Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Primes with a missing digit: distribution in arithmetic progressions and an application in sieve theory

Kunjakanan Nath

Abstract

We prove Bombieri-Vinogradov type theorems for primes with a missing digit in their $b$-adic expansion for some large positive integer $b$. The proof is based on the circle method, which relies on the Fourier structure of the integers with a missing digit and the exponential sums over primes in arithmetic progressions. Combining our results with the semi-linear sieve, we obtain an upper bound and a lower bound of the correct order of magnitude for the number of primes of the form $p=1+m^2+n^2$ with a missing digit in a large odd base $b$.

Primes with a missing digit: distribution in arithmetic progressions and an application in sieve theory

Abstract

We prove Bombieri-Vinogradov type theorems for primes with a missing digit in their -adic expansion for some large positive integer . The proof is based on the circle method, which relies on the Fourier structure of the integers with a missing digit and the exponential sums over primes in arithmetic progressions. Combining our results with the semi-linear sieve, we obtain an upper bound and a lower bound of the correct order of magnitude for the number of primes of the form with a missing digit in a large odd base .

Paper Structure

This paper contains 36 sections, 40 theorems, 280 equations.

Table of Contents

  1. Main results and outline of the proof
  2. Introduction
  3. Set-up and outline of the proof
  4. Sieve methods and their applications
  5. Preliminaries from sieve methods
  6. Proof of Theorem \ref{['Main TheoremApp']}
  7. Upper bound in Theorem \ref{['Main TheoremApp']}
  8. Lower bound in Theorem \ref{['Main TheoremApp']}
  9. Sieve set-up for the lower bound.
  10. Auxiliary results
  11. Proof of Proposition \ref{['PropAppS']}
  12. Proof of Proposition \ref{['PropAppT']}
  13. Exponential sums
  14. Preliminary estimates and Type I estimate
  15. Type I estimate
  16. ...and 21 more sections

Key Result

Theorem 1

Let $\delta>0$ and let $b$ be an integer that is sufficiently large in terms of $\delta$. Let $D\in [1, X^{1/3-\delta}]$ and let $r\in \mathcal{A}\cap[0, b)$ be an integer such that $(r, b)=1$. Then for any $A>0$, we have

Theorems & Definitions (93)

  • Remark
  • Theorem 1
  • Theorem 2
  • Definition 1.1: Well-factorable
  • Theorem 3
  • Remark
  • Theorem 4
  • Remark
  • Remark
  • Definition 3.1: Upper bound sieve
  • ...and 83 more