On the zeros of Riemann $\Xi(z)$ function

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.