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

In: International Symposium in Memory of Hua Loo Keng

Author

Listed:
  • Jingrun Chen

    (Academia Sinica, Institute of Mathematics)

  • Jianmin Liu

    (Academia Sinica, Institute of Mathematics)

Abstract

Let D be a large positive integer, (K, D) = 1, and P(D, K) the least prime p ≡ K (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).

Suggested Citation

  • Jingrun Chen & Jianmin Liu, 1991. "On the Least Prime in an Arithmetical Progression and Theorems Concerning the Zeros of Dirichlet’s L-Functions ( V )," Springer Books, in: Sheng Gong & Qi-Keng Lu & Yuan Wang & Lo Yang (ed.), International Symposium in Memory of Hua Loo Keng, pages 19-42, Springer.
    • Handle: RePEc:spr:sprchp:978-3-662-07981-2_3
    DOI: 10.1007/978-3-662-07981-2_3
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