Topics in Number Theory

数论专题

• 批准号: 0070523
• 负责人: Clinton McCrory
• 金额: $ 44.08万
• 依托单位: University of Georgia Research Foundation Inc
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2000
• 资助国家: 美国
• 起止时间: 2000-06-15 至 2005-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0070523&HistoricalAwards=false
• 关键词: Topics; Number; Theory

Abstract for Granville's proposal.This award will support the research of the investigator and severalgraduate students on various topics in number theory. In the last few years the investigator, in collaboration with Soundararajan, has been revisiting the central topic of mean values of multiplicative functions, and strengthening known results as well as adding several new perspectives. In particular they have shown that such questions are in some sense equivalent to problems about integral delay equations. Granville proposes continuing this work, particularly looking at the range of values taken and to solve some thorny optimization problems. This, in turn, should give new results on known problems about values of (Dirichlet and Hasse-Weil) L-functions at 1, bounds on leastmth power residues, and perhaps even distribution of zeros of L-functions.The investigator also proposes continuing his work, with Stark, on the connection between the abc-conjecture and Siegel zeros: On the one hand, looking further at Diophantine equations satisfied by modular functions;on the other hand, with Tucker, looking at whether the unproved abc-conjecture can be replaced by a weaker but feasibly provable hypothesis.The investigator, with collaborators, is looking at the finer aspects of thedistribution of multiplicative functions. The functions turn up in many different areas in mathematics, physics and cryptography.The investigator recently showed how an understanding of such questionsis closely related to the seemingly unrelated area of `integral delay equations'' which are analyzed extensively in understanding howvarious biological mechanisms work. Thus results and conjectures about mean values of multiplicative functions can be ``translated'' intoresults and conjectures about integral delay equations, so that the perspectives of one field can be brought to bear on the other, and vice-versa.This has already led to some new results and fresh questions in integral-delayequations, and the investigator has been asked to present his ideas in a series of lectures at a major conference in that area, to stimulate furtherinteraction. In addition the investigator has been working with students, particularly from the Deep South, to get their PhDs, and obtain good positions in research, education, industry and government, with a training that will help them bring a fresh perspective to their future professions. This group wasrecently ranked tenth in the US.

Granville的建议摘要。该奖项将支持研究者和几名研究生在数论的各个主题上的研究。在过去的几年里，研究者与Soundararajan合作，重新审视了乘法函数均值的中心主题，并加强了已知的结果，同时增加了一些新的观点。特别地，他们证明了这些问题在某种意义上等价于关于积分延迟方程的问题。Granville建议继续这项工作，特别是关注所取值的范围，并解决一些棘手的优化问题。反过来，对于已知的关于（Dirichlet和Hasse-Weil） l -函数在1处的值，最小次幂残数的界，甚至l -函数的零点分布的问题，这应该给出新的结果。研究者还建议与Stark一起继续研究abc猜想与西格尔零之间的联系：一方面，进一步研究由模函数满足的丢番图方程；另一方面，与塔克一起，研究未经证明的abc猜想是否可以被一个较弱但可行的可证明假设所取代。研究者和合作者正在研究乘法函数分布的更精细的方面。这些函数出现在数学、物理和密码学的许多不同领域。研究者最近展示了对这些问题的理解是如何与看似无关的“积分延迟方程”领域密切相关的，该领域在理解各种生物机制的工作原理方面得到了广泛的分析。因此，关于乘法函数平均值的结果和猜想可以“转换”为关于积分延迟方程的结果和猜想，这样一个领域的观点就可以被带到另一个领域，反之亦然。这已经导致了积分延迟方程的一些新结果和新问题，研究者被要求在该领域的一个主要会议上发表一系列演讲，以激发进一步的互动。此外，研究人员一直与学生，特别是来自南方腹地的学生合作，以获得博士学位，并在研究，教育，工业和政府中获得良好的职位，并提供培训，帮助他们为未来的职业带来新的视角。该集团最近在美国排名第十。