An explicit version of Bombieri's log-free density estimate and Sárközy's theorem for shifted primes

TL;DR

The paper delivers an explicit log-free zero-density estimate for Dirichlet $L$-functions by refining Gallagher’s large sieve approach and integrating Green’s unconditional machinery. It introduces an explicit zero-detection framework that connects near-line zeros to prime-sum estimates and then converts detection into explicit zero-count bounds via a pre-sifted large sieve. These explicit bounds, including an enhanced Deuring–Heilbronn phenomenon, drive a power-saving in Sárközy’s theorem for shifted primes: any $A\subseteq\{1,\dots,N\}$ with no two elements differing by $p-1$ satisfies $|A|\ll N^{1-1/10^{18}}$. The results provide fully explicit constants in the zero-density and sieve steps, improving prior non-explicit or conditional bounds and enhancing the applicability of zero-density methods to additive problems in shifted primes.

Abstract

We make explicit Bombieri's refinement of Gallagher's log-free "large sieve density estimate near $σ= 1$" for Dirichlet $L$-functions. We use this estimate and recent work of Green to prove that if $N\geq 2$ is an integer, $A\subseteq\{1,\ldots,N\}$, and for all primes $p$ no two elements in $A$ differ by $p-1$, then $|A|\ll N^{1-1/10^{18}}$. This strengthens a theorem of Sárközy.

Theorems & Definitions (58)

• Theorem 1.1
• Theorem 1.2
• Remark
• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Lemma 2.3
• proof
• Lemma 2.4