Normalized double sine functions

Source abstract

We express normalized double sine functions of integer periods $(N_1, N_2)$ via the standard double sine function of period $(1,1)$. As an application we give an Euler product expression using the di-logarithm for the double zeta function $\zeta(s, \mathbf{F}_{p^{N_1}}) \otimes \zeta(s, \mathbf{F}_{p^{N_2}})$ for a prime number $p$ and integers $N_1$, $N_2$.