Division values of multiple sine functions (Q1017347)

Assume the complex numbers \(1,\omega_1,\dots,\omega_r\) all lie in a given open half-plane in \(\mathbb C\). Then the multiple Hurwitz zeta series \[ \zeta_r(s,x,\omega)=\sum_{m_1,\dots,m_r\geq 0}(m_1\omega_1+\dots+m_r+\omega_r+x)^{-s} \] converges for \(\mathrm{Re}(s)>r\) and the ensuing function extends to a meromorphic function on the plane. The multiple \(\Gamma\)-function \[ \Gamma_r(x,\omega)=\exp(\zeta_r'(0,x,\omega)) \] is used to define the multiple sine function, \[ S_r(x,\omega)=\Gamma_r(x,\omega)^{-1}\Gamma_r(\omega_1+\dots+\omega_r-x,\omega)^{(-1)^r}. \] This function was introduced by \textit{N. Kurokawa} in [ Proc. Japan Acad., Ser. A 68, No. 9, 256--260 (1992; Zbl 0797.11053)]. Multiple sine functions are connected to special values of zeta functions. For example, set \(S_r(x)=S_r(x,(1,\dots,1))\). Then \[ \zeta(3)=4\pi^2\log S_3(1),\quad \zeta(5)=-\frac{4\pi^4}{3}\log\left( S_5(1)S_5(2)^{11}\right) \] and \[ \zeta(7)=\frac{8\pi^6}{45}\log\left(S_7(1)S_7(2)^{57}S_7(3)^{302}\right). \] In the present paper, some of Kurakawa's results are extended. For instance, it is shown that for \(N\geq 2\), \[ \prod_{1\leq k_1,\dots,k_4\leq N-1}S_4\left(\frac{k_1\omega_1+\dots+k_4\omega_4}{N},\omega\right)=1. \] There are other results of similar nature. In the applications, let me point out an interesting modularity assertion in three variables. Let \[ F(\tau_1,\tau_2,\tau_3)=\prod_{n_1,n_2,n_3=0}^\infty (1+q_1^{n_1+1/2}q_2^{n_2+1/2}q_3^{n_3+1/2}). \] Then for \(x=\frac{\tau_1+\tau_2+\tau_3}2\) one has \[ F(\tau_1,\tau_2,\tau_3)=e^\pi i\zeta_4(0,x,\tau)\frac{F\left(-\frac1{\tau_1},-\frac{\tau_2}{\tau_1},-\frac{\tau_3}{\tau_1}\right)F\left(\frac{\tau_1}{\tau_3},\frac{\tau_2}{\tau_3},-\frac1{\tau_3}\right)}{F\left(\frac{\tau_1}{\tau_2},-\frac1{\tau_2}-\frac{\tau_3}{\tau_2}\right).} \]