On the harmonic continuation of the Riemann xi function

TL;DR

This paper extends the harmonic continuation of the Riemann xi-function to multiple dimensions, solving the Dirichlet problem in higher-dimensional spaces and introducing a new expansion based on Duffin's work.

Contribution

It generalizes the harmonic continuation of the Riemann xi-function to n-dimensions and provides a novel expansion method using Duffin's expansion.

Findings

Harmonic continuation of Riemann xi-function in n-dimensions achieved.

Solution to the Dirichlet problem on _{+}^{n+1} provided.

New expansion for harmonic continuation based on Duffin's expansion introduced.

Abstract

We generalize the harmonic continuation of the Riemann xi-function to the $n$-dimension case, to obtain the solution to the Dirichlet problem on $\mathbb{R}_{+}^{n+1}.$ We also provide a new expansion for the harmonic continuation of the Riemann xi-function using an expansion given by R.J. Duffin.