| Streszczenie: | Recent developments in random matrix theory and the theory of log-correlated random variables have led to several interesting conjectures in analytic number theory. This conjectures primarily concern the asymptotic behavior of several quantities related to the moments of the zeta function. In this dissertation, we prove estimates of the conjectured order of magnitude for several of these quantities. First we will give upper bounds for the joint moments of the 2kth power of the zeta function with the 2hth power of its logarithmic derivative when k ∈ [1, 2] and h ∈ [0, 1]. Then in joint work with Andr´e Heycock, we extend these upper bounds to all 0 ≤ h ≤ k ≤ 2 unconditionally and for all 0 ≤ h ≤ k assuming the Riemann hypothesis. We also prove unconditional lower bounds for all 0 ≤ h ≤ k, demonstrating that our upper bounds are of the correct order of magnitude.
We then move on to study the problem of moments of moments of zeta and the closely related problem of estimating shifted moments of zeta on the half line. Moments of moments of zeta were introduced by Fyodorov and Keating [42] in order to understand the value distribution of zeta in short intervals on the critical line. By calculating moments of moments of random matrices, they formed a conjecture for the asymptotic behavior of the moments of moments of zeta and the distribution of the maximum of zeta in random short intervals. We prove upper bounds of the conjectured order for the (2, β)-moment of moments of zeta when 0 ≤ β ≤ 1 and lower bounds of the conjectured size for all β ≥ 0. In particular, we demonstrate that there is a phase transition that occurs at β = √1/2.
The key quantity we need to estimate to understand the (k, β)-moments of moments are averages of k shifted products of zeta raised to the power β. Unconditionally, we can only estimate these shifted moments when k = 1 and β ≤ 2 or when k = 2 and 0 ≤ β ≤ 1. Assuming the Riemann hypothesis, however, we give sharp upper and lower bounds for more general shifted moments of zeta. This improves upon previous work of Chandee [21] and Ng, Shen, and Wong [77]. Our bounds are especially interesting when the the shifts are unbounded and the random matrix theory model is no longer appropriate.
|
|---|