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.

黎曼假设认为黎曼函数的所有非平凡零点实部都等于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假设的非平凡零点的分数的界限，并找到了zeta函数在临界线上的一阶矩的渐近性。第二个时刻后来被英格汉姆发现了。Weyl-Hardy-Littlewood方法给出了指数和的边界，该边界可用于用Abel求和公式限定临界带上zeta函数的极值。Keating-Snaith猜想（后来由Conrey， Farmer， Rubinstein和Snaith扩展）给出了zeta函数的所有矩的精确渐近性。这个猜想是由随机矩阵理论激发的，特别是在临界线上的函数表现得像某些随机矩阵的特征多项式。关于zeta函数增长的一个关键界限是Lindelöf假设，它表明在临界线上，zeta函数的增长速度比虚部的任何正幂都要慢；我希望研究一下这个和Keating-Snaith猜想之间的相互作用，该猜想认为力矩是对数的幂次。因为对于较大的k值，第2k个矩被较大的k值所支配，我们可以预期Keating-Snaith猜想暗示的东西比Lindelöf假设强得多。在zeta函数的最大值和矩之间，有一个问题是确定一个区间的度量，在这个区间中，zeta函数的值高于某个阈值。Arguin-Bourgade-Radziwill的工作给出了zeta函数大的临界线上短间隔测度的衰减率，对于[T，2T]区间的衰减率是一致的，它比以前的边界强，并且与Fyodorov和Keating猜想的多重分形一致。关于黎曼ζ函数的许多定理可以推广到其他l函数；特别是狄利克雷l函数有许多类似的性质。在学习了Arguin， Bourgade和Radziwill在关于zeta函数生长的边界中所使用的工具后，我希望将它们应用到其他l -函数中。这是可以实现的，因为如果将具有非平凡特征的Dirichlet l -函数的周期相关项组合在一起，则结果和对于正实部是绝对收敛的，而对于Riemann zeta函数，只有实部大于1才能提供绝对收敛。这项研究将有助于EPRSC领域的数学分析，数学物理（因为基廷-斯奈思猜想是由物理学的考虑激发的）和数论。