Computation of Pair-Correlation Decay Constants for Riemann Zeta Zeros

\noindent\textbf{Description:}  
This notebook provides a reproducible framework for computing the decay constants $c_{1}$ and $c_{2}$ that quantify the pair-correlation behavior of the nontrivial zeros of the Riemann zeta function. The method is based on evaluating the exponential sum
\[
A(u;T) \;=\; \frac{1}{N(T)} \sum_{0<\gamma \leq T} e^{i\gamma u},
\]
where the $\gamma$ are ordinates of zeta zeros up to height $T$.

\noindent The algorithm proceeds as follows:
\begin{enumerate}
  \item \textbf{Zero Computation:} Compute the first $M$ nontrivial zeros of $\zeta(s)$ up to height $T$.
  \item \textbf{Exponential Sum Evaluation:} For a discretized grid of frequencies $u$, evaluate $A(u;T)$.
  \item \textbf{Threshold Selection:} Define thresholds $u_{\text{thresh}} = (\log T)^{-c_{1}}$ for fixed constants $c_{1}>0$.
  \item \textbf{Decay Measurement:} Record the supremum $\sup_{|u|\geq u_{\text{thresh}}} |A(u;T)|$.
  \item \textbf{Exponent Estimation:} Fit the decay law $\sup |A(u;T)| \ll (\log T)^{-\widehat{c}_{2}}$ to estimate the constant $c_{2}$.
\end{enumerate}

\noindent Both numerical tables and log--log plots are generated to demonstrate the decay pattern and verify consistency across different sample sizes. These computations provide empirical evidence in support of Montgomery's pair-correlation conjecture and are directly applicable to block cumulant factorization arguments in analytic number theory. The notebook is lightweight, self-contained, and executable on Google Colab with standard Python libraries.

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\noindent\textbf{Keywords:}  
Riemann zeta function; pair correlation; Montgomery conjecture; zeta zeros; exponential sums; decay constants; analytic number theory; numerical experiment.