Discrete moments of the Riemann zeta function

An important problem in number theory is to calculate the moments of the Riemann zeta function $\zeta (s)$. Moments have a wide range of applications, for example in calculating proportions of non-trivial zeros $\rho = \beta + i\gamma$ of $\zeta (s)$ that satisfy the Riemann Hypothesis, or that are simple. Shanks [280] noticed that $\zeta' (\rho)$ is real and positive on average, a strange result when we consider that this is a complex-valued function summed over complex points. Later, it was noticed that this peculiar behaviour continued to higher derivatives, where the sum remains real on average, but oscillates positive and negative depending on whether the order of the derivative is odd or even. Generalisations of this observation are considered throughout this thesis. These involve sums of the form \[ \sum_{0 < t \leq T} \zeta^{(n_1)} \left( \frac{1}{2} + it \right) \dots \zeta^{(n_k)} \left( \frac{1}{2} + it \right), \] where for integers $n_1,\dots,n_k$, $t$ ranges over either the non-trivial zeros of zeta or the zeros of the derivative of the Hardy $Z$-function, and $\zeta^{(n)}$ denotes the $n$\textsuperscript{th} derivative of $\zeta (s)$. After a comprehensive background on the zeta function, we begin with a simple heuristic for Shanks' observation and its generalisation, giving a clear reason for the oscillating behaviour before giving a rigorous proof of this fact with a full asymptotic expansion. A weighted first moment of this problem is then considered. We build on the analogy between characteristic polynomials of random matrices and $\zeta (s)$, first noted by Keating and Snaith [211]. We present new conjectures for full asymptotic expansions of the above summation over the non-trivial zeros of $\zeta (s)$ after giving other supporting evidence for the leading order behaviour of these sums. Finally we consider the above sum over the zeros $\lambda$ of the derivative of the Hardy $Z$-function, and show that the behaviour of this sum oscillates in the opposite way to that over the non-trivial zeros, that is, $\zeta' (1/2 + i\lambda)$ is real and negative on average.

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