Criteria equivalent to the Riemann Hypothesis

We give a brief overview of a few criteria equivalent to the Riemann Hypothesis. Next we concentrate on the Riesz and B{\'a}ez-Duarte criteria. We proof that they are equivalent and we provide some computer data to support them. It is not compressed to six pages version of the talk delivered by M.W. during the XXVII Workshop on Geometrical Methods in Physics, 28 June -- 6 July, 2008, Bia{\l}owie{\.z}a, Poland.Comment: It is not compressed to six pages version of the talk delivered by M.W. during the XXVII Workshop on Geometrical Methods in Physics, 28 June -- 6 July, 2008, Bia{\l}owie{\.z}a, Poland. New Fig.1 is include

The ternary Goldbach conjecture is true

The ternary Goldbach conjecture, or three-primes problem, asserts that every odd integer $n$ greater than $5$ is the sum of three primes. The present paper proves this conjecture. Both the ternary Goldbach conjecture and the binary, or strong, Goldbach conjecture had their origin in an exchange of letters between Euler and Goldbach in 1742. We will follow an approach based on the circle method, the large sieve and exponential sums. Some ideas coming from Hardy, Littlewood and Vinogradov are reinterpreted from a modern perspective. While all work here has to be explicit, the focus is on qualitative gains. The improved estimates on exponential sums are proven in the author's papers on major and minor arcs for Goldbach's problem. One of the highlights of the present paper is an optimized large sieve for primes. Its ideas get reapplied to the circle method to give an improved estimate for the minor-arc integral.Comment: 79 pages, 1 figure. Minimal change

Light-Weight Hierarchical Clustering Middleware for Public-Resource Computing

The goal of this work was to investigate ways to implement and improve a public-resource computing middleware. Specifically, to make hosting a public-resource computing project logistically simpler and to examine the affect of hierarchical clustering on bandwidth utilization at the central server. To this end, we present the architecture for our cross-platform, multithreaded public-resource computing middleware. Implementing and debugging the middleware proved far more challenging than initially anticipated. As hard as debugging multithreaded programs is, our experience has shown us that it can be leveraged to simplify system components. Our main contribution is the final system architecture.Computer Science Departmen