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On the harmonic continuation of the Riemann xi function

Alexander E. Patkowski

TL;DR

This work extends the harmonic continuation framework for the Riemann xi function to the $n$-dimensional setting, solving the Dirichlet problem on $\mathbb{R}_{+}^{n+1}$ by convolving boundary data $g(\mathbf{x})=\prod_{l=1}^{n} \frac{\Xi(x_l)}{x_l^2+\frac{1}{4}}$ with the Poisson-type kernel $K_y$ to obtain a harmonic function $u(\mathbf{x},y)=(g*K_y)(\mathbf{x})$, and provides an explicit boundary trace $u(0,y)$ as a multidimensional integral. The paper also derives an alternative, dimension-reducing route that recovers the $n=1$ Corollary 1.2 via a diagonalization approach and a Laplace-transform method. Additionally, a Duffin-type expansion is developed for the harmonic continuation of $f(x)=\frac{\Xi(x)}{x^2+\frac{1}{4}}$, giving an explicit double-series representation in terms of the Möbius function $\mu(m)$ and the function $\psi$, and discusses possible implications for Riemann Hypothesis criteria through the behavior of the continuation as $y\to0$ and the choice of $x$ at zeta zero ordinates.

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.

On the harmonic continuation of the Riemann xi function

TL;DR

This work extends the harmonic continuation framework for the Riemann xi function to the -dimensional setting, solving the Dirichlet problem on by convolving boundary data with the Poisson-type kernel to obtain a harmonic function , and provides an explicit boundary trace as a multidimensional integral. The paper also derives an alternative, dimension-reducing route that recovers the Corollary 1.2 via a diagonalization approach and a Laplace-transform method. Additionally, a Duffin-type expansion is developed for the harmonic continuation of , giving an explicit double-series representation in terms of the Möbius function and the function , and discusses possible implications for Riemann Hypothesis criteria through the behavior of the continuation as and the choice of at zeta zero ordinates.

Abstract

We generalize the harmonic continuation of the Riemann xi-function to the -dimension case, to obtain the solution to the Dirichlet problem on We also provide a new expansion for the harmonic continuation of the Riemann xi-function using an expansion given by R.J. Duffin.
Paper Structure (3 sections, 5 theorems, 24 equations)

This paper contains 3 sections, 5 theorems, 24 equations.

Table of Contents

  1. Introduction and main result
  2. Proof of the $n$-dimensional case and related comments
  3. The Duffin expansion of the harmonic continuation of the Riemann xi function

Key Result

Theorem 1.1

([9, Theorem 3.1]) The solution of Dirichlet's problem in the half plane, where $y\in\mathbb{R},$$x\ge0,$ initial condition $w(y,0)=h(y):=\bar{\gamma}(\frac{1}{2}+iy)L(\frac{1}{2}+iy),$$w(y,x)\rightarrow0,$ as $|y|\rightarrow\infty,$ is given by the Poisson integral. Furthermore, the solution also satisfies the condition $w(0,s-\frac{1}{2})=h(-i(s-\frac{1}{2}))-r(s),$ for

Theorems & Definitions (6)

  • Theorem 1.1
  • Corollary 1.1.1
  • Theorem 1.2
  • Theorem 3.1
  • Theorem 3.2
  • proof