Certain extensions of results of Siegel, Wilton and Hardy

Recently, Dixit et al. [24] established a very elegant generalization of Hardy's theorem concerning the infinitude of zeros that the Riemann zeta function possesses at its critical line.

By introducing a general transformation formula for the theta function involving the Bessel and modified Bessel functions of the first kind, we extend their result to a class of Dirichlet series satisfying Hecke's functional equation. In the process, we also find new generalizations of classical identities in Analytic number theory.