The Nexus of Analysis and Number Theory

提案摘要**本提案的问题是经典分析和数论的联系。这些工具是分析的，概率的，组合的和数论的，具有中心计算的味道。计算技术是复杂的，需要大量的计算资源（在某些情况下需要数百小时的网格计算），算法本身有时是价值因子问题的中心问题。**这个建议的核心问题通常是数论或组合学中的“老李子”。他们已经开了很多年了。它们也都是相关的，或者至少是潜在相关的。尽管存在明显的困难，但在每一种情况下都取得了部分进展，而且似乎可能取得更多进展。要了解更多细节，请参见[Borwein 2002]。所附CCV的论文几乎都涉及到这些问题中的至少一个。**最近在这一领域取得了许多令人兴奋的进展，特别是由于Green和Tao的研究，表明认真解决这些问题的一些棘手问题是可能的。同样，计算技术的进步可能使我们能够做出合理的推测。除其他外，以下两个问题将是我们建议的核心。***CHOWLA的余弦问题：**Ben Green在最近的数学评论中写道：这篇论文解决了审稿人最喜欢的问题之一，被称为“CHOWLA的余弦问题”：**这涉及到在一段时间内找到n个余弦和的最小值的负数。**已经有了各种各样的部分结果，但目前，在这个有趣的问题上，在可证明的和推测的之间存在着巨大的差距，事实上，正确的答案应该是什么也不清楚。***PADÉ ZETA函数的近似**探索（适当归一化）Riemann ZETA函数的pad<s:1>近似的零点和极点的位置。很明显Szegö-like曲线是存在的。这有点令人惊讶，这些曲线是什么还不清楚。图案引人注目，计算困难。可以证明的很少，但可以提出很多建议。当然，这与黎曼假设密切相关。**这些问题在数论中具有中心意义和重要性。数论当然是加拿大最强大的数学领域之一。

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.