## On the distribution of prime numbers: computational modelling, evaluation and analysis of the Riemann Zeta functionTools Chase, Leigh (2008)
## AbstractHow the primes are distributed amongst the natural numbers is one of the principal unresolved concerns of modern science. The issue has a broad impact ranging from the very foundations of mathematical number theory to the security of current and future information systems. This project focuses on generating new, original research into a specific subset of this problem-space; that of the Riemann Hypothesis and accompanying Zeta function. G.F.B. Riemann's now famous conjecture has remained resistant to either proof or disproof since its original formulation in his landmark paper of 1859. Some have described it as the "greatest unsolved problem in mathematics"; it was Hilbert's Eighth Problem and more recently named one of the Clay Mathematics Institute Millennium Problems. This work is the blending of techniques from mathematics, artificial intelligence and computer science to generate original methods for representing and analysing the behaviour of the Riemann Zeta function. The resulting artefact takes the form of a software agent designed to construct computational models of Zeta; evaluating the function and interpreting its output to facilitate knowledge discovery. The evaluation process incorporates the implementation of a polynomial time algorithm for computing Zeta with high precision, based on the Riemann-Siegel formula. The latter interpretation stage introduces an inference engine designed to operate on the results of its sister component. The inference engine combines techniques from statistics and artificial intelligence to develop coherent models of how the function behaves, based on a detailed analysis of its input-output sets. From this, predictions concerning the growth of the function with unseen complex arguments are made via the implementation of a stochastic process using existing 'known' data for training.
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