Real-rooted P\'olya-like approximations to the Riemann Xi-function

TL;DR

This paper develops improved Pólya-like approximations for the Riemann Xi-function's kernel, ensuring real zeros in the Fourier transform and better capturing the kernel's behavior at both ends.

Contribution

The authors introduce a refined approximation method for the kernel of the Riemann Xi-function that guarantees real zeros and matches the kernel at both zero and infinity.

Findings

The new approximation maintains real zeros in the Fourier transform.

It accurately captures the kernel's behavior at both t=0 and t→∞.

The method improves upon Pólya's original approximation.

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 $1/2$. P\'olya 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 $0\leqslant b<1,c\in\R$, $\cosh t+c \cosh(bt)$ goes to $\cosh t$. Thus we improve P\'olya'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 P\'olya'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 $\Phi(t)$.