On the zeros of Riemann's Xi Function

Abstract

We consider Riemann&#39;s Xi function $ξ(s)$ which is evaluated at $s = \frac{1}{2} + σ+ i ω$, given by $ξ(\frac{1}{2} + σ+ i ω)= E_{pω}(ω)$, where $σ, ω$ are real and compute its inverse Fourier transform given by $E_p(t)$. We study the properties of $E_p(t)$ and a promising new method is presented which could be used to show that the Fourier Transform of $E_p(t)$ given by $E_{pω}(ω) = ξ(\frac{1}{2} + σ+ i ω)$ does not have zeros for finite and real $ω$ when $0 < |σ| < \frac{1}{2}$, corresponding to the critical strip excluding the critical line.