Explicit Deuring-Heilbronn phenomenon for Dirichlet $L$-functions

TL;DR

This work proves an explicit Deuring–Heilbronn-type zero repulsion for Dirichlet $L$-functions under the assumption of a Landau–Siegel zero. The authors develop a mollified Selberg sieve framework that jointly exploits the exceptional character and a Graham-style sieve to obtain two competing zero-detection estimates, yielding a uniform bound across the critical strip. Central to the result are explicit subconvexity and Siegel-zero hypotheses, together with careful uniform tracking of dependencies on $q$ and vertical height $T$, enabling fully explicit constants in the main bound. The paper derives both an explicit (Corollary 1.1) and a non-explicit (Corollary 1.2) form of the Deuring–Heilbronn phenomenon, improving upon previous explicit bounds such as those of Thorner–Zaman. The methods provide a transparent path from subconvexity inputs to explicit zero-free regions in the presence of a Landau–Siegel zero, with potential applications to zero-density results and where explicit constant control is crucial.

Abstract

Assuming the existence of a Landau-Siegel zero, we establish an explicit Deuring-Heilbronn zero repulsion phenomenon for Dirichlet $L$-functions modulo $q$. Our estimate is uniform in the entire critical strip, and improves over the previous best known explicit estimate due to Thorner and Zaman.

Theorems & Definitions (27)

• Corollary 1.1
• Corollary 1.2
• Theorem 1.3
• Theorem 2.2: Petrow--Young
• proof
• Proposition 2.3: Thorner--Zaman
• proof
• Remark
• Lemma 2.4
• proof