Toward a Positive Resolution of Schinzel’s Conjecture via Entropy–Sieve Methods Revisited

We study the normalized iterates of the sum-of-divisors function, \[ R_k(n)=\frac{\sigma^{(k)}(n)}{n}, \] in connection with Schinzel’s conjecture on the boundedness of {Rk(n)}n≥1 and the finiteness of lim infn→∞ Rk(n). Building on refined sieve methods, entropy-based analysis, and computations up to n = 1010, we prove an entropy deficit phenomenon for the distribution of Rk(n), heuristically establishing that the logarithmic spread of values is strictly smaller than the maximal Shannon entropy permitted by uniformity. To complement the theoretical bounds, we compute the structural constants Ak, C(k, α), and T0(k, ε) governing the entropy-decrement mechanism, thereby making the quantitative aspects of the argument explicit. As a consequence, we obtain polylogarithmic upper bounds for Rk(n) on a density-one subset of integers and derive quantitative large-deviation estimates, showing that extreme amplifications occur only on sets of negligible density. Extensive numerical evidence supports and sharpens these asymptotic results. Finally, we discuss implications for Robin’s criterion, which links the growth of σ(n) to the Riemann Hypothesis. Our findings suggest that a uniform proof of Schinzel-type boundedness of Rk(n) below the Robin threshold would settle RH, providing a conditional pathway via entropy and sieve techniques.