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Prime numbers with a positive proportion of preassigned digits
Author(s) -
Swaenepoel Cathy
Publication year - 2020
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.12314
Subject(s) - mathematics , base (topology) , prime (order theory) , generalization , constant (computer programming) , zero (linguistics) , combinatorics , dirichlet distribution , prime number , arithmetic , discrete mathematics , mathematical analysis , linguistics , philosophy , computer science , boundary value problem , programming language
Bourgain [ Israel J. Math . 206 (2015) 165–182] estimated the number of prime numbers with a proportion c > 0 of preassigned digits in base 2 ( c is an absolute constant not specified). We establish a generalization of this result in any base g ⩾ 2 and we provide explicit admissible values for the proportion c depending on g . Our proof, which develops and enhances Bourgain's arguments, is based on the circle method and combines techniques from harmonic analysis together with results on zeros of Dirichlet L ‐functions, notably a very strong zero‐free region due to Iwaniec.