Eta Quotients and Rademacher Sums

  • Kevin Acres
  • David BroadhurstEmail author
Part of the Texts & Monographs in Symbolic Computation book series (TEXTSMONOGR)


Eta quotients on \(\varGamma _0(6)\) yield evaluations of sunrise integrals at 2, 3, 4 and 6 loops. At 2 and 3 loops, they provide modular parametrizations of inhomogeneous differential equations whose solutions are readily obtained by expanding in the nome q. Atkin–Lehner transformations that permute cusps ensure fast convergence for all external momenta. At 4 and 6 loops, on-shell integrals are periods of modular forms of weights 4 and 6 given by Eichler integrals of eta quotients. Weakly holomorphic eta quotients determine quasi-periods. A Rademacher sum formula is given for Fourier coefficients of an eta quotient that is a Hauptmodul for \(\varGamma _0(6)\) and its generalization is found for all levels with genus 0, namely for \(N = 1,2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 16, 18, 25\). There are elliptic obstructions at \(N = 11, 14, 15, 17, 19, 20, 21, 24, 27, 32, 36, 49,\) with genus 1. We surmount these, finding explicit formulas for Fourier coefficients of eta quotients in thousands of cases. We show how to handle the levels \(N=22, 23, 26, 28, 29, 31, 37, 50\), with genus 2, and the levels \(N=30,33,34,35,39,40,41,43,45,48,64\), with genus 3. We also solve examples with genera 4, 5, 6, 7, 8, 13.



The second author thanks KMPB for hospitality and colleagues at conferences in Zeuthen, Bonn, St. Goar and Les Houches for advice and encouragement that emboldened our joint effort to tackle eta quotients beyond the remit of genus zero so far encountered in massive Feynman diagrams. We especially thank Johannes Blümlein for his question on the possibility of obtaining an explicit formula for Fourier coefficients of the Hauptmodul of \(\varGamma _0(6)\) and Freeman Dyson for urging us to try to emulate the notable work by Rademacher on partition numbers [29]. We thank Yajun Zhou and an anonymous referee for helpful suggestions that improved our presentation.


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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Monash UniversityMelbourneAustralia
  2. 2.Open UniversityMilton KeynesUK

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