Heritage and Advancements in Lattice Point and Sampling Regularization Theory

This is the first book of two volumes dedicated to C. F. Gauss on the occasion of his 250th birthday. The objective of the two books is to demonstrate that the heritage of Gauss still has much to offer today in building a strong scientific bridge between mathematics and geoscience.Volume 1 presents a proof of Hardy’s conjecture for the Gauss circle problem in geometric number theory. In fact, the verification of this conjecture represents a consequence of the“transfer” of methods and frameworks from geomathematics to number theory. The essential ingredients originate from tools in mathematical (geo-)physics. Remarkably, the foundational techniques enable the recovery of significant topics in lattice point theory, including the Hardy–Landau identities and the planar non-uniform distribution of lattice points. Thus, Volume 1 can be characterized as a contribution to the geometric theory of numbers, strongly influenced by methods and procedures originating in geosystems mathematics. This book will be valuable to a broad audience, including mathematicians working in number theory and applied mathematics, as well as professionals in the geosciences, such as geophysics and geoengineering.