On the Least Prime in an Arithmetical Progression and Theorems Concerning the Zeros of Dirichlet’s L-Functions ( V )

  • Jingrun Chen
  • Jianmin Liu

Abstract

Let D be a large positive integer, (K, D) = 1, and P(D, K) the least prime pK (mod D). In 1934, S. Chowla conjectured that P(D, K) ≪ D 1+ε . Three years later, P. Turan proved that under the General Riemann Conjecture, the Chowla’s conjecture may hold for almost all modulo D. On the other hand, in 1949, Erdös obtained that, first, there is a constant number C 2= C 2 (C 1) and an infinity of integer D such that P(D, K) > (1+C 1) φ (D) log D for K’s value being at least C 2 φ (D); second, there is a constant C 4 = C 4 (C 3), such that P(D, K) ≤ C 3 φ (D)logD for K’s value of C 4 φ (D).

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    Pan Chengdong, On the least prime in an arithmetical progression, Sci. Record (N. S.), 1(1958), 311–313.Google Scholar
  2. [2]
    Chen Jingrun, On the least prime in an arithmetical progression, Sci. Sin.. 14(1964), 1868–1871.Google Scholar
  3. [3]
    Matti Jutila, Proc. of Sym. in Pure Math. AMS., (1971), 370.Google Scholar
  4. [4]
    Matti Jutila, A new estimate for Linnik’s constant, Ann. Acad. Sci. Fennicae 471(1970)8pp. Google Scholar
  5. [5]
    Chen Jingrun, On the least prime in an arithmetical progression and two theorems concerning the zeros of Dirichlet’s L-functions (I), Sci. Sin.,20(1977), 529-562.Google Scholar
  6. [6]
    Matti Jutila, On the Linnik’s constant, Math. Scand,41(1977), 54–62.Google Scholar
  7. [7]
    S. Graham, Applications of Sieve Methods, Ph. D. Thesis, University of Michigan, 1977.Google Scholar
  8. [8]
    S. Graham, On Linnik’s constant, Acta Arith.,39(1981), 163-179.Google Scholar
  9. [9]
    Chen Jingrun, On the least prime in an arithmetical progression and two theorems concerning the zeros of Dirichlet’s L-functions (II), Sci. Sin.,22(1979), 859–889.Google Scholar
  10. [10]
    Wang Wei, On the least prime in an arithmetic progression, Acta Math. Sin.,29(1986), 826-836.Google Scholar
  11. [11]
    Chen Jingrun & Liu Jianmin, On the least prime in an arithmetical progression and theorems concerning the zeros of Dirichlet’s L-functions (III), (IV), Sci. Sin.. (1989) to appear.Google Scholar
  12. [12]
    Chen Jingrun & Liu Jianmin, The exceptional set of Goldbach numbers (III), to appear.Google Scholar
  13. [13]
    Chen Jingrun, The exceptional set of Goldbach numbers (II), Sci. Sin.,26(1983), 714–731.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1991

Authors and Affiliations

  • Jingrun Chen
    • 1
  • Jianmin Liu
    • 1
  1. 1.Institute of MathematicsAcademia SinicaBeijingChina

Personalised recommendations