ABSTRACT
This chapter deals with the extension of the Hardy–Landau identities to regular regions and general Λ-lattice point weights. It is shown via the bivariate Euler-type summation formula that the validity of weighted Hardy–Landau identities may be verified for regular regions with boundaries, for which the technique of the stationary phase may be applied on their parameter representations. The chapter gives some insight into possible strategies and viable methodologies for the bivariate case. The overall intention in a number theoretic context is a twofold generalization of the classical 2D Hardy–Landau identity as far as possible in the following sense: the generalization from a “circle geometry” to a “general” geometry and the generalization from a constant to a “general”. A compromise in lattice point summation was already formulated by W. Freeden, where general weights for the circular geometry in spherical summation of the 2D series had been realized.
