Real-rooted P\'olya-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 $1/2$. 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 $\Phi(t)$.