3 problems

K2 residue conjectures. The following hold:

Pole and residue conjecture. The function $K_2(t,\pm1,\pm1)$ has simple poles precisely at $t=2$ and $t=-2m$, $m\in\mathbb{N}$; moreover, when $x=y\neq\pm1$, $K_2(t,x,x)$ has simpl…

Nonvanishing conjecture. The function $K_1(t,\varphi,\varphi)$ has simple poles exactly at $t=\frac12-m$, $m\in\mathbb{N}_0$; equivalently,