Moments of L-functions, correlation sums, and primes

L函数矩的研究是解析数论的一个重要研究领域。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认识到L函数的界具有重要的数论应用。Hardy和Littlewood研究了Riemann Zeta函数在临界线上的2k阶矩，记为I_k(T)，其中k为正。1918年，Hardy-Littlewood渐近地估计了二阶矩，1926年，Ingham渐近地计算了四阶矩。I_k(T)没有任何其他的k值。1998年，Kating和Snaith利用一个随机矩阵模型，给出了I_k(T)的渐近大小的一个精确猜想。1998年，Conrey和Gonek在k=3和k=4的情况下将I_k(T)与除数函数的相关和联系起来。猜想了I_k(T)的全主项渐近公式。2016年，我证明了三值除数函数的某些相关和的渐近公式蕴含了Zeta函数六阶矩的全部主项渐近。 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关系。特别是，我将重点讨论高阶因子函数的相关和。我的目的是在最近NSERC资助的关于Zeta函数六阶矩的工作的基础上，建立Riemann Zeta函数的八阶矩的渐近公式，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因子系数的长Dirichlet多项式的某些平均值来表示。与Alia Hamieh合作，我们的目标是证明这些平均值的一个更精确的渐近性，从而验证1998年Conrey-Gonek的猜想和最近Conrey-Kating猜想的特例。 利用我在zeta函数六阶矩方面的工作中的技巧，我还将研究L函数的其他平均值族。这包括Zeta函数导数的离散平均值和二次Dirichlet L-函数的平均值。 该提案的最后一项内容是探讨素数计数函数的大小和相关的误差项。蒙哥马利对素数定理中误差项的大小做了一个深刻的猜测。我的目标是假设线性独立猜想来证明这一点。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。其中一些目标将适用于本科生和研究生。

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.