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

Andrew Pearce-Crump

doi:10.1142/s1793042125500174

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

Andrew Pearce-Crump^*

^*Corresponding author for this work

School of Mathematics

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Abstract

We obtain a full asymptotic for the sum of ζ⁽ⁿ⁾(ρ)X^ρ, where ζ⁽ⁿ⁾(s) denotes the n^th derivative of the Riemann zeta function, X is a positive real number, and ρ denotes a nontrivial 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.

Original language	English
Journal	International Journal of Number Theory
Early online date	7 Nov 2024
DOIs	https://doi.org/10.1142/s1793042125500174
Publication status	E-pub ahead of print - 7 Nov 2024

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title = "A Further Generalisation of Sums of Higher Derivatives of the Riemann Zeta Function",

abstract = "We obtain a full 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 number, and ρ denotes a nontrivial 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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AB - We obtain a full 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 number, and ρ denotes a nontrivial 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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DO - 10.1142/s1793042125500174

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JO - International Journal of Number Theory

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

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