Lattice Points, the Gauss Circle Problem, and the Riemann Zeta-Function
The investigator studies the properties of lattice points close to smooth curves and surfaces and applications of these properties to classical number theory problems. One objective is to improve on the Iwaniec-Bombieri method and to obtain new results in the Gauss circle problem and on the order of the Riemann zeta-function on the critical line. Another aim is to improve on the Fouvry-Iwaniec method for estimating exponential sums with monomials and to obtain new applications on the distribution of almost-prime numbers in short intervals and B-free numbers in short intervals. The investigator and his colleagues are also using Pade approximation polynomials to obtain new bounds on the fractional parts of rational numbers and applications to diophantine equations and on the irreducibility of polynomials. Another line of research is on the convergence of Minmod-type numerical methods for hyperbolic differential equations.
The Gauss circle problem and the problem of the order of the Riemann zeta-function on the critical line are two of the most famed problems in number theory. The Riemann zeta-function is closely related to the distribution of prime numbers. Knowledge about prime numbers plays a crucial role in the areas of cryptography and data encryption which are important both for national security and e-commerce. Hyperbolic differential equations find applications in problems involving flow (some examples are gases around aircraft, wake of aircraft, underground contamination, weather patterns). In most cases it is impossible to solve exactly the relevant differential equation, so numerical schemes for an approximate solution are applied. The investigator and his colleagues have achieved progress and are working toward completely solving an important open problem in the mathematical theory of numerical methods for hyperbolic differential equations.
