Siegel zeros and small gaps between zeros of the Riemann zeta function

Technical Summary: Siegel Zeros and Small Gaps Between Zeros of the Riemann Zeta Function

Problem Statement
The paper addresses the vertical distribution of the non-trivial zeros of the Riemann zeta function, $\zeta(s)$. Specifically, it investigates the existence of "small gaps" between consecutive ordinates $\gamma_n$ and $\gamma_{n+1}$. The distribution is normalized by the average spacing $2\pi/\log \gamma_n$. The authors define the quantity $\mu = \liminf_{n\to\infty} (\gamma_{n+1} - \gamma_n) / (2\pi/\log \gamma_n)$.

A longstanding conjecture posits that $\mu = 0$. While the Riemann Hypothesis (RH) is assumed, the best known unconditional lower bound for $\mu$ is not the issue; rather, the paper focuses on the upper bound. Prior to this work, the best result on RH was $\mu < 0.50895$ (Inoue). There has been a prevailing sentiment that the value $1/2$ represents a natural barrier for $\mu$ due to the potential existence of Siegel zeros (exceptional real zeros of Dirichlet $L$-functions). Previous work by Heath-Brown and Conrey–Iwaniec suggested that if Siegel zeros exist, they might enforce a rigidity in the zeta zeros, preventing gaps smaller than $1/2$ from occurring frequently (the "Alternative Hypothesis").

Methodology
The authors employ the Montgomery–Odlyzko method, which compares two weighted integrals to detect small gaps. The core innovation lies in the choice of the "resonator" function $R(t)$ and the length of the associated Dirichlet polynomial.

1. Long Dirichlet Polynomials: Traditional applications of the Montgomery–Odlyzko method are limited to "short" Dirichlet polynomials (length $L \ll T$) to avoid intractable off-diagonal terms involving additive correlations of coefficients. This paper breaks this barrier by utilizing long Dirichlet polynomials with length $L \approx T^{17/14}$.
2. Resonator Construction: The resonator is chosen as $R(t) = \sum_{n \le L} \chi(n)G(n)n^{-1/2-it}$, where $\chi$ is a primitive quadratic Dirichlet character associated with a Siegel zero. The coefficients involve a smooth weight $G(n)$ and the character $\chi$.
3. Handling Off-Diagonal Terms: The primary technical challenge is estimating the off-diagonal correlations $\sum \chi(m)\chi(km+r)$ and $\sum \Lambda(k)\chi(km+r)$. The authors develop a new scheme to handle these correlations with power-saving error terms, splitting the analysis into three regimes based on the size of the prime variable $k$:
   • Small $k$: Handled via bounds on correlation sums using Lemma 4 (an extension of Heath-Brown's work).
   • Intermediate $k$: Addressed using second-moment estimates for shifted character sums (Cauchy–Schwarz combined with additive divisor problem techniques).
   • Large $k$: Solved by separating variables in $\chi(km+r)$ via Fourier inversion (generalized Jacobi sums) and applying zero-density estimates for Dirichlet $L$-functions (specifically the Chen–Gupta–Li estimate) to bound the sum over characters.

Key Results
The paper establishes the following main theorem under the assumption of the Riemann Hypothesis:

Theorem 1: Assume RH and that there exists an exceptional sequence of Siegel zeros with quality $E = \inf_j E_j$ sufficiently large. Then:
$$\mu < 0.4733$$
Specifically, the authors show that $\liminf_{n\to\infty} (\gamma_{n+1} - \gamma_n) \frac{\log \gamma_n}{2\pi} < 0.4733$.

Corollary 2: Under the same assumptions, an asymptotic distribution where normalized gaps are restricted to the set $\frac{1}{2}\mathbb{Z}_{\ge 1} + o(1)$ cannot exist. This refutes a strong form of the "Alternative Hypothesis" which posits that Siegel zeros force all gaps to be half-integers.

Significance and Claims
The authors claim that their result is surprising because it demonstrates that Siegel zeros do not act as a barrier to proving $\mu < 1/2$; rather, the existence of such zeros can be used to prove the existence of gaps smaller than $1/2$.

• Refutation of Strong Alternative Hypotheses: The result refutes the idea that Siegel zeros imply a rigid distribution where almost all gaps are $\ge 1/2$. Instead, it shows that if Siegel zeros exist, there must be a clustering of zeros with normalized gaps strictly less than $0.4733$.
• Methodological Breakthrough: The paper highlights that the limitation of the Montgomery–Odlyzko method to $\mu < 0.508$ was due to the restriction to short polynomials. By successfully managing the off-diagonal terms of long polynomials (length $T^{17/14}$) using Siegel zeros to structure the coefficients, the authors bypass this computational barrier.
• Technical Nuance: The authors clarify that their result does not contradict Heath-Brown or Conrey–Iwaniec. Those works established that Siegel zeros imply few gaps smaller than $1/2$ (a quantitative proportionality result). This paper establishes the existence of at least one gap smaller than $0.4733$ (a liminf result) without claiming a positive proportion of such gaps.

The paper concludes that while Siegel zeros remain a barrier for establishing a positive proportion of small gaps (due to the need for higher moments which restrict polynomial length further), they are not a barrier to establishing the existence of gaps below $1/2$. The authors note that further refinements in coefficient choice or zero-density estimates (e.g., the Density Hypothesis) could potentially lower the bound further, perhaps to $\mu < 0.467$, but the current work focuses on the mechanism of using Siegel zeros to break the $1/2$ barrier.