A Further Generalisation of Sums of Higher Derivatives of the Riemann Zeta Function

We prove an asymptotic for the sum of ζ⁽ⁿ⁾ (ρ)Xρ where ζ⁽ⁿ⁾ (s) denotes the nth derivative of the Riemann zeta function, X is a positive real and ρ 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 → ∞. 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.