I posted [this question][1] at math.stackexchange.com and was told that it is more appropriate to post this research related question here at mathoverflow.

So I re-post it below.

Riemann $\Xi(z)$ function is related to Riemann $\zeta(s)$ function via ($s=1/2+i z$):

$$\Xi(z)=\frac{1}{2}s(s-1)\pi^{-s/2}\Gamma(s/2)\zeta(s)$$

The functional equation for $\zeta(s)$ is equivalent to $\Xi(z)=\Xi(-z)$.

Riemann $\Xi(z)$ function can be expressed as a Fourier transformation:

$$\Xi(z)=2\int_0^{\infty}\Phi(u)\cos(z u){\rm d}u$$

where $$\Phi(u)=\sum_{n=1}^{\infty}\left(4\pi^2n^4\exp(9u/2)-6\pi n^2\exp(5u/2)\right)\exp\left(-\pi n^2 \exp(2u)\right)=\Phi(-u)$$

(1) Polya approximated $\Phi(u)$ with $\Phi_{*}(u)$ and $\Phi_{**}(u)$:

$$\Phi_{*}(u)=8\pi^2\cosh(9u/2)\exp\left(-2\pi \cosh(2u)\right)$$

$$\Phi_{**}(u)=\left(8\pi^2\cosh(9u/2)-6\pi\cosh(5u/2)\right)\exp\left(-2\pi \cosh(2u)\right)$$

This is because he noticed that when $u\to\infty$, $\Phi(u)\to\Phi_{*}(u)$ and $\Phi(u)\to\Phi_{**}(u)$.

Polya proved that the resulting $\Xi_*(z)$ and $\Xi_{**}(z)$ have real zeros only.

(2) de Bruijn approximated $\Phi(u)$ with $\Phi_d(u)$:

$$\Phi_d(u)=2\cosh(5u/2)\left(2\pi^3-3\pi+4\pi^2\cosh(u)\right)\exp\left(-2\pi \cosh(2u)\right)$$

de Bruijn proved that the resulting $\Xi_d(z)$ has real zeros only.

(3) de Bruijn also approximated $\Phi(u)$ with $\Phi_\lambda(u)$:

$$\Phi_\lambda(u)=\exp(\lambda u^2)\Phi(u)$$

de Bruijn proved that when $\lambda\ge \frac{1}{8}$,the resulting $\Xi_\lambda(z)$ has real zeros only.

Newman showed that when $\lambda\lt 0$,$\Xi_\lambda(z)$ has non-real zeros as well.

(@SylvainJULIEN pointed out that) The so-called de Bruijn-Newman constant $\Lambda$ is defined in such a way that $4\lambda \ge \Lambda$ implies $\Xi_\lambda(z)$ has only real zeros.

(4) Hejhal approximated $\Phi(u)$ with $\Phi_N(u)$:

$$\Phi_N(u)=\sum_{n=1}^{\infty}\left(8\pi^2n^4\cosh(9u/2)-12\pi n^2\cosh(5u/2)\right)\exp\left(-2\pi n^2 \cosh(2u)\right)$$

Hejhal proved that almost all the zeros of the resulting $\Xi_N(z)$ are real. However when $N\to\infty$ $\Phi_N(u) \not\to \Phi(u)$.

For more details please refer to two review papers by Dimitrov and Rusev [1] and Ki [2] and references therein.

[1]: Dimitrov and Rusev, The zeros of entire Fourier transforms, EAST JOURNAL ON APPROXIMATIONS Volume 17, Number 1 (2011), 1-108

[2]: Ki, The Zeros of Fourier Transforms

Question 1: Are there any new research results on approximating Riemann $\Xi(z)$ by Fourier transforms?

Question 2: Why approximating Riemann $\Xi(z)$ by Fourier transforms does not seem to be an active research field towards a possible proof of Riemann hypothesis$?

Best regards- mike