On a generalisation of the Riemann $\xi$-function

TL;DR

This paper introduces a new entire function generalizing the Riemann xi-function, derived from higher derivatives of Hardy's Z-function, and explores its properties.

Contribution

It presents a novel entire function generalizing the Riemann xi-function based on higher derivatives of Hardy's Z-function and investigates its properties.

Findings

Defined a new entire function generalizing the Riemann xi-function.

Established properties of the new function.

Connected the function to higher derivatives of Hardy's Z-function.

Abstract

It is known that we can construct the meromorphic function $Z_k(s)$ associated with the higher derivative of Hardy's $Z$-function. In this paper, we introduce the entire function derived from $Z_k(s)$, a generalisation of the Riemann $\xi$-function and prove some properties.