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GENERALIZED D. H. LEHMER PROBLEM OVER SHORT INTERVALS

Published online by Cambridge University Press:  13 December 2010

PING XI  and
YUAN YI
PING XI
Affiliation:
School of Science, Xi'an Jiaotong University, Xi'an 710049, P. R. China email: xprime@163.com
YUAN YI
Affiliation:
School of Science, Xi'an Jiaotong University, Xi'an 710049, P. R. China and Department of Mathematics, The University of Iowa, Iowa City, IA 52242-1419, USA email: yuanyi@mail.xjtu.edu.cn
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Abstract

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Let n ≥ 2 be a fixed positive integer, q ≥ 3 and c, ℓ be integers with (nc, q)=1 and ℓ|n. Suppose and consist of consecutive integers which are coprime to q. We define the cardinality of a set: The main purpose of this paper is to use the estimates of Gauss sums and Kloosterman sums to study the asymptotic properties of N(, , c, n, ℓ; q), and to give an interesting asymptotic formula for it.

Keywords

Primary 11A0711N37Secondary 11L05
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 53 , Issue 2 , May 2011 , pp. 293 - 299
Copyright
Copyright © Glasgow Mathematical Journal Trust 2010

References

REFERENCES

1.Estermann, T., On Kloosterman's sum, Mathematika 8 (1961), 83–86.CrossRefGoogle Scholar
2.Guy, R. K., Unsolved Problems in Number Theory, 3rd. edn (Springer-Verlag, New York, 2004).CrossRefGoogle Scholar
3.Lu, Y. and Yi, Y., On the generalization of the D. H. Lehmer problem, Acta Math. Sinica, English Ser. 9 (2009), 1269–1274.CrossRefGoogle Scholar
4.Pan, C. D. and Pan, C. B., Goldbach Conjecture (Science Press, Beijing, 1981).Google Scholar
5.Shparlinski, I. E., On a generalised Lehmer problem for arbitrary powers, East-West J. Math. Special Vol. (2008), 197–204.Google Scholar
6.Shparlinski, I. E., On a generalisation of a Lehmer problem, Math. Zeitschrift 263 (2009), 619–631.CrossRefGoogle Scholar
7.Yi, Y. and Zhang, W., On the generalization of a problem of D. H. Lehmer, Kyushu J. Math. 56 (2002), 235–241.Google Scholar
8.Zhang, W., On D. H. Lehmer problem, Chin. Sci. Bull. 21 (1992), 1765–1769.Google Scholar
9.Zhang, W., A problem of D. H. Lehmer and its generalization, Compos. Math. 86 (1993), 307–316.Google Scholar
10.Zhang, W., A problem of D. H. Lehmer and its generalization(II), Compos. Math. 91 (1994), 47–56.Google Scholar