Selberg's sieve of irregular density

TL;DR

This paper advances the Selberg sieve method to handle thin prime sets, providing new lower bound results and a novel proof of Linnik's theorem in cases with exceptional zeros.

Contribution

It introduces refined lower bound sieve techniques tailored for thin prime sets and applies them to give a new proof of Linnik's theorem involving exceptional zeros.

Findings

New lower bound sieve results for thin prime sets

A sieve-based proof of Linnik's theorem with exceptional zeros

Enhanced understanding of Selberg's sieve in irregular density contexts

Abstract

We study certain aspects of the Selberg sieve, in particular when sifting by rather thin sets of primes. We derive new results for the lower bound sieve suited especially for this setup and we apply them in particular to give a new sieve-propelled proof of Linnik's theorem on the least prime in an arithmetic progression in the case of the presence of exceptional zeros.