Real-rooted Pólya-like approximations to the Riemann Xi-function

Abstract

The Riemann $\Xi (z)$ function admits a Fourier transform of a even kernel $\Phi (t)$. The latter is related to the derivatives of Jacobi theta function $\theta (z)$, a modular form of weight $1/2$. Pólya noticed that when $t$ goes to infinity, $e^t$ goes to $e^t+e^{-t}=2\cosh t$. He then approximated the kernel $\Phi (t)$ by ${\Phi }_P(t)$ that contained only the leading term and with $\exp t,\exp (9t/4)$ replaced by $2\cosh t,2\cos (9t/4)$. This procedure captured almost all of the contribution from the tail part (i.e., $t\to \infty$) of the kernel $\Phi (t)$. We realize that when $t$ goes to infinity and $0⩽b<1,c\in \mathbb{R}$, $\cosh t+c\cosh (bt)$ goes to $\cosh t$. Thus we improve Pólya's approximation by replacing $\cosh (9t/4)$ with $\cosh (9t/4)+b{\sum }_{k=0}^{m-1}{b}_k\cosh (9kt/(4m))$ and adjusting the parameters $b,b_k,m$ such that (A) the approximated kernel ${\Phi }_S(b,b_k,m;t)$ goes to $\Phi (t)$when $t$ goes to infinity;(B) ${\Phi }_S(b,b_k,m;t)$ is identical to $\Phi (t)$ at $t=0$; (C) the Fourier transform of ${\Phi }_S(b,b_k,m;t)$,like in Pólya's case, has only real zeros. Since this procedure also captures almost all of the contribution from the head part (i.e., near $t=0$) of the kernel $\Phi (t)$, we are able to anchor both ends of the kernel $\Phi (t)$.

21 pages, 17 figures

References in corpus (1)

• Lee-Yang measures and wave functions

Cited by in corpus (1)

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