On the zeros of Riemann $Ξ(z)$ function (1706.08868v1)

Abstract: The Riemann $\Xi(z)$ function (even in $z$) admits a Fourier transform of an even kernel $\Phi(t)=4e^{9t/2}\theta''(e^{2t})+6e^{5t/2}\theta'(e^{2t})$. Here $\theta(x):=\theta_3(0,ix)$ and $\theta_3(0,z)$ is a Jacobi theta function, a modular form of weight $\frac{1}{2}$. (A) We discover a family of functions ${\Phi_n(t)}{n\geqslant 2}$ whose Fourier transform on compact support $(-\frac{1}{2}\log n, \frac{1}{2}\log n)$, ${F(n,z)}{n\geqslant2}$, converges to $\Xi(z)$ uniformly in the critical strip $S_{1/2}:={|\Im(z)|< \frac{1}{2}}$. (B) Based on this we then construct another family of functions ${H(14,n,z)}{n\geqslant 2}$ and show that it uniformly converges to $\Xi(z)$ in the critical strip $S{1/2}$. (C) Based on this we construct another family of functions ${W(n,z)}{n\geqslant 8}:={H(14,n,2z/\log n)}{n\geqslant 8}$ and show that if all the zeros of ${W(n,z)}{n\geqslant 8}$ in the critical strip $S{1/2}$ are real, then all the zeros of ${H(14,n,z)}{n\geqslant 8}$ in the critical strip $S{1/2}$ are real. (D) We then show that $W(n,z)=U(n,z)-V(n,z)$ and $U(n,z^{1/2})$ and $V(n,z^{1/2})$ have only real, positive and simple zeros. And there exists a positive integer $N\geqslant 8$ such that for all $n\geqslant N$, the zeros of $U(n,x^{1/2})$ are strictly left-interlacing with those of $V(n,x^{1/2})$. Using an entire function equivalent to Hermite-Kakeya Theorem for polynomials we show that $W(n\geqslant N,z^{1/2})$ has only real, positive and simple zeros. Thus $W(n\geqslant N,z)$ have only real and imple zeros. (E) Using a corollary of Hurwitz's theorem in complex analysis we prove that $\Xi(z)$ has no zeros in $S_{1/2}\setminus\mathbb{R}$, i.e., $S_{1/2}\setminus \mathbb{R}$ is a zero-free region for $\Xi(z)$. Since all the zeros of $\Xi(z)$ are in $S_{1/2}$, all the zeros of $\Xi(z)$ are in $\mathbb{R}$, i.e., all the zeros of $\Xi(z)$ are real.