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  • Open Access

Resonance gaps, discrepancies, and lines

Rogelio Tomás García

  • CERN, Esplanade des Particules, Meyrin, Switzerland

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Phys. Rev. Accel. Beams 28, 114001 – Published 24 November, 2025

DOI: https://doi.org/10.1103/2gfw-xckn

Abstract

The sequence formed by the fractional tune resonances up to order n exactly coincides with the Farey sequence, Fn. In circular accelerators, it is important to place the tunes in regions free of resonances. From numerical evaluations, it is well known that the largest resonance-free gaps happen next to low order resonances. In this report, we provide a mathematical demonstration of this statement for any order n∈N showing that the resonance gaps are strictly ordered according to the order of the adjacent lowest order resonance up to some given value. We present a connection between the resonance sequence and one of the most important unsolved problems in mathematics, the Riemann hypothesis (RH). If RH is true, it implies that the resonance sequence is significantly more regularly spaced than if it were built at random; however, this has not yet been demonstrated. A new analytical estimate of the number of resonance lines up to order n in the two-dimensional tune diagram is also derived and compared to previous estimates.

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Phys. Rev. Accel. Beams 28, 114001– Published 24 November, 2025
  • Received 19 August 2025
  • Accepted 31 October 2025
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