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 $\Phi (t)=4e^{9t/2}{\theta }^{\prime \prime }(e^{2t})+6e^{5t/2}{\theta }^{\prime }(e^{2t})$. 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 $\{{\Phi }_n(t){\}}_{n⩾2}$ whose Fourier transform on compact support $(-\frac{1}{2}\log n,\frac{1}{2}\log n)$, $\{F(n,z){\}}_{n⩾2}$, 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 $\{H(14,n,z){\}}_{n⩾2}$ 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 $\{W(n,z){\}}_{n⩾8}:=\{H(14,n,2z/\log n){\}}_{n⩾8}$ and show that if all the zeros of $\{W(n,z){\}}_{n⩾8}$ in the critical strip $S_{1/2}$ are real, then all the zeros of $\{H(14,n,z){\}}_{n⩾8}$ 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 $N⩾8$ such that for all $n⩾N$, 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 $W(n⩾N,z^{1/2})$ has only real, positive and simple zeros. Thus $W(n⩾N,z)$ 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.

5 figures. arXiv admin note: text overlap with arXiv:1107.5483 by other authors

References in corpus (3)

• Numerical analysis of minimal beta-sequences associated with a family of entire functions
• Analysis of Riemann Zeta-Function Zeros using Pochhammer Polynomial Expansions
• Riemann's Zeta Function: The alternating Xi-Function Xia(s)