This is a naive attempt to show that the Riemann $\zeta $ function is non-vanishing over line $\text{Re}(s)=0$:
Let us devise an integral representation for the $\zeta$ function which works over a set containing the $\text{Re}(s)=0$ line. For $\text{Re}(s)>0$ we have $$ \zeta(s)=\left(1-\frac{2}{2^s}\right)^{-1}\sum_{n\geq 1}\frac{(-1)^{n+1}}{n^s}=\left(1-\frac{2}{2^s}\right)^{-1}\frac{1}{\Gamma(s)}\int_{0}^{+\infty}\frac{t^{s-1}}{e^t+1}\,dt $$ and by integration by parts $$ \zeta(s) = \left(1-\frac{2}{2^s}\right)^{-1}\frac{1}{\Gamma(s+1)}\int_{0}^{+\infty}\frac{t^s}{(e^{t/2}+e^{-t/2})^2}\,dt $$ is a valid integral representation over half-plane $\text{Re}(s)>-1$. Assuming $s=iT$, the first two factors are clearly non-vanishing, hence the problem boils down to showing that the integral $$ \int_{0}^{+\infty}\frac{\exp\left(iT\log u\right)}{\cosh^2 u}\,du = \int_{-\infty}^{+\infty}\frac{\exp(iTz)}{e^{-z}\cosh^2 e^z}\,dz=\mathscr{F}\left(\frac{e^z}{\cosh^2 e^z}\right)(T)$$ is non-vanishing. The Fourier transform of the similarly shaped function $\frac{e^z}{\exp(2e^z)}$ is explicitly given by $\frac{2^{-iT}\Gamma(1+iT)}{2\sqrt{2\pi}}$, which is blatantly non-vanishing. A better approximation is $\exp\left(z-2e^z\right)\left[4-2\exp\left(-\frac{e^z}{2}\right)\right]$, whose Fourier transform is given by $\frac{1}{5\sqrt{2\pi}}2^{1-iT}\left(5-2^{1+2iT}5^{-iT}\right)\,\Gamma(1+iT)$. This is also non-vanishing for $T\in\mathbb{R}$.
Q1. Is it possible to find a sequence of approximations for $\frac{e^z}{\cosh^2 e^z}$ such that the uniform error is convergent to zero and the Fourier transforms of such approximations maintain the same blatantly-non-vanishing-over-$\mathbb{R}$ structure? I would be happier to avoid the Poisson summation formula, since invoking it is essentially equivalent to invoking the reflection formula for the $\zeta$ function, and I am already aware of classical proofs exploiting such principle.
Q2. Given a function $f$ in the Schwartz space $\mathcal{S}(\mathbb{R})$, what is known about conditions ensuring that $\widehat{f}$ is non-vanishing over the real line? A discrete analogue is the Fejér-Riesz theorem, also mentioned in this previous question.
