An extension to the complex plane of the Riemann-Siegel Z function

TL;DR

This paper extends the Riemann-Siegel Z function into the complex plane, creating a new complex function related to the Riemann Xi function that maintains similar error bounds across the critical strip.

Contribution

It introduces a novel complex extension of the Riemann-Siegel Z function, broadening its applicability beyond the critical line.

Findings

The new function depends on t and the distance from the critical line.

It maintains error bounds comparable to the original Z(t) on the critical line.

The extension covers at least the entire critical strip.

Abstract

The usual Riemann-Siegel Z(t) is a real-valued function. We construct a complex function depending from t and from distance from critical line. It is linked to Riemann Xi(s) function by the same real scaling factor of the usual Riemann-Siegel Z(t) on critical line. Errors are not greater than the errors of Riemann-Siegel Z(t) on the critical line, while this result covers at least the whole critical strip.