Are there any new results on approximating Riemann $\Xi$ function by Polya-like Fourier transforms?

Question

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

Answer 1

I've been hoping someone else would take a stab at Question 2. For myself, the answer relates to the work of de Bruijn (and Newman) mentioned in (3) above; see this question The Riemann zeros and the heat equation for more on the de Bruijn-Newman constant $\Lambda$. Random matrix models for the spacing of the Riemann zeros lead me to believe there are infinitely many 'Lehmer pairs': closer than average pairs of Riemann zeros. (You may read about this in the section on 'Lehmer pairs' in some of Odlyzko's preprints on very high Riemann zeros). And the work of Csordas and others would then mean the de Bruijn-Newman constant is $0$: the Riemann hypothesis comes as close to failing as it possibly could. So $\zeta(s)$ is extremely sensitive to deformation, which discourages me from this approach.

Answer 2

Ki has also written a paper called 'The Riemann Xi-function under repeated differentiation" which uses the Fourier transform to calculate derivatives of the Xi-function. I am currently trying to improve upon his work for my PhD. I think that someone has calculated the number of zeros on the real axis as being something like 1+O(x^(-n)) as n is the nth derivative, but I can't find where I read that.