Iterating sum of power divisor function and New equivalence to the Riemann hypothesis

Abstract

This paper investigates the dynamics of the iterated sum-of-divisors function $\sigma_k(m)$ and its behaviour modulo $m$, motivated by classical questions on perfect and multiperfect numbers and by the congruences $\sigma_k(m) \equiv 0 \pmod m$. Perfect and multiperfect numbers remain extremely rare; odd perfect numbers are still unknown and must be astronomically large. Here, the emphasis is on the dynamical and statistical structure of the iterates rather than on isolated examples. Three main results are obtained. First, it is proved that no integer $m>1$ can satisfy $\sigma_k(m) \equiv 0 \pmod m$ for all $k \ge 0$, thereby ruling out the existence of "metaperfect" numbers and showing that the iteration of $\sigma$ cannot remain permanently trapped in the residue class $0$ modulo $m$. Second, for certain explicit integers such as $m=6,12,24$, the sequence $\sigma_k(m) \bmod m$ is strictly periodic with small period dividing $L=\mathrm{lcm}(e_i+1)$, where the $e_i$ are the prime exponents of $m$. Bifurcation plots and distributional analysis reveal a transition from rigid two-cycle structure to more complex residue dynamics as $m$ increases. Third, a new equivalence with the Riemann Hypothesis is established: RH holds if and only if, for every even non-squarefree $m \ge 5041$ containing a prime fifth power, \[ \frac{\sigma_k(m)}{\sigma_{k-1}(m)\log\log\sigma_{k-1}(m)} \le e^\gamma, \] and the sequence $\sigma_k(m) \bmod m$ is eventually periodic, uniformly in $k \ge 0$. Extensive computations support these periodicity phenomena, yield non-normal discrete distribution models for the residues, and suggest a connection with a newly proposed Schrodinger-type "Caceres" operator whose spectrum numerically reproduces key statistical features of the nontrivial zeros of the Riemann zeta function.