Two theorems on sums over zeros of the Riemann zeta function.

This journal-style thesis presents two new results concerning discrete moments of derivatives of the Riemann zeta function.

In Chapter2 we establish a generalisation of the Landau-Gonek Theorem, in particular proving asymptotics uniform in $X$ and $T$ for \begin{equation*}
S(X,T)=\sum_{T<\Im(\rho)\le 2T}\chi(\rho)X^{\rho},
\end{equation*}
where $\rho=\beta+i\gamma$ are the non-trivial zeta zeros and $\chi$ is the factor from the functional equation $\zeta(s)=\chi(s)\zeta(1-s)$. This allows one to evaluate sums of approximate functional equations evaluated at the non-trivial zeta zeros, and as a consequence of this we are able to provide a new proof of the Generalised Shanks conjecture.\\

In Chapter 3 we consider sums of the form $$I(\mu,\nu)=\sum_{0<\Im(\rho)\le T}\zeta^{(\mu)}(\rho)\zeta^{(\nu)}(1-\rho).$$ These sums were first considered by Gonek in 1984, whereby a leading asymptotic was established. We extend this to a full asymptotic by establishing all of the lower order terms in the asymptotic expansion. As a corollary we recover a 2008 theorem due to Milinovich which provides a full asymptotic for $\sum_{0<\Im(\rho)\le T}|\zeta'(\rho)|^2$, and we go further by establishing the full asymptotic for $\sum_{0<\Im(\rho)\le T}|\zeta^{(\nu)}(\rho)|^2$ for all positive integers $\nu$. Our theorem is entirely unconditional, but we provide sharper bounds on the assumption of the Riemann Hypothesis.