Fast algorithms, computational complexity, and subconvexity bounds in analytic number theory

本提案的主题属于数论领域。这是数学的一个基本领域对理解整数感兴趣。在某些被称为l函数的解析对象中编码有关于整数的深层信息。在l函数和计算方法之间有许多卓有成效的相互作用。这种相互作用始于黎曼ζ函数，黎曼用数值方法计算了这个函数。从那时起，计算在解析数论中的作用不断增强。作者将从计算的角度研究某些l函数。这包括研究它们的计算复杂性，新的快速算法的推导，应用于学习整数分解，以及将计算作为研究l函数的实验工具。申请者将研究分析数论和计算数论中的几个独立课题。主要的主题是研究Riemann zeta函数的次凸估计与计算复杂度之间的联系。这里的新奇之处在于，这两个方面通常是弱相关的。然而，对于临界带中的Riemann zeta函数，以及对于幂满模的Dirichlet字符和，甚至对于更多的数论对象，它们都是强连接的。该项目的一个目标是翻译分析和计算观点之间的最新进展。另一个主题涉及具有辛对称或正交对称的l函数“族”的全矩猜想。该提议的目的是推导出这些族的全矩猜想的一致渐近性，扩展了先前与Michael Rubinstein在酉情况下的联合工作。第三个主题是开发用于检测无平方数的最新算法，该算法是与Andrew Booker和Jon Keating共同开发的，特别是与质数上字符和增长率的下界有关。

The topics in this proposal fall within the area of number theory. This is a fundamental area in mathematics interested in understanding the integers. There is deep information about the integers encoded in certain analytic objects called L-functions. And there have been many fruitful interactions between L-functions and computational methods. Such interactions started with the Riemann zeta function which, famously, Riemann computed numerically. The role of computation in analytic number theory has continued to grow since then. The proposer will investigate certain L-functions from a computational viewpoint. This includes investigating their computational complexity, the derivation of new fast algorithms, with an application to learning about integer factorization, and the use of computation as an experimental tool in the study of L-functions.The proposer will study several independent topics in analytic and computational number theory. The main topic is to investigate connections between subconvexity estimates of the Riemann zeta function, on the one hand, and its computational complexity on the other. Part of the novelty here is that these two aspects are weakly related in general. However, they connect strongly for the Riemann zeta function in the critical strip, and also for Dirichlet character sums to a power-full modulus, perhaps even for more number-theoretic objects. One goal of the project is to translate recent progress between the analytic and computational viewpoints. Another topic concerns the full moment conjectures of ``families'' of L-functions with symplectic or orthogonal symmetry. The proposer aims to derive uniform asymptotics for the full moment conjectures for such families, extending previous joint work with Michael Rubinstein in the unitary case. A third topic is to develop the recent algorithm for detecting squarefree numbers that was derived in joint work with Andrew Booker and Jon Keating, especially in relation to lower bounds for the growth rate of character sums over the primes.