On certain multiple series with functional equation in a totally real number field. II (Q1382840)

Let \(K\) be a totally real field of algebraic numbers of degree \(n\), let \(A,B\) be ideals of its ring of integers, choose a positive integer \(k\) and let \(\tau_1,\dots,\tau_n\) be non-zero complex numbers whose arguments lie in the interval \((-\pi/2k,\pi/2k)\). For \(l\) prime to \(k\) put \[ M(\tau;A,B;k,l)=\sum_\mu{1\over| N(\mu)| }\sum_{0\neq\nu\in B} \exp \Biggl( -2\pi\sum_{q=1}^n| \nu_j| ^{k/l}| \mu_j| \tau_j^k \Biggr), \] where the first sum is taken over non-zero elements \(\mu\) of \(A\), no two of them having ratio which is a \(k\)-th power of a unit and \(\nu_j\), \(\mu_j\) denote the \(j\)-th conjugate of \(\nu\) and \(\mu\), respectively. With the aid of that series the author defines a rather complicated function of \(n\) complex variables, depending also on \( A,B\) and \(k,l\), and shows that it satisfies a functional equation. Its form is in the general case too complicated to be reproduced here. When \(K\) is the field of rationals, this function is defined for \(s\) satisfying \(| \arg(s)| <\pi/2kl\) and has the form \[ \begin{multlined} P(s;a,b;k,l)=s^{-kl/4}(2\pi a)^{-l/2} \exp\left(a(2\pi)^{-k/l}b^{k/l}\Gamma(1+k/l) \zeta(1+k/l){\sin(\pi k/2l)\over \sin(\pi/2l)}s^k\right)\\ \times\prod_{j=0}^{l-1}\prod_{n-1}^\infty \left(1-\exp(-2\pi (bn)^{k/l}as^ke^{i\pi(l-1-2j)/2l})\right)^{-1}. \end{multlined} \] It satisfies the functional equation \[ P(s;a,b;k,l)= P(1/s;1/b,1/a;l,k). \] The case \(k=l=1\) was earlier treated by the author (for \(K\) totally real in [\textit{T. Mitsui}, Tokyo J. Math. 18, 49-60 (1995; Zbl 0840.11046) and for \(K\) totally complex ibid., 61-74 (1995; Zbl 0840.11047)]).