Prime numbers with a positive proportion of preassigned digits | Zendy

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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.

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