FRG: Averages of L-functions and Arithmetic Stratification

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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.