Frontiers of Mathematics in China

, Volume 12, Issue 2, pp 261–280 | Cite as

Large sieve inequality with sparse sets of moduli applied to Goldbach conjecture

Research Article
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Abstract

We use the large sieve inequality with sparse sets of moduli to prove a new estimate for exponential sums over primes. Subsequently, we apply this estimate to establish new results on the binary Goldbach problem where the primes are restricted to given arithmetic progressions.

Keywords

Three primes theorem exponential sums over primes sparse set of modules 

MSC

11F32 11F25 

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References

  1. 1.
    Baier S, Zhao L Y. Bombieri-Vinogradov type theorems for sparse sets of moduli. Acta Arith, 2006, 125(2): 287–301MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Balog A, Perelli A. Exponential sums over primes in an arithmetic progression. Proc Amer Math Soc, 1985, 93: 578–582MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Bauer C. The binary Goldbach conjecture with restrictions on the primes. Far East Asian J Math Sci, 2012, 70(1): 87–120MathSciNetMATHGoogle Scholar
  4. 4.
    Bauer C, Wang Y H. On the Goldbach conjecture in arithmetic progressions. Rocky Mountain J Math, 2006, 36(1): 35–66MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Bauer C, Wang Y H. The binary Goldbach conjecture with primes in arithmetic progressions with large modulus. Acta Arith, 2013, 159.3: 227–243MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Bombieri E. Le grand crible dans la théorie analytique des nombres. Astérique, 18, 1974Google Scholar
  7. 7.
    Gallagher P X. A large sieve density estimate near σ = 1. Invent Math, 1970, 11: 329–339MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Ireland K, Rosen M. A Classical Introduction to Modern Number Theory. 2nd ed. Grad Texts in Math, Vol 84. Berlin: Springer, 1990Google Scholar
  9. 9.
    Liu J Y, Zhan T. The ternary Goldbach problem in arithmetic progressions. Acta Arith, 1997, 82: 197–227MathSciNetMATHGoogle Scholar
  10. 10.
    Liu J Y, Zhan T. The exceptional set in Hua’s theorem for three squares of primes. Acta Math Sin (Engl Ser), 2005, 21(2): 335–350MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Liu M C, Zhan T. The Goldbach problem with primes in arithmetic progressions. In: Motohashi Y, ed. Analytic Number Theory. London Math Soc Lecture Note Ser, Vol 247. Cambridge: Cambridge University Press, 1997, 227–251MathSciNetMATHGoogle Scholar
  12. 12.
    Norton K K. Numbers with Small Prime Factors, and the Least kth Power Non-residue. Mem Amer Math Soc, No 106. Providence: Amer Math Soc, 1971Google Scholar
  13. 13.
    Ren X M. On exponential sum over primes and application in Waring-Goldbach problem. Sci China Ser A, 2005, 48(6): 785–797MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Higher Education Press and Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  1. 1.Dolby LaboratoriesBeijingChina

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