On Zeta and Dirichlet Beta Function Families as Generators of Generalized Mathieu Series, Providing Approximation and Bounds

Integral representations for a generalized Mathieu series and its companions are used to undertake analysis leading to novel insights for Zeta and Dirichlet Beta function families. The bounds are procured using sharp bounds of Zeta and Dirichlet family bounds to procure approximating and bounds utilising integral representation of generalized Mathieu series results using in particular Hardy-type upper bounds.