Moments of L-functions, correlation sums, and primes

L 函数的矩、相关和和素数

• 批准号: RGPIN-2020-06032
• 负责人: Ng, Nathan
• 金额: $ 1.97万
• 依托单位: University of Lethbridge
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2021
• 资助国家: 加拿大
• 起止时间: 2021-01-01 至 2022-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=740989
• 关键词: Moments; functions; correlation; sums; primes

An important field of research in analytic number theory is the study of moments of L-functions. At the beginning of the twentieth century, researchers including Bohr, Landau, Hardy, and Littlewood realized that bounds for L-functions had important number theoretic applications. Hardy and Littlewood studied the 2k-th moments of the Riemann zeta function on the critical line, denoted I_k(T) where k is positive. In 1918 Hardy-Littlewood asymptotically evaluated the second moment and in 1926 Ingham asymptotically evaluated the fourth moment. I_k(T) has not been evaluated for any other value of k. In 1998 Keating and Snaith, using a random matrix model, gave a precise conjecture for the asymptotic size of I_k(T). In 1998 Conrey and Gonek linked I_k(T) to correlation sums of divisor functions in the cases k=3 and k=4. In 2006 Conrey et al. conjectured the full main term asymptotic formula for I_k(T). In 2016, I gave a proof in that an asymptotic formula for certain correlation sums of ternary divisor functions implies the full main term asymptotic for the sixth moment of the zeta function. The main theme of this proposal is to study the connection between mean values of L-functions and correlation sums of arithmetic functions. In particular, I shall focus on correlations sums of higher divisor functions. I aim to establish an asymptotic formula for the eighth moment of the Riemann zeta function based on my recent NSERC funded work on the sixth moment of the zeta function and also establish asymptotics for related moments of the zeta function. The moments of I_k(T) can be modelled by certain mean values of long Dirichlet polynomials with higher divisor coefficients. In collaboration with Alia Hamieh we aim to prove a more precise asymptotic for these mean values thus verifying a 1998 conjecture of Conrey-Gonek and special cases of a recent conjecture of Conrey-Keating. Using techniques from my work on the sixth moment of the zeta function, I will also study other families of mean values of L-functions. This includes the discrete mean values of the derivative of the zeta function and mean values of quadratic Dirichlet L-functions. A final element of the proposal is to explore the size of prime counting functions and related error terms. Montgomery has made a deep conjecture regarding the size of the error term in the prime number theorem. I aim to prove this assuming the Linear Independence conjecture. Also I shall study the size of the sum of the Mobius function, M(x), attempting to exhibit large values. A number of these objectives will be suitable for undergraduate and graduate students.

解析数论的一个重要研究领域是l -函数矩的研究。在二十世纪初，包括玻尔、朗道、哈代和利特尔伍德在内的研究人员意识到l函数的界在数论上有重要的应用。Hardy和Littlewood研究了Riemann zeta函数在临界线上的第k个矩，记作I_k(T)，其中k为正。1918年Hardy-Littlewood渐近地求出了第二个矩，1926年Ingham渐近地求出了第四个矩。I_k(T)没有对k的任何其他值进行评估。1998年，Keating和Snaith使用随机矩阵模型，给出了I_k(T)的渐近大小的精确猜想。1998年，Conrey和Gonek将I_k(T)与k=3和k=4情况下的除数函数的相关和联系起来。2006年Conrey等人推测了I_k(T)的完整主项渐近公式。在2016年，我证明了某些三元除数函数的相关和的渐近公式暗示了zeta函数的第六阶矩的完整主项渐近。本文的主题是研究l函数的均值与算术函数的相关和之间的联系。特别地，我将着重于高因子函数的相关性和。我的目标是基于我最近NSERC资助的关于zeta函数的第六矩的工作，建立黎曼zeta函数的第八矩的渐近公式，并建立zeta函数的相关矩的渐近性。I_k(T)的矩可以用具有较高除数系数的长狄利克雷多项式的某些平均值来表示。在与Alia Hamieh的合作中，我们的目标是证明这些平均值的更精确的渐近，从而验证1998年Conrey-Gonek猜想和最近Conrey-Keating猜想的特殊情况。利用我在zeta函数的第六阶矩上的工作技巧，我还将研究l函数的其他均值族。这包括zeta函数导数的离散均值和二次狄利克雷l函数的均值。该建议的最后一个要素是探索素数计数函数的大小和相关的误差项。Montgomery对素数定理中误差项的大小做了一个深刻的猜想。我的目的是在线性无关猜想的前提下证明这一点。我还将研究莫比乌斯函数M(x)的和的大小，试图展示大的值。其中一些目标将适合本科生和研究生。