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Primes with restricted digits

James Maynard

Abstract

Let $a_0\in\{0,\dots,9\}$. 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.

Primes with restricted digits

Abstract

Let . We show there are infinitely many prime numbers which do not have the digit 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.

Paper Structure

This paper contains 17 sections, 33 theorems, 373 equations.

Table of Contents

  1. Introduction
  2. Outline
  3. Notation
  4. Structure of the paper
  5. Basic estimates
  6. Sieve Decomposition and proof of Theorem \ref{['thrm:MainTheorem']}
  7. Sieve Asymptotics
  8. Type I estimate
  9. Type II estimate
  10. Fourier Estimates
  11. Major arcs
  12. Generic minor arcs
  13. Exceptional minor arcs
  14. Lattice Estimates
  15. Line Estimates
  16. ...and 2 more sections

Key Result

Theorem 1.1

Let $X\ge 4$ and $\mathcal{A}=\{\sum_{0\le i\le k}n_i10^i< X:\, n_i\in\{0,\dots,9\}\backslash\{a_0\},\,k\ge 0\}$ be the set of numbers less than $X$ with no digit in their decimal expansion equal to $a_0$. Then we have

Theorems & Definitions (35)

  • Theorem 1.1
  • Theorem 1.2
  • Remark
  • Proposition 6.1: Sieve asymptotic terms
  • Proposition 6.2: Type II terms
  • Proposition 7.1: Type I estimate
  • Proposition 7.2: Type II estimate
  • Lemma 7.3: Type II terms, alternative formulation
  • Lemma 7.4: Fundamental Lemma
  • Lemma 8.1: Large sieve estimate
  • ...and 25 more