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

Andriy Bondarenko; Winston Heap

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arXiv · 2608.07399

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

Andriy Bondarenko·Winston Heap

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Abstract

On assuming the Riemann Hypothesis, we show that Siegel zeros imply the existence of gaps between the zeros of the Riemann zeta function less than 1/2 the normalised length. Specifically, we show that an infinite family of Siegel zeros implies $\liminf_{n\to\infty}(\gamma_{n+1}-\gamma_n)\log(\gamma_n)/2\pi< 0.4733$ on RH. This refutes the existence of certain strong alternative hypotheses under these assumptions. Our arguments incorporate long Dirichlet polynomials of length $T^{17/14-\varepsilon}$ into the Montgomery--Odlyzko method.

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Andriy Bondarenko, Winston Heap. 2026-08-07. Siegel zeros and small gaps between zeros of the Riemann zeta function. https://arxiv.org/abs/2608.07399

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