On certain multiple series with functional equation in a totally real number field. I (Q1903371)

The author constructs a generalization \(M(\tau_1,\dots, \tau_n, {\mathfrak a}, {\mathfrak b})\) (with \(\text{Re }\tau_i> 0\) and \(\mathfrak a\), \(\mathfrak b\) being fractional ideals) of the series \[ f(\tau)= \sum^\infty_{m= 1} \sum^\infty_{n= 1} \exp(- 2\pi mn\tau)/m \] in a totally real extension of the rationals of degree \(n\) and proves a functional equation for it. The proof uses a transformation formula of Hecke and Rademacher [\textit{H. Rademacher}, Math. Z. 27, 321-426 (1928; JFM 53.0154.03)] which leads to a representation of \(M\) as a series of integrals of functions which are essentially products of Hecke's zeta-functions with the gamma function. The case of a totally imaginary field is treated by the author in another paper [Tokyo J. Math. 18, 61-74 (1995; see the following review, Zbl 0840.11047)].