Scatter Plot Analysis of Iterative Divisor-Sum Ratios $R_k(n)=\sigma^{(k)}(n)/n$

This repository contains the Jupyter notebook \texttt{scatter\_Rk\_density.ipynb}, which computes and visualizes the normalized iterative divisor-sum ratios:
\[
R_k(n) = \frac{\sigma^{(k)}(n)}{n},
\]
where $\sigma(n)$ is the sum-of-divisors function and $\sigma^{(k)}(n)$ denotes its $k$-fold iterate.

The notebook implements an \emph{efficient sieve-based algorithm} to calculate $\sigma(n)$ and its iterates for large ranges of $n$, making it suitable for high-resolution empirical studies. It produces a \emph{scatter plot} of $R_k(n)$ versus $n$ and saves the figure in \textbf{PDF format} for direct inclusion in \LaTeX{} documents.

\subsection*{Key Features}
\begin{itemize}
    \item Efficient computation of $\sigma(n)$ using a \textbf{sieve-based algorithm}.
    \item Iterative computation of $\sigma^{(k)}(n)$ for arbitrary $k \ge 1$.
    \item Visualization of $R_k(n)$ up to large $n$.
    \item Scatter plot highlighting \emph{measure density} and \emph{asymptotic behavior}.
    \item Generates a \textbf{publication-ready PDF} figure for research papers.
\end{itemize}

\subsection*{Applications}
\begin{itemize}
    \item Empirical analysis of \emph{iterated arithmetic functions}.
    \item Investigation of \emph{measure density} and \emph{boundedness} properties.
    \item Numerical support for conjectures related to divisor-sum iterates (e.g., Schinzel’s conjecture).
\end{itemize}

\subsection*{Output}
\begin{itemize}
    \item \texttt{scatter\_density\_Rk.pdf} — scatter plot visualization.
    \item Ready-to-use \LaTeX{} figure integration.
\end{itemize}

\subsection*{Requirements}
\begin{itemize}
    \item Python 3.x
    \item \texttt{numpy} and \texttt{matplotlib}
    \item Optional: \texttt{sympy} (if using direct divisor enumeration)
\end{itemize}