Spectral functions of zeros for $q$-Bessel functions

Zeta functions $\zeta_\nu(z;q)=\sum_{n=1}^{\infty}\bigl[j_{\nu n}(q)\bigr]^{-z}$ and partition functions $Z_\nu(t;q)=\sum_n\exp[-tj_{\nu n}^2 (q)]$ related to the zeros $j_{\nu n}(q)$ of the $q$-Bessel functions $J_\nu(x;q)$ and $J_\nu^{(2)}(x;q)$ are studied. Explicit formulas for $\zeta_\nu(2n;q)$ at $n=\pm 1,\pm 2,\ldots$ are obtained. Poles of $\zeta_\nu(z;q)$ in complex plane and corresponding residues are found. Asymptotics of the partition functions $Z_\nu(t;q)$ as $t \downarrow 0$ is derived.