The Nexus of Analysis and Number Theory

提案摘要**本提案的问题存在于经典分析和数论的结合点。这些工具是分析、概率、组合和数论，带有中心计算的味道。计算技术很复杂，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)，而算法本身有时是核心问题，就像优点因子问题一样。**这个建议的核心问题通常是数论或组合学中的老问题。它们已经开了很多年了。它们也都是相关的，或者至少是潜在的相关。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，而且似乎有可能取得更多进展。更多细节见[Borwein 2002]。所附CCV的论文几乎都与这些问题中的至少一个有关。**这一一般领域最近取得的令人振奋的进展，特别是格林和陶渊明的进展，表明认真解决这些问题的一些棘手问题是可能的。同样，计算上的进步可能会让我们做出合理的猜测。以下两个问题将是我们提案的核心。*乔拉余弦问题：**本·格林在最近的一篇数学评论中写道：这篇论文讨论了评论者最喜欢的问题之一，被称为“乔拉余弦问题”：**这涉及到求一段时间内n个余弦和的最小值的负数。**已经有了各种不同的部分结果，但目前，在这个有趣的问题中，什么是可证明的和什么是猜测之间存在着巨大的差距，实际上，还不清楚正确的答案应该是什么。*Zeta函数的Padé近似**探索Padé近似的零点和极点对(适当标准化的)Riemann Zeta函数的位置。似乎很明显，存在类似于Szegö的曲线。这有点令人惊讶，目前还不清楚这些曲线是什么。**图案醒目，计算困难。几乎没有什么可以证明的，但有很多建议。当然，这与黎曼假设是密切相关的。**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.