I am sorry that this is long post. But it might be of interest to you.

This post is related to [zeros of partial sum of Taylor series of $e^x-1$] 1.

Entire functions $e^z$, $\cos(z)$, and $\sin(z)$ can be represented by the following Taylor series: $$ e^z = \sum_{k=0}^\infty \frac{1}{k!}z^{k} $$ $$ \cos(z) = \sum_{k=0}^\infty \frac{(-1)^k}{(2k)!}z^{2k} $$ $$ \sin(z) = \sum_{k=0}^\infty \frac{(-1)^k}{(2k+1)!}z^{2k+1} $$

Let us define the corresponding partial sums (polynomials in $z$) as: $$ e_n(z) = \sum_{k=0}^n \frac{1}{k!}z^{k} $$ $$ \cos_n(z) = \sum_{k=0}^{[n/2]} \frac{(-1)^k}{(2k)!}z^{2k} \text{ (n even)}$$ $$ \sin_n(z) = \sum_{k=0}^{[(n-1)/2]} \frac{(-1)^k}{(2k+1)!}z^{2k+1} \text{ (n odd)}$$

Szegő (1924) showed that the zeros of polynomials $e_n(nz)$ clustered on a Jordan curve (now is refered as Szegő curve in the literature): $|ze^{1-z}|=1, |z|≤1$.

Szegő also showed that zeros of polynomials $\sin_n(nz)$ and $\cos_n(nz)$ clustered on the joint of curve (1) $|-ize^{1+iz}|=1, |z|≤1, Im(z)\ge 0$, curve (2) $|ize^{1-iz}|=1, |z|≤1, Im(z)\le 0$ and line segment $(-\frac{1}{e},+\frac{1}{e})$.

For detailed information, please refer to papers by Kappert (1996) "On the zeros of the partial sums of $\cos(z)$ and $\sin(z)$" and by Varga and Carpenter etc "Zeros of partial sums of $\cos(z)$ and $\sin(z)$ I(2000), II(2001), III(2010)" and references therein.

The Riemann $\Xi(z)$ function is an entire function related to the Riemann $\zeta(s)$ function ($s=1/2+iz$) via (Titchmarsh, p16):

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

$\Xi(z)$ can be expressed as a Fourier transform:

$$\Xi(z)=2\int_0^{\infty}\Phi(u)\cos(z u){\rm d}u$$ where $$\Phi(u)=\sum_{n=1}^{\infty}\left(4\pi^2n^4\exp(9u/2)-6\pi n^2\exp(5u/2)\right)\exp\left(-\pi n^2 \exp(2u)\right)=\Phi(-u)$$

Expanding $\cos(zu)$ as a Taylor series turned function $\Xi(z)$ into a Taylor series as well:

$$ \Xi(z) = \sum_{n=0}^\infty \frac{(-1)^k}{(2k)!}b_{2k}z^{2k} $$

$$b_{k}=2\int_0^{\infty}u^{k}\Phi(u){\rm d}u$$

For details and the numerical values of the first few coefficients $b_{2k}$, please refer to paper by CSORDAS, NORFOLK , VARGA (1986) "THE RIEMANN HYPOTHESIS" AND THE TURAN INEQUALITIES, TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 296, Number 2.

I saw a paper on the internet that has plots of $\Xi_n(nz)$ with a few values of $n$. I verified it by myself. The pattern of zeros are very much like that for $\cos_n(nz)$.

The similarity of the patterns of zeros between the Taylor series of $\Xi(z)$ and of $\cos(z)$ leads us to define the counterparts of $\sin(z)$ and $e^z$ as:

$$ \Psi(z) = \sum_{n=0}^\infty \frac{(-1)^k}{(2k+1)!}b_{2k+1}z^{2k+1} $$

$$ E(z) = \sum_{n=0}^\infty \frac{1}{k!}b_{k}z^{k} $$

The corresponding Fourier transforms for these two functions are given by:

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

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

Question (1): Does the zeros of polynomials $E_n(nz)$ cluster on a Szegő curve: $|zE^{1-z}|=1, |z|≤1$?

Question (2): Do the zeros of polynomials $\Psi_n(nz)$ and $\Xi_n(nz)$ cluster on the joint of curve (1) $|-izE^{1+iz}|=1, |z|≤1, Im(z)\ge 0$, curve (2) $|izE^{1-iz}|=1, |z|≤1, Im(z)\le 0$ and line segment $(-\frac{1}{E(1)},+\frac{1}{E(1)})$?