**Polya-Hurwitz** program.

This may become more interesting in light of the recent progress in the **Polya-Jensen** program by Griffin, Ono, Rolen, Zagier.

We will first provide definitions of some functions involved.

The Riemann Xi-function $\Xi(z)$ is related to the Riemann zeta-function $\zeta(s)$ via ([A], [B]): $\Xi(z)=\xi(\tfrac{1}{2}+iz)$,
$\xi(s)=\tfrac{1}{2}s(s-1)\pi^{-s/2}\Gamma\left(\frac{s}{2}\right)\zeta(s)$.

Riemann $\Xi(z)$ can be expressed as a Fourier transform of a positive, fast decaying, and even kernel $\Phi(t)$ ([A], [B]):

\begin{equation}
\Xi(z)=2\int_0^{\infty}\Phi(t)\cos(zt)\mathrm{d}t,\tag{1}
\end{equation}

where

\begin{equation}
\Phi(t)=\sum_{k\geqslant 1}\phi_k(t)=\Phi(-t),\tag{2}\\
\end{equation}

\begin{equation}
\phi_k(t)=\left(4\pi^2 k^4 e^{9t/2}-6\pi k^2e^{5t/2}\right)\exp\left(-\pi k^2 e^{2t}\right)\tag{3}.
\end{equation}

The Polya aspect of this approach is the following:

Truncate the Kernel $\Phi(t)$ of (2) and/or the integration range in (1) such that the resulting Fourier transform leads to a family of entire functions which only have real roots.

One such candidate is given in [C]:
\begin{equation}
\Phi_{\color{red}n}(t)=(1/2)\sum_{1\leqslant k\leqslant {\color{red}n}}\left(\phi_k(t)+\phi_k(-t)\right)=\Phi_{\color{red}n}(-t)\tag{4}
\end{equation}

\begin{equation}
\Xi_{\color{red}n}(z)=2\int_0^{(1/2)\log {\color{red}n}}\Phi_{\color{red}n}(t)\cos(zt)\mathrm{d}t=\Xi_{\color{red}n}(-z),\tag{5}
\end{equation}

We refer to [D] and [E] for a near complete review on the zeros of entire functions as Fourier transforms.

The Hurwitz aspect of this approach is the following:

**Corollary of Hurwitz's theorem** in complex analysis (applied to our case) [F]:

If $\Xi(z)$ and $\{\Xi_n(z)\}$ are analytic functions on a domain $S_{1/2}(z)=\{z: 0<Im(z)<1/2\}$, $\{\Xi_n(z)\}$ converges to $\Xi(z)$ uniformly on compact subsets of $S_{1/2}(z)$, and all but finitely many $\Xi_n(z)$ have no zeros in $S_{1/2}(z)$, then either $\Xi(z)$ is identically zero or $\Xi(z)$ has no zeros in $S_{1/2}(z)$.

The functional equation for $\zeta(s)$ becomes $\Xi(-z)=\Xi(z)$. The candidate of $\Xi_n(z)$ in (5) automatically satisfies this functional equation.

Another benefit of Polya-Hurwitz approach is that the entrance barrier is relatively low (comparing to other approaches that usually require the advance knowledge of analytical number theory).

To get started, one only needs to know Fourier transform, basic complex analysis, some knowledge of entire functions, polynomials etc. So anyone who has math training with the college undergraduate math major may start to work on the Polya-Hurwitz approach and learn other necessary new math as he/she goes.

The most difficult part of Polya-Hurwitz approach seems to be the following:(for example,) proving that all the zeros of $\Xi_n(z)$ in (5) are real in $S_{1/2}(z)$.

One may need to have several iterations: guess one form of the Kernel like $\Phi_n(1,t)$ and complete the integration to get explicit expression for $\Xi_n(1,z)$. If all the zeros of $\Xi_n(1,z)$ are found not to be all real in $S_{1/2}(z)$, then move on to $\Phi_n(2,t)$ and $\Xi_n(2,z)$...

References:

[A] Titchmarsh,"The Theory of the Riemann Zeta-Function",

(1986).

[B] Edwards, "Riemann's Zeta Function", (1974).

[C] Shi, "On the zeros of Riemann Xi-function", (2017) arXiv:1706.08868.

[D] Dimitrov and Rusev, “ZEROS OF ENTIRE FOURIER TRANSFORMS” (2001), 108 page review paper.

[E] Hallum, “ZEROS OF ENTIRE FUNCTIONS REPRESENTED BY FOURIER TRANSFORMS” (2014), Master thesis.

[F] Conway, "Functions of One Complex Variable",(1978)

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