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

2021/11/04 by Andrew Pearce‐Crump, Pearce-Crump, Andrew
Mathematics · #Analytic Number Theory Research #Meromorphic and Entire Functions #Advanced Mathematical Theories

paper · pdf · doi:10.48550/arxiv.2111.02756

Abstract

We prove an asymptotic for the sum of ζ⁽ⁿ⁾ (ρ)X^ρ 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.

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