On the non homogeneous quadratic Bessel zeta function

Abstract

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