The Riemann zeta function

黎曼 zeta 函数

• 批准号: 2879204
• 金额: --
• 依托单位: University of Oxford
• 依托单位国家: 英国
• 项目类别: Studentship
• 财政年份: 2023
• 资助国家: 英国
• 起止时间: 2023 至 无数据
• 项目状态: 未结题
• 来源: https://gtr.ukri.org/projects?ref=studentship-2879204
• 关键词: Riemann; zeta; function

The Riemann Hypothesis asserts that all of the non-trivial zeros of the Riemann zeta function have real part equal to 1/2. Much progress has been made recently on the moments and extreme values of the Riemann zeta function on the line Res=1/2 (the critical line). Hardy and Littlewood established bounds on the fraction of the non-trivial zeroes satisfying the Riemann Hypothesis and found the asymptotics for the first moment of the zeta function on the critical line. The second moment was later found by Ingham. The Weyl-Hardy-Littlewood method gives bounds for sums of exponentials, which can be used to bound the extremal values of the zeta function in the critical strip using the Abel summation formula. The Keating-Snaith conjecture (later extended by Conrey, Farmer, Rubinstein and Snaith) gives precise asymptotics for all moments of the zeta function. This conjecture is motivated by random matrix theory, and specifically by the idea that on the critical line the zeta function behaves like the characteristic polynomial of certain random matrices.A key bound on the growth of the zeta function is the Lindelöf Hypothesis, which states that on the critical line, the zeta function grows slower than any positive power of the imaginary part; I hope to investigate the interplay between this and the Keating-Snaith conjecture, which states that the moments are a power of log. Since for large values of k, the 2k-th moment is dominated by its large values, we would expect that the Keating-Snaith conjecture implies something far stronger than the Lindelöf hypothesis.In between the maximum values and the moments of the zeta functions, there is the problem of determining the measure of an interval where the zeta function takes values above a certain threshhold. The work of Arguin-Bourgade-Radziwill gives decay rates for the measure of short intervals on the critical line where the zeta function is large, uniform for intervals in [T,2T], which are stronger than previous bounds and are consistent with the multifractality conjectured by Fyodorov and Keating. Many theorems about the Riemann zeta function can be extended to other L-functions; Dirichlet L-functions in particular have many similar properties. Having learnt the tools used in the work of Arguin, Bourgade and Radziwill in their bounds on the growth of the zeta function, I hope to apply them to other L-functions. This may be achievable, since if the terms relating to a period of a Dirichlet L-function with non-trivial character are grouped together, the results sum is absolutely convergent for positive real part, whereas for the Riemann zeta function, absolute convergence can only be provided for real part greater than 1. The research will contribute to the EPRSC areas of mathematical analysis, mathematical physics (Since the Keating-Snaith conjecture is motivated by considerations from Physics) and number theory.

Riemann假设认为Riemann Zeta函数的所有非平凡零点的实部都等于1/2。最近关于Riemann Zeta函数在res=1/2(临界线)上的矩和极值的研究取得了很大进展。Hardy和Littlewood建立了满足Riemann假设的非平凡零点分式的界，并找到了Zeta函数在临界线上一阶矩的渐近性。第二个时刻后来被英厄姆发现。Weyl-Hardy-Littlewood方法给出了指数和的界，它可以利用Abel求和公式来确定Zeta函数在临界带上的极值。基廷-斯奈斯猜想(后来由康瑞、法默、鲁宾斯坦和斯奈斯推广)给出了Zeta函数所有矩的精确渐近性。这个猜想的动机是随机矩阵理论，特别是在临界线上Zeta函数的行为类似于某些随机矩阵的特征多项式的想法。Zeta函数增长的一个关键限制是Lindelöf假设，该假说指出在临界线上，Zeta函数的增长慢于虚部的任何正幂；我希望研究这与基廷-斯奈特猜想之间的相互作用，该猜想表明矩是对数的幂。由于对于大的k值，2k阶矩被它的大值所支配，我们可以预期基廷-斯奈斯猜想暗示的东西比林德洛夫假设强得多。在Zeta函数的最大值和矩之间，存在着确定Zeta函数取值于某一阈值以上的区间的度量的问题。Arguin-Borgade-Radziwill的工作给出了临界线上Zeta函数较大的短区间度量的衰减率，对于[T，2T]中的区间是一致的，这比以前的界更强，并与Fyodorov和Kating的多重分形性猜想一致。关于黎曼Zeta函数的许多定理可以推广到其他L函数，特别是Dirichlet L函数具有许多类似的性质。在学习了Arguin，Borgade和Radziwill关于zeta函数增长的工作中所使用的工具后，我希望将它们应用于其他L函数。这是可能实现的，因为如果将具有非平凡性质的狄利克雷L函数的周期项组合在一起，结果和对于正实部是绝对收敛的，而对于Riemann Zeta函数，只有大于1的实部才能绝对收敛。这一研究将有助于EPRSC的数学分析、数学物理(由于基廷-斯奈斯猜想是出于物理学的考虑)和数论。