Landau-Siegel Zero in Dirichlet L-functions

• Landau-Siegel zero is a hypothetical real zero of a Dirichlet L-function near s=1, potentially violating standard zero-free regions.
• Recent studies use explicit computations and zero repulsion phenomenons, like the Deuring–Heilbronn effect, to constrain its potential location.
• Assuming its existence, conditional results influence prime distribution, class numbers, and analytic ranks, steering future research directions.

A Landau–Siegel zero refers to a hypothetical real zero β very near s = 1 of a Dirichlet L-function attached to a real primitive character modulo q. If it exists, such a zero would violate commonly conjectured zero-free regions for L-functions and would have profound implications for analytic and algebraic number theory—affecting prime distribution, class numbers, and behavior of arithmetic functions. Theoretical and computational developments have focused both on ruling out the existence of Landau–Siegel zeros and on understanding their conditional consequences.

1. Historical and Theoretical Framework

The classical studies of Landau and Siegel introduced the concept of an "exceptional" or Landau–Siegel zero: a real zero β of a Dirichlet L-function L(s, χ) for a real primitive character χ modulo q, with β = 1 – 1/(n log q) where n > 0 quantifies the "quality" of the zero (Jaskari et al., 16 Sep 2024). Such zeros are predicted to be absent under the Generalized Riemann Hypothesis (GRH), but their possible existence remains a critical conditional in many analytic estimates. The effect of these "exceptional zeros" is captured in density theorems, lower bounds (e.g., Siegel's ineffective bound L(1, χ) ≫_ε D^{-ε}), and in zero repulsion phenomena such as Deuring–Heilbronn, which forces other zeros away from the real axis in the presence of a Landau–Siegel zero (Stopple, 2011).

2. Explicit and Quantitative Zero-Free Regions

Recent advances have provided increasingly explicit bounds on the location of possible Landau–Siegel zeros. For Dirichlet L-functions attached to quadratic characters modulo an odd prime q ≤ 10⁷, explicit computations yield L(1, χ) > c₁·log q and β < 1 – (c₂/ log q), where c₁ ≈ 0.0125 and c₂ ≈ 0.0092 (Languasco, 2023). Under various hypotheses (e.g., a uniform abc-conjecture for number fields), more refined regions are available, such as β < 1 – (√5·φ + o(1))/log|D| (Táfula, 2019). Zero-free regions for automorphic Rankin–Selberg L-functions have also been extended to GL_n, with at most one Landau–Siegel zero permitted; all other zeros are "repelled" with effective bounds explicit in terms of the analytic conductor (Humphries et al., 2020, Thorner, 13 Aug 2025).

3. Deuring–Heilbronn Phenomenon and Zero Repulsion

The existence of a Landau–Siegel zero induces the Deuring–Heilbronn phenomenon: a measurable "repulsion" of other nontrivial zeros in the critical strip. Explicit versions show that if there is a real zero β₁ > 1 – 1/(10 log q), all other zeros ρ = β + iγ with β > ½ and |γ| ≤ T satisfy

$$\beta < 1 - \frac{\log\Bigl(\theta/4(1-\beta_1)\log M\Bigr)}{\log M},$$

where M depends on q, T, and explicit parameters (Benli et al., 8 Oct 2024). The strength of the repulsion improves with the proximity of β₁ to 1, and is captured in refined sieve and mollification techniques tailored for the distribution of zeros in Dirichlet L-functions (Stopple, 2011, Benli et al., 8 Oct 2024).

4. Conditional Consequences for Arithmetic Functions and Conjectures

Assuming the existence of a Landau–Siegel zero enables conditional progress on otherwise intractable problems. For the Chowla conjecture, which asserts that $$\sum_{n \leq x} \lambda(n+h_1)\cdots\lambda(n+h_k) = o(x)$$ for the Liouville function, conditional bounds show rapid decay of k-point correlations under a Landau–Siegel zero—specifically, $$\sum_{n \leq x} \lambda(n+h_1)\cdots\lambda(n+h_k) \ll x \exp(-c v \log n)$$ for x in intervals depending on q and n (Jaskari et al., 16 Sep 2024). These bounds improve on prior work by Germán–Kátai, Chinis, and Tao–Teräväinen, exploiting a refined beta-sieve in place of Selberg’s method.

In the context of analytic ranks of automorphic L-functions, the presence of a Siegel zero forces almost all odd newforms of weight 2 and level q to have analytic rank 1 and even newforms to have rank at most 2, in line with the Brumer–Murty conjecture for Jacobians of modular curves (Bui et al., 2021).

5. Extensions to Artin L-functions and Dedekind Zeta Functions

Stark’s work on effective bounds for Dedekind zeta residues at s=1 highlighted that any Landau–Siegel phenomenon in number fields is contributed by a unique quadratic “exceptional” character. Extensions to arbitrary Artin L-functions now provide unconditional bounds for the leading term of the Laurent expansion at s=1, encompassing potential Landau–Siegel zeros via contributions from exceptional characters (Cho et al., 2 Oct 2025). The key bounds are of the form

$$|κ(χ)| \ll_{[k:\mathbb{Q}], |G|, \chi(1)} (\log D_K)^{\tilde{\chi}(1) - \langle \chi, \mathrm{id} \rangle},$$

and, in the presence of an exceptional zero, more precise lower bounds featuring explicit contributions from the exceptional character. These generalizations employ truncated Euler products and bypass holomorphy assumptions, making them robust even in contexts lacking RH or Artin’s conjecture.

6. Conditional Links with Other Conjectures and Dynamical Models

Assumptions on Landau–Siegel zeros facilitate conditional connections to other conjectures, such as the Goldbach conjecture (Goldston et al., 2021). Weak forms of Goldbach imply explicit upper bounds on the proximity of Landau–Siegel zeros to 1. Dynamical systems inspired by Zhang’s results have also explored the connection between chaotic maps and the distribution of zeros, with Lyapunov exponents and phase diagrams reflecting sensitivity, bifurcation, and underlying arithmetic regularity (Rafik, 2023, Rafik et al., 2023). These models numerically corroborate that when analytic inputs such as L(1, χ) are effectively bounded away from zero, the system’s phase space displays regularity, echoing the absence or tightly constrained existence of Landau–Siegel zeros.

7. Remaining Open Problems and Future Directions

Despite increasingly stringent unconditional zero-free regions for L-functions (both abelian and automorphic), the possibility of a Landau–Siegel zero remains a core open problem. Current research continues to tighten effective bounds, improve mollification and sieve techniques, generalize results to higher-rank cases, and connect conditional assumptions to major conjectures in analytic number theory. Ongoing work seeks further unconditional progress through explicit zero-density estimates, asymptotic expansions for auxiliary functions (Reyna, 7 Jun 2024), and refined repulsion phenomena. The resolution of the Landau–Siegel zero question is expected to yield pervasive advances in arithmetic statistics, prime distribution, and the analytic theory of L-functions.