A Note on the Riemann $ξ$-Function

Abstract

This note investigates a number of integrals of and integral equations satisfied by Riemann's ξ−function. A different, less restrictive, derivation of one of his key identities is provided. This work centers on the critical strip and it is argued that the line s = 3/2 + it , e.g., contains a holographic image of this region.

An Integral Equation for Riemann's Zeta Function and its Approximate Solution

“…The author comments: The fundamental result of this paper is a proof that ζ(s 1 ) is related to the same function ζ(s 2 ) integrated over a contour through the intermediary of a specific integral operator. This is a rigorous result, which, as noted earlier, has been shown [15,Theorem 1] to be equivalent to the Cauchy Contour Integral Theorem. In this paper, a specific example has been chosen such that s 2 covers the 1-line through the intermediary of the functional equation (2.7).…”

“…Figure(15) This Figure details the properties of MI (σ + it, v) when t = 100, σ = 0.1 and v < 0 (left) and v > 0 (right). Shown is the value of the extremum (dotted) as well as the location of one of its half-width intercepts (dashed) on the v−axis to first (asymptotic) order in t −1 .…”