Topics related to the theory of numbers: integer points close to convex hypersurfaces, associated magic squares and a zeta identity

Abstract

Let C be the boundary surface of a strictly convex d-dimensional body. Andrews obtained an upper bound in terms of M for the number of points on MC, the M-fold enlargement of C. We consider the integer points within a distance 5 of the hypersurface MC. Introducing S requires some uniform approximability condition on the surface C, involving determinants of derivatives. To obtain an asymptotic formula (main term the volume of the search region) requires the Fourier transform with conditions up to the Gd-th derivative. We obtain an upper bound subject to a Curvature Condition that re quires only first and second derivatives, that MC has a tangent hyperplane everywhere, and each two-dimensional normal section has radius of curvature in the range cqM +1/2 3), satisfying the Curvature Condition at size M. Then the total number, N, of integer points lying within a distance 6 of MC is bounded by the sum of two terms, one from Andrews's bound, the other from the hypervolume of the search region, with explicit constant factors involving 6, cq and c . In the body of the thesis, to simplify the notation, we use C for the enlarged surface called MC in this summary. In Part II we enumerate a class of special magic squares. We observe a new identity between values of the zeta functions at even integers