Frontiers of Mathematics in China >

2012 , Vol. 7 >Issue 4: 765 - 783

DOI: https://doi.org/10.1007/s11464-012-0222-5

RESEARCH ARTICLE

Estimate for exponential sums and its applications

  • Weili YAO , 1,2
Expand
  • 1. Department of Mathematics, College of Sciences, Shanghai University, Shanghai 200444, China
  • 2. School of Mathematics, Shandong University, Jinan 250100, China

Received date: 29 Mar 2012

Accepted date: 12 May 2012

Published date: 01 Aug 2012

Copyright

2014 Higher Education Press and Springer-Verlag Berlin Heidelberg
Fold

Abstract

In this paper, we establish a new estimate on exponential sums by using the Bombieri-type theorem and the modified Huxley-Hooley contour. We also generalize the famous Goldbach-Vinogradov theorem, via different argument from that of Vinogradov. In particular, our major arcs are quite large and these enlarged major arcs are treated by the estimate we have established.

Key words： Exponential sum; zero-density estimate; circle method

Cite this article

Weili YAO . Estimate for exponential sums and its applications[J]. Frontiers of Mathematics in China, 2012 , 7(4) : 765 -783 . DOI: 10.1007/s11464-012-0222-5

References

Publishing order | Descend order by publishing year | Descend order by cited within
1
Davenport H. Analytic Methods for Diophantine Equations and Diophantine Inequalities. Cambridge: Cambridge University Press, 2005

2
Karatsuba A A. Basic Analytic Number Theory. Berlin: Springer-Verlag, 1993

3
Liu Jianya. A large sieve estimate for Dirichlet polynomials and its applications. Ann Univ Sci Budapest Sect Comput, 2007, 27: 91-110

4
Liu Jianya, Zhan Tao. The ternary Goldbach problem in arithmetic progressions. Acta Arith, 1997, 82: 197-227

5
Meng Xianmeng. On sums of three integers with a fixed number of prime factors. J Number Theory, 2005, 114: 37-65

DOI

6
Pan Chengdong, Pan Chengbiao. Goldbach Conjecture. Beijing: Science Press, 1984 (in Chinese)

7
Ramachandra R. Some problems of analytic number theory. Acta Arith, 1976, 31: 313-324

8
Titchmarsh E C. The Theory of the Riemann Zeta-function. Oxford: Oxford University Press, 1986

9
Vinogradov I M. Representation of an odd numbers as the sum of three primes. Dokl Akad Nauk SSSR, 1937, 16: 139-142

10
Wolke D, Zhan Tao. On the distribution of integers with a fixed number of prime factors. Math Z, 1993, 213: 133-147

DOI

11
Yao Weili. A Bombieri-type theorem for exponential sums. Acta Math Sin (Engl Ser) (to appear)

Options
Outlines

/

〈 〉