FRG: Averages of L-functions and Arithmetic Stratification

所有数学中的一些最困难的挑战，如黎曼假设和伯奇和斯温纳顿-戴尔猜想，自然是用L函数来表述的。这些函数对信息进行编码，例如有多少素数可达给定的大小，或某些方程的有理数解的频率，或曲面上特殊点的分布，所有这些在数论中都很重要。L-函数通常在被称为族的集合中被研究。在这个项目中，我们将使用一种称为“分层”的新方法来研究L函数的值在族中的分布。近年来，研究人员发现了关于黎曼Zeta函数和其他L函数族的值和零点的统计量的非常精确的猜想。对于低阶矩，这些猜想源于关于广义因子函数相关性的精确知识或猜想。但在更高的时刻，这种联系一直缺失。这个项目的主要目的是完成这幅图，并证明L函数族的矩猜想源于因子相关的知识，这等价于某些变种的指定区域中的计点数。我们还将研究相同的情况，但对于除数相关被关于素数组的一般Hardy-Littlewood猜想取代的族中L函数的比率的平均值。对于Riemann Zeta-Function，该项目将遵循两个PI最近的工作中概述的方法。对于L函数的其他家庭来说，还需要另一种创新。这个项目将受到马宁关于通过识别起作用的分层亚种并计算这些亚种上的点来计算品种上的理性点数的想法的启发。我们还将调查在计算这些亚类的点数时自然产生的指数和。这一奖励反映了NSF的法定使命，并通过使用基金会的智力优势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.