On the zeros of Riemann's Xi Function

Abstract

We consider Riemann's Xi function $\xi (s)$ which is evaluated at $s=\frac{1}{2}+\sigma +i\omega$, given by $\xi (\frac{1}{2}+\sigma +i\omega )=E_{p\omega }(\omega )$, where $\sigma ,\omega$ 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 $E_{p\omega }(\omega )=\xi (\frac{1}{2}+\sigma +i\omega )$ does not have zeros for finite and real $\omega$ when $0<|\sigma |<\frac{1}{2}$, corresponding to the critical strip excluding the critical line.

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