FRG: Averages of L-functions and Arithmetic Stratification

数学中一些最困难的挑战，比如黎曼假设、伯奇猜想和斯温纳顿-戴尔猜想，自然都是用l函数来表达的。这些函数编码了一些信息，比如在给定的量级下有多少素数，或者某些方程的有理数解的频率，或者曲面上特殊点的分布，所有这些在数论中都很重要。l函数通常在称为族的集合中进行研究。在这个项目中，我们将使用一种叫做“分层”的新方法来研究l函数值在家庭中的分布。近年来，研究人员对黎曼ζ函数和其他l函数族的值和零的统计数据进行了非常精确的推测。对于低阶矩，这些猜想来自于对广义除数函数相关性的精确认识或猜想。但在更高的时刻，这种联系就消失了。本项目的主要目标是完成这幅图，并证明l函数族的矩猜想来自于除数相关的知识，这相当于在某些品种的特定区域中计数点。我们还将研究相同的情况，但对于用关于素数元组的一般Hardy-Littlewood猜想取代除数相关的族中l函数比率的平均值。对于黎曼ζ函数，该项目将遵循两位pi最近工作中概述的方法。对于l函数的其他族，需要另一个创新。该项目将借鉴Manin的思想，通过识别起作用的分层亚品种并计算这些品种上的点数来计算品种上的有理点。我们还将研究在这些子变种的点数中自然产生的指数和。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

Some of the most difficult challenges in all of mathematics, such as the Riemann Hypothesis and the Birch and Swinnerton-Dyer conjecture, are naturally phrased in terms of L-functions. These functions encode information such as how many primes there are up to a given magnitude, or the frequency of rational number solutions to certain equations, or the distribution of special points on a surface, all of which are important in number theory. L-functions are often studied in collections called families. In this project we will use a new approach called "stratification" to study the distribution of the values of L-functions in families.In recent years researchers have found very precise conjectures about the statistics of values and zeros of the Riemann zeta function and other families of L-functions. For low order moments these conjectures follow from precise knowledge or conjectures about correlations of generalized divisor functions. But for higher moments this linkage has been missing. The main goal of this project is to complete this picture and prove that the moment conjectures for families of L-functions follow from knowledge of divisor correlations, which is equivalent to counting points in specified regions of certain varieties. We will also investigate the same scenario but for averages of ratios of L-functions in families with the divisor correlations replaced by the general Hardy-Littlewood conjectures about prime tuples. For the Riemann zeta-function the project will follow the method outlined in recent work by two of the PIs. For other families of L-functions another innovation is required. This project will be informed by Manin's ideas for counting rational points on varieties by identifying the stratified subvarieties that play a role and counting the points on these. We will also investigate the exponential sums that naturally arise in counting points on these subvarieties.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.