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

乘性混沌的特征和矩和黎曼 zeta 函数矩

• 批准号: EP/V055755/1
• 负责人: Adam Harper
• 金额: $ 22.24万
• 依托单位: University of Warwick
• 依托单位国家: 英国
• 项目类别: Research Grant
• 财政年份: 2022
• 资助国家: 英国
• 起止时间: 2022 至 无数据
• 项目状态: 未结题
• 来源: https://gtr.ukri.org/projects?ref=EP%2FV055755%2F1
• 关键词: Moments; character; sums; Riemann; zeta

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.

在解析数论中，我们研究“乘法”问题（例如有关素数分布的问题）的一些最强大的工具是具有乘法性质的生成函数和特征。这种生成函数最著名的例子是黎曼 zeta 函数，它对乘法信息进行编码，因为它是由某个半平面中素数的乘积定义的。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Legendre 符号 mod $q$。理解此类函数和字符的行为的强大哲学是它们的行为类似于合适的随机模型对象。例如，黎曼 zeta 函数被认为在不同的设置下表现得像带有随机系数的素数的欧拉积，或者像随机矩阵的特征多项式。狄利克雷特征被认为表现得像随机幺模乘法函数。在最近的工作中，我通过将这些矩与随机欧拉积的短积分矩连接起来，证明了随机乘法函数和的所有矩（即幂平均值）的尖锐上限和下界。这些短积分与数学物理和概率中的乘性混沌概念相关，并且可以使用乘性混沌研究的思想进行分析。完成了随机方面的分析，很自然地想要“去随机化”并获得狄利克雷特征和黎曼 zeta 函数短积分的相应结果。到目前为止，这个去随机化的几个步骤已经成功完成。我证明了低幂平均值的两个问题（字符和问题和短积分问题）的猜想尖锐上限。较高功率平均值的相应结果以及相应的下限尚不清楚。在短积分方面，Arguin-Ouimet-Radziwill证明了一些相关结果，但并不清晰。最近在字符和问题的下界方面也取得了进展，例如由于 La Bret'eche、Munsch 和 Tenenbaum 的贡献，其中所建立的界限可能并不尖锐。在这两种设置中，与上限和下限相比，关于限制分布结果知之甚少。该提案的目标是解决去随机化的一些缺失步骤，并将其应用于值分布以及字符和和黎曼 zeta 函数的不消失。