Three two-dimensional Weyl steps in the circle problem I. The Hessian determinant

1. Summary. In a sequence of three papers we study the circle problem and its generalization involving the logarithmic mean. Most of the deeper results in this area depend on estimates of exponential sums. For the circle problem itself Chen has carried out such estimates using three two-dimensional Weyl steps with complicated techniques. We make the same Weyl steps but our approach is simpler and clearer. Crucial is a good understanding of the Hessian determinant that appears and a simple estimate of certain exponential integrals. In Part I we determine the order of magnitude of the Hessian as well as that of the maximum of the second derivatives for the functions h, which are third order differences of the two-dimensional Euclidean vector norm.