For all problems we considered this produces better bounds than any bandlimited construction.

Background
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Let $\zeta(s)$ be Riemann’s zeta-function. It is well-known that all non-trivial zeros of $\zeta(s)$ are located in the critical strip $0<{{\rm Re}\,}s<1$, and the Riemann Hypothesis (RH) is the statement that all these zeros are aligned in the line ${{\rm Re}\,}s=1/2$. Let $N(T)$ count the number of zeros $\rho = \beta + i \gamma$ of $\zeta(s)$, repeated according the multiplicity, such that $0<\beta<1$ and $0 <\gamma \leq T$. The Riemann-von Mangoldt formula (in its weaker form) states that $$\begin{aligned}
\label{RvMformula}
N(T) =  (1+o(1)) \frac{T}{2\pi} \log T.\end{aligned}$$ Let $$N^{*}(T):=\displaystyle\sum_{0<\gamma\leq T}m_{\rho},$$ where the sum is over the non-trivial zeros of $\zeta(s)$ counting multiplicities[^1] and $m_\rho$ is the multiplicity of $\rho$. In addition to RH, it is also conjectured that all zeros of $\zeta(s)$ are simple, and therefore it is [*conjectured*]{} that $$\begin{aligned}
 \label{NstarNconjecure}
N^{*}(T) \sim N(T).\end{aligned}$$ To study the distribution of the zeros of zeta, Montgomery defined the pair correlation function $$\label{paircorrelfunction}
N(x,T):=\sum_{\substack{0< {\gamma},{\gamma}'\leq T \\ 0<{\gamma}'-{\gamma}\leq \frac{2\pi x}{\log T}}} 1$$ and [*conjectured*]{} that $$N(x,T) \sim N(T)\int_0^x \bigg(1-\frac{\sin^2(\pi y)}{(\pi y)^2}\bigg){\,\text{\rm d}}y.$$ Note that by the average gap between zeros is $\frac{2\pi}{\log T}$, hence $N(x,T)$ is counting zeros 