The average length of a trajectory in a certain billiard in a flat two-torus.
The distribution and moments of the error term in the Dirichlet divisor problem
The distribution of powers of integers in algebraic number fields
For an arbitrary (not totally real) number field of degree , we ask how many perfect powers of algebraic integers in exist, such that for each embedding of into the complex field. ( a large real parameter, a fixed integer, and for any complex .) This quantity is evaluated asymptotically in the form , with sharp estimates for the remainder . The argument uses techniques from lattice point theory along with W. Schmidt’s multivariate extension of K.F. Roth’s result on the approximation...
The Ehrhart polynomial of a lattice -simplex.
The hyperbola
We include several results providing bounds for an interval on the hyperbola containing lattice points.
The inner periodic structure of a function.
The lattice points of an n-dimensional tetrahedron.
The lattice points of an n-dimensional tetrahedron. (Summary).
The number of solutions of a homogeneous linear congruence
The Piltz divisor problem in number fields: An improved lower bound by Soundararajan's method
The polynomial captures its primes.
The rational points close to a curve
The rational points close to a curve II
The rational points close to a curve III
The representation of primes by cubic polynomials
The two parameter ellipse problem
The two parameter hyperbola problem
Théorie additive et géométrie des nombres
Theorie der Gitterpunkte.
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...