Exceptional Landau–Siegel Zero

• Exceptional Landau–Siegel zero is defined as a real zero extremely near s = 1 in Dirichlet L-functions, impacting error estimates and analytic number theory.
• Its potential presence triggers the Deuring–Heilbronn phenomenon, repelling other zeros and refining predictions for prime distributions in arithmetic progressions.
• Research shows that strict bounds on these zeros directly influence class number estimates and the behavior of automorphic and higher-rank L-functions.

An exceptional Landau–Siegel zero, often called simply a Landau–Siegel zero or Siegel zero, is a hypothetical real zero β very near s = 1 of a real (typically quadratic) Dirichlet L-function or, more generally, certain L-functions associated with real characters or self-dual automorphic representations. The (non-)existence and properties of such zeros influence the analytic behavior of L-functions, error terms in classical problems such as the distribution of primes in arithmetic progressions, class number estimates, and the distribution of short gaps between zeros of L-functions. Though their existence is widely disbelieved and excluded in many settings, the consequences of their hypothetical presence are central to understanding both obstructions and phenomena in analytic number theory.

1. General Definition and Classical Context

An exceptional Landau–Siegel zero is a real zero β = 1 − ε, for ε ≪ 1/ log q, of a Dirichlet L-function L(s, χ) associated with a real, primitive Dirichlet character χ modulo q. The existence of such zeros is neither known nor expected, but various results in analytic number theory depend critically upon whether they might exist extremely close to s = 1. If such a zero exists, the associated L(1, χ) must be abnormally small, and this in turn disrupts bounds in class number problems and the distribution of primes.

In broader settings (Rankin–Selberg products, symmetric power L-functions), exceptional zeros are defined analogously: a real zero extremely close to s = 1, usually only possible when an abelian (real) factor divides the L-function. Their presence is tightly constrained in higher-rank situations (Thorner, 13 Aug 2025, Thorner, 9 Apr 2024, Humphries et al., 2020).

2. Impact on the Distribution of Zeros and the Deuring–Heilbronn Phenomenon

The presence of a Landau–Siegel zero gives rise to the Deuring–Heilbronn phenomenon: all other nontrivial zeros of the relevant L-function (and, in some contexts, other related L-functions) are repelled from the line Re(s)=1. The zero-free region for other zeros is effectively widened, leading to explicit lower bounds on the real parts β < 1 − c/[ (1–β₁) log M ] for other zeros ρ = β + iγ (with M depending on q, T, and other parameters) (Benli et al., 8 Oct 2024).

This repulsion is now quantified with completely explicit, uniform in the critical strip, effective constants. For example, if a Landau–Siegel zero β₁ exists with β₁ > 1 − (1/10 log q), then every other zero ρ = β + iγ (with β > 1/2, |γ| ≤ T) satisfies:

$$β < 1 - \frac{\log( \theta / [4(1 - β₁) \log M]) }{ \log M }$$

where $M = K q^{8θ + 2ε} T^{4θ}$ and constants are completely specified in (Benli et al., 8 Oct 2024). This improves on earlier bounds by providing smaller constants and full uniformity over the critical strip.

This result is both a tool and an obstruction: the presence of β₁ forces a structural change in the distribution of zeros, quantifiably pushing them away from the edge of the critical strip (Stopple, 2011, Thorner, 2022).

3. Consequences for Class Numbers and Arithmetic

A Landau–Siegel zero affects lower bounds for class numbers of imaginary quadratic fields and the least prime in an arithmetic progression. For quadratic L-functions, the Dirichlet class number formula relates L(1, χ) to the class number h(–q), so a small L(1, χ) leads to small h(–q) and renders classical lower bounds ineffective (Zaman, 2015, Languasco, 2023).

Recent work has achieved explicit unconditional lower bounds of the form $L(1, \chi) \gg (\log D)^{-2022}$ (Zhang, 2022), ruling out Landau–Siegel zeros closer than a corresponding bound, and explicit upper bounds such as β < 1 – c/ log q (with c computed numerically) (Languasco, 2023). These results—supported by large computations—demonstrate that in the vast majority of accessible cases, the critical zero-free region is maintained and class numbers remain robustly large.

Furthermore, for primes in arithmetic progressions, the existence of a Siegel zero implies improvement in the level of distribution for the Chebyshev function across moduli, as error terms can be “turned to advantage” if L(1, χ) is exceptionally small (Wright, 2023).

