On the Discrete Mean of the higher Derivative of Hardy's $Z$-Function

TL;DR

This paper generalizes Yıldırım's asymptotic formula for the discrete moments of Hardy's Z-function's higher derivatives over its zeros, extending the analysis to broader cases.

Contribution

It provides a generalized asymptotic formula for the discrete moments of the higher derivatives of Hardy's Z-function over its zeros.

Findings

Derived an asymptotic formula for the discrete moments of higher derivatives of Hardy's Z-function.

Extended Yıldırım's results to a more general setting.

Contributed to understanding the distribution of zeros of derivatives of Hardy's Z-function.

Abstract

Y\i ld\i r\i m obtained an asymptotic formula of the discrete moment of $|\zeta(\frac{1}{2}+it)|$ over the zero of the higher derivatives of Hardy's $Z$-function. We give a generalization of his result on Hardy's $Z$-function.