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

Jingrun Chen and Jianmin Liu
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Jingrun Chen: Academia Sinica, Institute of Mathematics
Jianmin Liu: Academia Sinica, Institute of Mathematics

A chapter in International Symposium in Memory of Hua Loo Keng, 1991, pp 19-42 from Springer

Abstract: 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).

Date: 1991
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DOI: 10.1007/978-3-662-07981-2_3

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