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: 2022
• 资助国家: 加拿大
• 起止时间: 2022-01-01 至 2023-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=758730
• 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。二十世纪初，玻尔、朗道、哈代和利特伍德等研究人员意识到 L 函数的界限具有重要的数论应用。 Hardy 和 Littlewood 研究了黎曼 zeta 函数在临界线上的第 2k 个矩，表示为 I_k(T)，其中 k 为正。 1918 年 Hardy-Littlewood 渐近评估了二阶矩，1926 年 Ingham 渐近评估了四阶矩。 I_k(T) 尚未针对 k 的任何其他值进行评估。 1998年Keating和Snaith利用随机矩阵模型对I_k(T)的渐近大小给出了精确的猜想。 1998 年，Conrey 和 Gonek 将 I_k(T) 与 k=3 和 k=4 情况下除数函数的相关和联系起来。 2006 年，康利等人。猜想了 I_k(T) 的完整主项渐近公式。 2016年，我证明了三元除数函数的某些相关和的渐近公式隐含了zeta函数六阶矩的完整主项渐近。 该提案的主题是研究L函数的平均值与算术函数的相关和之间的联系。特别是，我将重点关注高除数函数的相关和。我的目标是基于最近 NSERC 资助的 zeta 函数第六矩的工作，建立黎曼 zeta 函数第八矩的渐近公式，并建立 zeta 函数相关矩的渐近公式。 I_k(T) 的矩可以通过具有较高除数系数的长狄利克雷多项式的某些平均值来建模。我们与 Alia Hamieh 合作，旨在证明这些平均值的更精确渐近，从而验证 Conrey-Gonek 的 1998 年猜想以及 Conrey-Keating 最近猜想的特殊情况。利用我在 zeta 函数六阶矩方面的研究成果，我还将研究 L 函数平均值的其他族。这包括 zeta 函数导数的离散平均值和二次狄利克雷 L 函数的平均值。 该提案的最后一个要素是探索素数计数函数的大小和相关误差项。蒙哥马利对素数定理中误差项的大小做出了深刻的猜想。我的目的是假设线性无关猜想来证明这一点。我还将研究莫比乌斯函数 M(x) 之和的大小，尝试展示大值。其中许多目标适合本科生和研究生。