ABSTRACT
This chapter starts from the Euler-type summation formula over increasing balls in Euclidean spaces. Spherically-reflected convergence criteria are formulated in adaptation to the alternating summands. Hardy–Landau-type lattice point identities are derived from Poisson-type summation formulas in Euclidean spaces. The interest is in investigating the asymptotic behavior with increasing radii to derive sufficient criteria for the validity of Poisson-type summation formulas over the whole Euclidean space, which do not necessarily imply the absolute convergence of the infinite multivariate sums on both sides and, therefore, are of particular importance in the context of multivariate alternating sum convergence. As essential tools one need some integral mean asmptotics for iterated Euler–Green functions, which are formulated in similarity to the bivariate deductions. The developed convergence criteria are applied to higher-dimensional extensions of the Hardy–Landau identity.
