Generalisations of the Landau--Gonek Theorem and applications to mean values of zeta

The Landau--Gonek Theorem evaluates $X^ρ$ summed over the non-trivial zeros of the Riemann zeta function. Their result shows great sensitivity to the arithmetic nature of $X$. We prove a related result concerning the sum of $χ(ρ) X^ρ$ over the zeros of zeta, where $χ(s)$ is the term arising in the functional equation for the zeta function. Again, this result depends deeply on whether $X$ is an integer or not. We show the result splits into three cases, depending on whether $X$ is smaller than $T$, about the same size as $T$, or bigger than $T$. The reason this result is useful is that it easily permits the calculation of discrete moments of the Riemann zeta function via the approximate functional equation. As an application of this result, we provide an alternative proof of Shanks' conjecture.