Moments of character sums and of the Riemann zeta function via multiplicative chaos

在解析数论中，我们研究“乘法”问题（例如关于素数分布的问题）的一些最强大的工具是生成具有乘法性质的函数和字符。这种生成函数最著名的例子是黎曼ζ函数，它编码了乘法信息，因为它是由某半平面上质数的乘积定义的。众所周知的乘法字符的例子是狄利克雷字符mod $q$的集合，例如勒让德符号mod $q$。理解这些函数和角色行为的一个强有力的理念是，它们的行为就像合适的随机模型对象。例如，黎曼ζ函数被认为在不同的情况下表现得像随机系数质数上的欧拉积，或者像随机矩阵的特征多项式。狄利克雷字符被认为表现得像随机的单模乘法函数。在最近的工作中，我通过将这些矩与随机欧拉积的短积分的矩联系起来，证明了随机乘法函数和的所有矩（即幂平均）的清晰的上界和下界。这些短积分与数学物理和概率论中的乘法混沌概念有关，并且可以使用乘法混沌研究中的思想进行分析。在完成了随机方面的分析之后，很自然地想要“去随机化”并获得狄利克雷特征和黎曼ζ函数的短积分的相应结果。到目前为止，这种非随机化的一些步骤已经成功完成。我从理论上证明了两个问题（字符和问题和短积分问题）对于低功率平均值的明显上界。更高功率平均值的相应结果，以及相应的下界，目前还不知道。在短积分方面，Arguin- Ouimet- radzii将会证明一些相关的结果，但是这些结果并不尖锐。最近在特征和问题的下界方面也取得了进展，例如由于La Bret\ eche、Munsch和Tenenbaum，在这些问题中，既定的边界可能也不是很明显。在这两种情况下，我们对极限分布结果所知甚少，而不是上界和下界。本提案的目标是通过应用于值分布和字符和和Riemann zeta函数的不消失来解决非随机化的一些缺失步骤。

In analytic number theory, some of our most powerful tools for studying "multiplicative" problems (e.g. problems about the distribution of prime numbers) are generating functions and characters having multiplicative properties. The most famous example of such a generating function is the Riemann zeta function, which encodes multiplicative information because it is defined by a product over primes in a certain half plane. Well known examples of multiplicative characters are the collection of Dirichlet characters mod $q$, e.g. the Legendre symbol mod $q$.A powerful philosophy for understanding the behaviour of such functions and characters is the idea that they behave like suitable random model objects. For example, the Riemann zeta function is believed to behave in different settings like an Euler product over primes with random coefficients, or like the characteristic polynomial of a random matrix. Dirichlet characters are believed to behave like random unimodular multiplicative functions.In recent work, I proved sharp upper and lower bounds for all the moments (that is, the power averages) of sums of random multiplicative functions, by connecting these moments with moments of short integrals of random Euler products. These short integrals are connected with the notion of multiplicative chaos from mathematical physics and probability, and can be analysed using ideas from the study of multiplicative chaos. Having completed the analysis on the random side, it is natural to want to "derandomise" and obtain the corresponding results for Dirichlet characters and for the short integrals of the Riemann zeta function.So far, a few steps of this derandomisation have been successfully completed. I proved conjecturally sharp upper bounds for both problems (the character sum problem and the short integral problem) for low power averages. The corresponding results for higher power averages, and the corresponding lower bounds, are not yet known. On the short integral side, Arguin--Ouimet--Radziwill have proved some related results, which however are not sharp. There has also been recent progress on lower bounds in the character sum problem, for example due to La Bret\`eche, Munsch and Tenenbaum, where again the established bounds are presumably not sharp. Very little is known about limiting distributional results, as opposed to upper and lower bounds, in either setting.The goal of this proposal is to work out some of these missing steps of the derandomisation, with applications to the value distribution and non-vanishing of character sums and of the Riemann zeta function.