Year: 2019
Type: article
Abstract: Let $$a_0\in \{0,\ldots ,9\}$$ . We show there are infinitely many prime numbers which do not have the digit $$a_0$$ in their decimal expansion. The proof is an application of the Hardy–Littlewood circle method to a binary problem, and rests on obtaining suitable ‘Type I’ and ‘Type II’ arithmetic information for use in Harman’s sieve to control the minor arcs. This is obtained by decorrelating Diophantine conditions which dictate when the Fourier transform of the primes is large from digital conditions which dictate when the Fourier transform of numbers with restricted digits is large. These estimates rely on a combination of the geometry of numbers, the large sieve and moment estimates obtained by comparison with a Markov process.
Source: Inventiones mathematicae
Author: James Maynard
Institution: University of Oxford
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $3,140
OpenAlex ID: W2341163736