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

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 $\liminf_{n\to\infty}(γ_{n+1}-γ_n)\log(γ_n)/2π< 0.4733$ 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.

Disclosure

“result regarding a positive proportion of small gaps is due to Chirre–Gonçalves–de Laat [6] at 0.6039 with refinements due to Bui–Goldston– Milinovich–Montgomery [1]. Acknowledgments. The authors gratefully acknowledge the assistance of OpenAI’s ChatGPT, whose exploratory discussions, calculations and numerics helped stimulate the development of this paper. 2. The Montgomery–Odlyzko setup with long polynomials 2.1. Weight choices and the small gaps condition. Our aim is to compute”

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Computational experiments or data processing

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3

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2026-08-23T05:00:05.464641Z

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