A Prime-Based Continued Fraction Constant ρ = [2; 3, 5, 7, 11, …]: Convergence, Prime-Indexed Engel/Egyptian Expansions, and Conditional Estimates Involving the Riemann Hypothesis

This paper studies the constant \(\rho = {\lbrack 2;3,5,7,11,13,17,\ldots\rbrack}\) whose partial quotients are the prime numbers. Our analysis emphasizes convergence properties and prime-indexed Engel/Egyptian expansions, together with conditional estimates that depend on known error bounds for the Chebyshev function \(\vartheta{(x)}\) under the Riemann Hypothesis. The growth of the convergent denominators is framed by primorial products and by \(\vartheta\), and the standard inequality for convergents implies an irrationality exponent of 2. Under the classical conditional error term for \(\vartheta{(x)}\), the leading asymptotic for \(\log Q_{n}\) is sharpened at a square-root scale. The role of Engel and Egyptian expansions is twofold: they provide prime-driven decompositions of the fractional part of \(\rho\) and a baseline for comparing denominator growth against continued-fraction convergents. We supply computable upper bounds for \(|{\rho - {P_{n}/Q_{n}}}|\), error audits for the first 25 convergents, high-precision digits of \(\rho\), and comparative plots linking \(\log Q_{n}\) to \(\vartheta{(p_{n + 1})}\). The contribution is conceptual—an elementary framework that connects prime-indexed partial quotients to \(\vartheta\) and to classical Diophantine estimates—and empirical, via a compact reproducibility bundle. All statements that reference the Riemann Hypothesis are strictly conditional applications of known estimates for \(\vartheta{(x)}\).