Integrality of the higher Rademacher symbols (Q6594394)

\textit{W. Duke} [Exp. Math. 33, No. 4, 624--643 (2023; \url{doi:10.1080/10586458.2023.2219071}] reinterpreted the work of \textit{C. L. Siegel} [Nachr. Akad. Wiss. Gött., II. Math.-Phys. Kl. 1968, 7--38 (1968; Zbl 0273.12002)] and defined the \textit{higher Rademacher symbols} \(\Psi_{n}:\mathrm{SL}_{2}(\mathbb{Z})\rightarrow\mathbb{Q}\). The most important property of the higher Rademacher symbols is that the values at nonpositive integers of the zeta function for a narrow ideal class of a real quadratic field,\N\[\N\zeta(1-n,\mathcal{A})=\frac{\Psi_{n}(A)}{j_{n}},\N\]\Nwhere \(j_{n}\) is the denominator of \(\zeta(1-2n)=-\frac{B_{2n}}{2n}\) (here \(B_{2n}\) is the Bernoulli number), and \(\mathcal{A}\) is a narrow ideal class of a real quadratic field which assigns a hyperbolic element \(A\in\mathrm{SL}_{2}(\mathbb{Z})\). Similar to the usual Rademacher symbols, it was conjectured that the higher Rademacher symbols are also integer-valued. The paper in review establishes the conjecture through the detailed study of the divisibility of Bernoulli numbers. The author notes that the similar techniques may be applied to show the integrality of the Rademacher symbols for general Fuchsian groups, as defined in [\textit{C. Burrin}, J. Théor. Nombres Bordx. 34, No. 3, 739--753 (2023; Zbl 1515.11048)].