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(例如素数的分布问题)的一些最强大的工具是生成具有乘法性质的函数和特征标。这种生成函数最著名的例子是Riemann Zeta函数，它对乘法信息进行编码，因为它是由某个半平面上素数的乘积定义的。乘法字符的众所周知的例子是Dirichlet字符的集合mod$Q$，例如勒让德符号mod$Q$。用于理解这些函数和字符的行为的强大哲学是它们的行为就像合适的随机模型对象的想法。例如，Riemann Zeta函数被认为在不同的设置中的行为类似于具有随机系数的素数上的欧拉积，或者类似于随机矩阵的特征多项式。Dirichlet特征标具有随机么模乘函数的性质，在最近的工作中，我通过将随机乘函数和的矩与随机欧拉积的短积分的矩联系起来，证明了这些矩(即幂平均)的精确上下界。这些短积分与数学物理和概率中的乘性混沌概念有关，可以用乘性混沌研究中的思想进行分析。在完成了随机方面的分析后，自然想要对Dirichlet字符和Riemann Zeta函数的短积分进行去随机化并获得相应的结果。到目前为止，这种去随机化的一些步骤已经成功完成。我证明了两个问题(特征和问题和短积分问题)在低幂平均下的猜想上界是精确的。对于更高的功率平均，相应的结果和相应的下界尚不清楚。在短积分方面，Arguin-Ouimet-Radzi将证明一些相关结果，但这些结果并不尖锐。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，这些问题的既定界限大概也不是很尖锐。在这两种情况下，关于限制分布结果，而不是上下界，我们知之甚少。这个建议的目的是解决去随机化中缺失的一些步骤，并将其应用于特征和和的值分布和非零化以及Riemann Zeta函数。