Rigorous characterization of the size of \(\mathfrak{F}(N)\)

Prove rigorously the empirically observed behavior of the cardinality \(|\mathfrak{F}(N)|\), where \(\mathfrak{F}(N)\) is the set of integers \(r\) for which there exists a divisor \(f\) of \(Nr+1\) satisfying \([\mathcal{X}(N,r,f)]>0\), particularly the reported relation for prime \(N\) involving \((\phi(N)-2)/(2(\log\log N-C_N))\).

Background

The paper defines $\mathfrak{F}(N)$ as the set of distinct integers $r$ for which some divisor $f$ of $Nr+1$ yields a set $\mathcal{X}(N,r,f)$ consisting entirely of positive integers. For prime $N$, empirical data suggest that $|\mathfrak{F}(N)|$ is substantially smaller than the elementary upper bound $(\phi(N)-2)/2$ as $N$ grows.

The authors introduce a quantity $C_N$ through the empirical relation $|\mathfrak{F}(N)|=(\phi(N)-2)/(2(\log\log N-C_N))$ and report that $C_N$ appears to increase with $N$. Establishing this observed behavior rigorously is explicitly identified as unresolved.