Type: Article
Publication Date: 1999-01-01
Citations: 29
DOI: https://doi.org/10.4064/aa-89-2-163-187
x. This result has been improved consecutively by many authors. The best estimate known to date is very far from the expected result. The historical records are as follows: P (x) x 0.625 by Ramachandra [20, II], P (x) x 0.662 by Graham [8], P (x) x 0.692 by Jia [16, I], P (x) x 0.7 by Baker [1], P (x) x 0.71 by Jia [16, II], P (x) x 0.723 by Jia [16, III] and Liu [18], P (x) x 0.728 by Jia [16, IV], P (x) x 0.732 by Baker and Harman [2]. We note that the last two papers are independent. In both, the same estimates for exponential sums were used. But Baker and Harman [2] introduced the alternative sieve procedure, developed by Harman [10] and by Baker, Harman and Rivat [3], to get a better exponent. In this paper we shall prove a sharper lower bound. Theorem 1. We have P (x) x 0.738 for suciently large x.