positive integer , define the following set: let denote the Dirichlet character modulo , define the sums as follows:

Xi and Yi [5] studied the problem for the nonprincipal Dirichlet character modulo , and got where was a constant and was the divisor function. Before this, Wenpeng [6] got an asymptotic formula for the case that was the principal Dirichlet character modulo .

On the other hand, for each integer with and , we know that there exists one and only one with such that . Let be the number of solutions of the congruent equation for , in which and are of opposite parity, this can be expressed as follows:

Richard [7] asks us to find or at least to say something nontrivial about it. About this problem, a lot of scholars have studied it [812]. Now we let be another integer with and let denote the number of all pairs of integers , satisfying , , , and . Lu and Yi [13] have obtained the asymptotic formula of generalized D. H. Lehmer problem as follows: where the constant only depends on .

In this paper, let be an odd prime and let be a fixed prime with , define the set for such that and , that is,

As another case of (7), we will consider the mean value of Dirichlet character sums as follows: and get an interesting estimate. That is, we will prove the following theorem.

Theorem 1. Let be an odd prime and let be a fixed prime with , and let denote the Dirichlet character modulo . Let denote the following set: then, for any nonprincipal Dirichlet character , we have the following estimate: where the constant only depends on .
From this Theorem we can get
For any integer and fixed integer such that , whether or not there exists an estimate for is still an open problem.

2. Some Lemmas

In this section, we will give several lemmas which are necessary in the proof of the theorem.

Lemma 2. Let be an integer, and let be a primitive character modulo . Then, for any real number and with , we have where and are Gauss sums.
Especially, let , we have a slight modification

Proof. (See [1]).

Lemma 3. Let be a prime, let be an integer with , and let be a primitive character modulo , then, we have where are the Dirichlet -functions corresponding to .

Proof. From Lemma 2, we take , and and get
When , we have
Let , then, we have
The case of can be treated in the same way. This proves Lemma 3.

Lemma 4. Let be odd primes and let be the Dirichlet characters modulo and , respectively, such that , denote , , and is the Dirichlet character modulo , the famous Gauss sums are defined as follows: where . Hence, we have
When , we denote ; therefore, we have

Proof. (See [14]).

Lemma 5. Let , and be integers and let be prime, let denote the Dirichlet character modulo , the generalized Kloosterman sums are defined by where and .
Then, we have the following estimate: where denotes the gcd of , and .

Proof. (See [15]).

Lemma 6. Let be an odd prime, let , be a Dirichlet character modulo , and . For any odd prime with , let be any Dirichlet characters with , and , respectively, then, no matter is odd character or even character modulo , we have where the constant only depends on .

Proof. For any integer with ( is any positive integer), we have
Now let and let . Then, from the Pólya-Vinogradov inequality, we obtain
Hence, from Abel’s identity, for any Re, we can easily get We will take (25), for example, to prove this lemma. For , from the definition of and (31), since is not the principle character modulo and is not the principle character modulo , we have where . From Lemma 5, we can easily obtain where the constant only depends on . Therefore, this completes the proof of Lemma 6.

Lemma 7. Let be a fixed odd prime and let be a prime with , let denote the nonprincipal Dirichlet character modulo , and denote the Dirichlet character modulo , then, we have where the constant only depends on .

Proof. For primes and , and are nonprincipal and primitive characters modulo , hence, from Lemmas 3 and 6, when , we can get where the constant is only concerned with .
When , by the similar method, we can also obtain where the constant is only concerned with . Combining (35) and (36), we can obtain Lemma 7. This completes the proof of Lemma 7.

Lemma 8. Let be a fixed odd prime and let be a prime with , let denote the nonprincipal Dirichlet character modulo , let denote the Dirichlet character modulo respectively, then, we have where the constant is only concerned with .

Proof. According to the properties of Dirichlet character, we can get
For primes and , let be a nonprincipal and primitive character modulo , is also a primitive character modulo , so is a primitive character modulo ; therefore, from Lemmas 3, 4, and 6 and from (38), it is clear that when , we can obtain where the constant is only concerned with .
When , in the similar way, we can also obtain where the constant is only concerned with .
Therefore, from (39) and (40), we can easily get Lemma 8. This completes the proof of Lemma 8.

3. Proof of Theorem

In this section, we will complete the proof of the theorem. According to the orthogonality relation for character sums, we have

Note that for any nonprincipal Dirichlet character modulo ( is an positive integer), we have , hence, we obtain From (41) and (42) and Lemmas 7 and 8, we get where the constant only depends on . This completes the proof of Theorem.

Acknowledgments

The authors greatly appreciate the referees for their kind comments and suggestions. This work is supported by the Basic Research Fund of Northwestern Polytechnical University of P. R. China (JC201123).

References

  1. G. Pólya, “Über die Verteilung der quadratische Reste und Nichtreste,” Quadratischer Nichtrest, pp. 21–29, 1918.
  2. L. K. Hua and S. H. Min, “On a double exponential sum,” Science Record, vol. 1, pp. 23–25, 1942.
  3. D. A. Burgess, “On character sums and L-series. II,” Proceedings of the London Mathematical Society, vol. 13, pp. 524–536, 1963.
  4. D. A. Burgess, “The character sum estimate with r=3,” Journal of the London Mathematical Society, vol. 33, pp. 219–226, 1986.
  5. P. Xi and Y. Yi, “On character sums over flat numbers,” Journal of Number Theory, vol. 130, no. 5, pp. 1234–1240, 2010. View at Publisher · View at Google Scholar · View at Scopus
  6. Z. Wenpeng, “On the distribution of inverses modulo n,” Journal of Number Theory, vol. 61, no. 2, pp. 301–310, 1996. View at Publisher · View at Google Scholar · View at Scopus
  7. K. G. Richard, Unsolved Problems in Number Theory, Springer, 1981.
  8. Z. Xu and W. Zhang, “On a problem of D. H. Lehmer over short intervals,” Journal of Mathematical Analysis and Applications, vol. 320, no. 2, pp. 756–770, 2006. View at Publisher · View at Google Scholar · View at Scopus
  9. W. Zhang, X. Zongben, and Y. Yuan, “A problem of D. H. Lehmer and its mean square value formula,” Journal of Number Theory, vol. 103, no. 2, pp. 197–213, 2003. View at Publisher · View at Google Scholar · View at Scopus
  10. W. Zhang, “A problem of D. H. Lehmer and its generalization,” Compositio Mathematica, vol. 86, pp. 307–316, 1993.
  11. R. Ma and Y. Zhang, “On a kind of generalized Lehmer problem,” Czechoslovak Mathematical Jounal, vol. 62, no. 137, pp. 1135–1146, 2012.
  12. W. Zhang, “On the difference between a D. H. Lehmer number and its inverse modulo q,” Acta Arithmetica, vol. 68, no. 3, pp. 255–263, 1994.
  13. Y. M. Lu and Y. Yi, “On the generalization of the D. H. Lehmer problem,” Acta Mathematica Sinica, vol. 25, no. 8, pp. 1269–1274, 2009. View at Publisher · View at Google Scholar · View at Scopus
  14. P. Chengdong and P. Chengbiao, Elements of the Analytic Number Theory, Science Press, Beijing, China, 1991, (Chinese).
  15. A. V. Malyshev, “A generalization of Kloosterman sums and their estimates,” Vestnik Leningrad University, vol. 15, no. 13, pp. 59–75, 1960.