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The Distribution of α p Modulo One
Author(s) -
HeathBrown D.R.,
Jia C.
Publication year - 2002
Publication title -
proceedings of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.899
H-Index - 65
eISSN - 1460-244X
pISSN - 0024-6115
DOI - 10.1112/plms/84.1.79
Subject(s) - mathematics , exponent , irrational number , riemann hypothesis , modulo , kloosterman sum , number theory , exponential sum , distribution (mathematics) , trigonometry , exponential function , prime number , discrete mathematics , combinatorics , pure mathematics , mathematical analysis , geometry , philosophy , linguistics
In this paper the authors prove the following result. Let α be an irrational number. Then for any ε > 0, there are infinitely many prime numbers p such that ∣ α p ∣ < p −16/49+ε . The exponent 16 49 improves on 9 28 , which was obtained recently by the second author [ Sci. China Ser. A 43 (2000) 703‐721]. The result is very close to the exponent 1 3 , which can be obtained under the Generalized Riemann Hypothesis. Previous approaches to this problem have all used the same basic estimates for the trigonometric sums that arise. However, the present proof uses new bounds, which depend on the Kloosterman sum and also on a counting problem in the geometry of numbers. In addition new techniques for the sieve method are applied. The most significant feature of the new approach is that, unlike previous methods, the exponential sum estimates remain non‐trivial for the exponent 1 3 . This gives one hope for an unconditional result as good as that available under the Generalized Riemann Hypothesis. 2000 Mathematical Subject Classification : 11N36, 11J71.