Series expansions for Maass forms on the full modular group from the Farey transfer operators. (English) Zbl 1458.11071
Summary: We deepen the study of the relations previously established by D. H. Mayer [Bull. Am. Math. Soc., New Ser. 25, No. 1, 55–60 (1991; Zbl 0729.58041)],
J. Lewis and D. Zagier [Ann. Math. (2) 153, No. 1, 191–258 (2001; Zbl 1061.11021)], and the authors, among the eigenfunctions of the transfer operators of the Gauss and the Farey maps, the solutions of the Lewis-Zagier three-term functional equation and the Maass forms on the modular surface \(\mathrm{PSL}(2, \mathbb{Z}) \backslash \mathcal{H} \). In particular we introduce an “inverse” of the integral transform studied by Lewis and Zagier [loc. cit.], and use it to obtain new series expansions for the Maass cusp forms and the non-holomorphic Eisenstein series restricted to the imaginary axis. As corollaries we obtain further information on the Fourier coefficients of the forms, including a new series expansion for the divisor function.
MSC:
| 11F37 | Forms of half-integer weight; nonholomorphic modular forms |
| 11M36 | Selberg zeta functions and regularized determinants; applications to spectral theory, Dirichlet series, Eisenstein series, etc. (explicit formulas) |
| 37C30 | Functional analytic techniques in dynamical systems; zeta functions, (Ruelle-Frobenius) transfer operators, etc. |
Software:
DLMFReferences:
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