Resonance gaps, discrepancies, and lines

The sequence formed by the fractional tune resonances up to order <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"> <a:mi>n</a:mi> </a:math> exactly coincides with the Farey sequence, <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"> <c:msub> <c:mi>F</c:mi> <c:mi>n</c:mi> </c:msub> </c:math> . 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 <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"> <e:mi>n</e:mi> <e:mo>∈</e:mo> <e:mi mathvariant="double-struck">N</e:mi> </e:math> 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 <h:math xmlns:h="http://www.w3.org/1998/Math/MathML" display="inline"> <h:mi>n</h:mi> </h:math> in the two-dimensional tune diagram is also derived and compared to previous estimates.