Riemann's Zeta Function: The alternating Xi-Function Xia(s)

TL;DR

This paper explores the alternating zeta function and its associated entire function, deriving formulas for numerical evaluation and verifying their accuracy, thereby contributing to the understanding of related special functions.

Contribution

It introduces the entire function {\xi}_a(s) related to the alternating zeta function and develops formulas for its numerical evaluation based on incomplete gamma functions.

Findings

Derived formulas for evaluating {\xi}_a(s) numerically

Verified formulas through numerical examples

Established the functional equation for {\xi}_a(s)

Abstract

As well known, the study of Riemanns zeta function {\zeta}(s) involves the related entire function {\xi}(s). A close relative of {\zeta}(s) is the alternating zeta function {\eta}(s). Similar to {\zeta}(s), also {\eta}(s) has a corresponding entire function {\xi}_a (s). After establishing its definition and a related functional equation, formulas based on incomplete gamma functions are worked out, allowing to numerically evaluate {\xi}_a (s). Examples verifying the obtained formulas are included.