4. Interplay with Automorphic and Higher-Rank L-Functions

In higher-degree L-functions (e.g., Rankin–Selberg products, symmetric power lifts), exceptional zeros are generically ruled out except in special abelian “self-dual” cases. Recent advances establish strong zero-free regions and non-existence of exceptional zeros for families such as

$$L(s, \mathrm{Sym}^2(\pi)\times(\mathrm{Sym}^2 (\pi')\otimes\chi))$$

provided the L-function is not divisible by an abelian factor (Thorner, 9 Apr 2024, Thorner, 13 Aug 2025). These results extend previous knowledge beyond the self-dual or diagonal case, using auxiliary Dirichlet series with nonnegative coefficients and analytic arguments generalizing zero density methods.

Such nonexistence results have immediate applications to effective rates in Sato–Tate distributions for Hecke eigenangles and joint Sato–Tate distributions (Thorner, 9 Apr 2024), and reinforce the analytic prerequisites for functoriality and the purity of the automorphic spectrum.

5. Quantitative and Conditional Results

Explicit bounds on the proximity of exceptional zeros have been numerically calculated for all quadratic characters modulo primes up to q = 10⁷, with the smallest normalized L-value (L(1, χ)/(log q)) observed being approximately 0.0125, and potential zeros bounded by β < 1 – 0.0092/ log q (Languasco, 2023).

Conditional approaches, for example those assuming a uniform version of the abc-conjecture, can force strong zero-free regions near s = 1 for L(s, χ_D), with explicit constants emerging from Diophantine geometric invariants such as the height of singular moduli (Táfula, 2019).

Similarly, under weak Goldbach-type hypotheses or other conditional conjectures, an exceptional zero is forced to recede from 1 at rate at least C / log² q (Goldston et al., 2021). If one assumes all non-real zeros remain on Re(s)=1/2 in a neighborhood, even sharper results are possible (Basak et al., 24 Apr 2024).

6. Broader Arithmetic and Additive Consequences

Assuming the existence of a Landau–Siegel zero has far-reaching conditional implications for additive problems and correlation sums. For Chowla’s conjecture, which predicts cancellation in short sums of shifted Liouville functions, the existence of a Siegel zero allows for strong bounds on correlations:

$$\sum_{n \le x} \lambda(n+h_1)\cdots \lambda(n+h_k) \ll_{k} \left( \eta + x\exp(-c V\log\eta ) \right)$$

for $x = q^V$ in specified ranges, with $\eta$ tied to the zero’s proximity to 1, reflecting the impact of “exceptional” (Siegel) zeros on multiplicative randomness (Jaskari et al., 16 Sep 2024).

Moreover, their existence influences least prime bounds in arithmetic progressions and the structure of zeros in automorphic L-functions, as well as conditional nonvanishing results for Dirichlet L-functions at the critical point (Bui et al., 2020), analytic ranks of automorphic forms (Bui et al., 2021), and even aspects of quantum unique ergodicity (Thorner, 2022).

7. Connections to Dynamical and Statistical Properties

Novel work has translated qualitative and effective lower bounds on L(1, χ) into discrete dynamical systems whose stability properties reflect the distribution of zeros. For maps inspired by the Dirichlet class number formula, the presence of a Siegel zero translates into nearly nonchaotic, predictable dynamics, while absence or “distance” increases chaos (as measured by Lyapunov exponents and entropy) (Rafik, 2023, Rafik et al., 2023). This dynamical lens uncovers additional structure—connecting fixed points and bifurcations in iterative maps to analytic data of L-functions.

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In conclusion, the exceptional Landau–Siegel zero, while hypothetical, remains a central analytic obstruction and organizing principle in number theory. Its purported existence or absence is intricately connected to upper and lower bounds for L-values, zero-free regions, the distribution and repulsion of zeros of L-functions (in both abelian and nonabelian contexts), arithmetic statistics such as class numbers, and deeper phenomena in arithmetic geometry and ergodic theory. The cumulative research—from explicit zero repulsion and uniform Deuring–Heilbronn theorems (Benli et al., 8 Oct 2024), through effective non-existence in broad automorphic settings (Thorner, 13 Aug 2025, Thorner, 9 Apr 2024), to conditional implications for correlation sums and additive conjectures (Jaskari et al., 16 Sep 2024)—has both sharpened the analytic toolkit for bounding exceptional zeros and clarified their arithmetic and statistical ramifications across number theory.