Certain Extensions of Results of Siegel, Wilton and Hardy

Abstract

Recently, Dixit et al. 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.

arXiv commentMore details are given in Subsection 3.1 regarding the method used in the proof of Theorem 1.1; We added Lemma 3.1 which clarifies an assertion made on page 36 of the previous version; Minor typos in the text corrected