Numbers with a large prime factor

Abstract

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.