Topics in Number Theory

格兰维尔的提案摘要。该奖项将支持调查员和几个研究生在数论中的各种主题的研究。 在过去的几年里，研究人员与Soundararajan合作，重新审视了乘法函数平均值的中心主题，加强了已知的结果并添加了一些新的观点。特别是他们已经表明，这些问题在某种意义上相当于积分延迟方程的问题。Granville建议继续这项工作，特别是研究取值范围，并解决一些棘手的优化问题。 这反过来又会对已知的关于（Dirichlet和Hasse-Weil）L-函数在1处，最小m次幂剩余的界，甚至可能是L-函数零点的分布.研究者还建议继续他的工作，与Stark，在abc-猜想和Siegel零点之间的联系：一方面，进一步研究模函数满足的丢番图方程;另一方面，与塔克一起，研究未被证明的abc猜想是否可以被一个较弱但可行的可证明假设所取代。研究者与合作者正在研究乘法函数分布的更精细方面。这些函数出现在数学、物理学和密码学的许多不同领域。研究人员最近展示了对这些问题的理解是如何与“积分延迟方程”这一看似无关的领域密切相关的，后者在理解各种生物机制如何工作时被广泛分析。这样，关于乘性函数均值的结果和定理就可以“移植”为关于积分延迟方程的结果和定理，从而使一个领域的观点可以与另一个领域的观点相联系，反之亦然，这已经在积分延迟方程中产生了一些新的结果和新的问题，研究者被要求在该领域的一个主要会议上的一系列讲座中提出他的想法，以促进进一步的互动。此外，调查人员一直在与学生，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，培训将帮助他们为未来的职业带来新的视角。这个群体最近在美国排名第十。

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.