On the non homogeneous quadratic Bessel zeta function

Abstract

We study the non homogeneous quadratic Bessel zeta function ${\zeta }_{RB}(s,\nu ,a)$ defined as the sum of the square of the positive zeros of the Bessel function $J_{\nu }(z)$ plus a positive constant. In particular, we give explicit formulas for the main associated zeta invariants, namely poles and residua, ${\zeta }_{RB}(0,\nu ,a)$ and ${\zeta }_{RB}^{\prime }(0,\nu ,a)$.