The Nexus of Analysis and Number Theory

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SUMMARY OF PROPOSAL The questions of this proposal live at the nexus of classical analysis and number theory. The tools are analytic, probabilistic, combinatorial and number theoretic with a central computational flavour. The computational techniques are sophisticated, require significant computational resources (hundreds of hours of grid computing in some instances) and the algorithms themselves are sometimes the central issue as with the Merit Factor Problem. The problems which are at the heart of this proposal are often “old plums” in number theory or combinatorics. They have been open for many years. They are also all related, or at least potentially related. Despite their clear difficulty, in each case partial progress has been made and more appears possible. For much more detail, see [Borwein 2002]. The papers of the attached CCV almost all relate to at least one of these problems. Much exciting recent progress in this general area, particularly due to Green and Tao, suggests that seriously attacking some of the intractabilities of these problems may be possible. Likewise computational advances may allow us to make reasonable conjectures. The following two problems will be central to our proposal, among others. CHOWLA’S COSINE PROBLEM: Ben Green writes in a recent Math Review: This paper addresses one of the reviewer’s favourite questions, which has been referred to as “Chowla’s cosine problem”: This involves finding the negative of the minimum of a sum of n cosines over a period. There have been various partial results, but currently, there is an enormous gap between what is provable and what is conjectured in this intriguing problem, and indeed, it isn’t clear what the right answer should be. PADÉ APPROXIMATION OF THE ZETA FUNCTION Explore the location of the zeros and poles of the Padé approximations to the (suitably normalized) Riemann Zeta function. It seems clear that Szegö-like curves exist. This is somewhat surprising and it isn’t clear what these curves are. The patterns are striking and the computations are difficult. Little is possible to prove, but much is suggested. This, of course, is intimate to the Riemann hypothesis. These problems are of central interest and importance in number theory. Number theory is of course one the strongest areas of mathematics in Canada.