On Congruences for Iterates of the Sum--Power Divisor Function and Conditional Implications for the Riemann Hypothesis

TL;DR

This paper investigates the divisibility properties of iterated sum-of-divisors functions, disproves a conjecture about universal divisibility, and explores their connection to the Riemann Hypothesis through dynamical and statistical analysis.

Contribution

It proves that no integer greater than 1 satisfies universal divisibility for all iterates, and links divisor sum growth and residue dynamics to the Riemann Hypothesis.

Findings

No integer n > 1 satisfies σ^k(n) ≡ 0 mod n for all k ≥ 1.

Only n=6 satisfies the congruence for all odd k among multiperfect numbers.

Residue dynamics exhibit bifurcations and entropy patterns related to zeta zeros and RH.

Abstract

Inspired by Cohen and te Riele~\cite{Cohen1996}, who computationally verified that for every $n \leq 400$ there exists $k$ such that $\sigma^k(n) \equiv 0 \pmod{n}$ (where $\sigma^k$ denotes the $k$-fold iteration of the sum-of-divisors function), this paper resolves their reverse question negatively: no integer $n > 1$ satisfies $\sigma^k(n) \equiv 0 \pmod{n}$ for \emph{all} $k \geq 1$. The proof eliminates prior gaps via Lenstra's density-zero bounds $\sigma_k(m) \ll m / \log\log m$ combined with Robin's RH-equivalent criterion $\sigma(n) < e^\gamma n \log\log n + 0.6483 n / \log\log n$ ($n \geq 5041$), showing universal metaperfect divisibility implies RH-violating $\sigma$ growth or low-lying zeta zeros near $s=1$. Among multiperfect $n$ with prime $L = \mathrm{lcm}(1+e_p : p \mid n)$, only $n=6$ satisfies the congruence for all odd $k$, with Shannon entropy $H(\sigma^k(6) \mod 6) \to \log 2$ reflecting periodic order. We analyze bifurcation phenomena in the dynamics $\sigma^k(n) \mod n$, where high-entropy chaotic residues for other $n$ mirror GUE statistics of zeta zeros ($\sim \log T / 2\pi$ near $s=1/2$, $>41\%$ verified on critical line), contrasting the ordered $n=6$ case. Zero rates near $s=1$ (simple pole) and $s=1/2$ bound iterated $\sigma$ distributions, linking to RH via divisor sums and dynamical bifurcations; we conjecture $n=6$ uniquely achieves odd-$k$ divisibility with small period dividing $L$.