## Eta quotients and Rademacher sums. (arXiv:1810.07478v1 [math.NT])

Eta quotients on \$\Gamma_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 \$\Gamma_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.查看全文

## Solidot 文章翻译

 你的名字 留空匿名提交 你的Email或网站 用户可以联系你 标题 简单描述 内容 Eta quotients on \$\Gamma_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 \$\Gamma_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.
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