The Nexus of Analysis and Number Theory

提案摘要 ** 这个提案的问题存在于经典分析和数论的联系中。这些工具是分析，概率，组合和数论与中央计算味。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），算法本身有时是核心问题，就像优异因子问题一样。这个建议的核心问题通常是数论或组合学中的“老李子”。他们已经开放了很多年。它们也都是相关的，或者至少是潜在的相关。尽管有明显的困难，但在每一个情况下都取得了部分进展，而且似乎有可能取得更多进展。更多细节请参见[Borwein 2002]。所附《共同核心价值观》的文件几乎都至少涉及其中一个问题。最近在这一领域取得了许多令人振奋的进展，特别是由于绿色和陶，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预测。除其他外，以下两个问题将是我们建议的核心。* 乔拉的余弦问题：** 本绿色写道，在最近的数学评论：本文解决了一个评论家最喜欢的问题，这已被称为“乔拉的余弦问题”：** 这涉及到寻找负的最小值的总和n余弦超过一个时期。** 已经有各种各样的部分结果，但目前，在这个有趣的问题中，可证明的和被证明的之间存在巨大的差距，事实上，不清楚正确的答案应该是什么。Zeta函数的Padé近似 ** 探索（适当归一化的）Riemann Zeta函数的Padé近似的零点和极点的位置。很明显，类似塞格的曲线是存在的。这有点令人惊讶，也不清楚这些曲线是什么。** 这些模式是惊人的，计算是困难的。几乎没有什么可以证明的，但有很多建议。当然，这与黎曼假设密切相关。这些问题是中央利益和重要性数论。数论当然是加拿大数学最强的领域之一。

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.