A further generalisation of sums of higher derivatives of the Riemann Zeta Function

Abstract

We prove an asymptotic for the sum of ${\zeta }^{(n)}(\rho )X^{\rho }$ where ${\zeta }^{(n)}(s)$ denotes the $n$th derivative of the Riemann zeta function, $X$ is a positive real and $\rho$ denotes a non-trivial zero of the Riemann zeta function. The sum is over the zeros with imaginary parts up to a height $T$, as $T\to \infty$. We also specify what the asymptotic formula becomes when $X$ is a positive integer, highlighting the differences in the asymptotic expansions as $X$ changes its arithmetic nature.