On the Non Homogeneous Quadratic Bessel Zeta Function

From the point of view of differential geometry and mathematical physics, the Riemann zeta function appears as the operator zeta function associated to the Laplacian operator on the line segment [18] [17] [5] [6] [14]. A natural generalisation of this setting, is to consider a Sturm Liouville operator instead, i.e. a singularity at one of the end points [9] [10] [11] [7] [8][16]. This leads again to a concrete zeta function, namely the Bessel zeta function, where the sum is extended on the positive zeros of the Bessel function Jν(z), and reduces for the opportune choice of ν to the classical Riemann case. Such a function was first considered and studied by Stolarsky in [21], where formulas for poles and residua are given, and more recently by other authors, who calculated the associated zeta invariants by different methods [1] [16]. In these notes, we study the non homogeneous version of this function. We determinate his poles and give formulas for the residua. In particular, we introduce two simple but quite general methods to calculate the value of the derivative at the origin, and therefore the regularized determinant of the associated Sturm-Liouville singular operator [3] [4] [12] [19].