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

Zeraoulia Rafik, Pedro Caceres

TL;DR

<3-5 sentence high-level summary> The paper proves that there is no integer m>1 with σ^k(m) ≡ 0 (mod m) for every k, thereby ruling out universal metaperfect numbers. It also shows that among multiperfect n with prime L, the unique example is n=6, which exhibits a stable, low-entropy dynamics for odd iterations. Adopting a dynamical-systems view of the map x ↦ σ(x) mod n, the authors contrast ordered (low-entropy) behavior at n=6 with chaotic, high-entropy dynamics at other moduli and relate these patterns to Robin-type bounds and zeta-zero statistics (GUE). They formulate a conditional entropy-based RH conjecture, linking the asymptotic equidistribution of residue orbits to RH and zero-spacing phenomena, thereby bridging divisor-sum growth, dynamical bifurcations, and zeta-zero theory.

Abstract

Inspired by Cohen and te Riele~\cite{Cohen1996}, who computationally verified that for every $n \leq 400$ there exists $k$ such that $σ^k(n) \equiv 0 \pmod{n}$ (where $σ^k$ denotes the $k$-fold iteration of the sum-of-divisors function), this paper resolves their reverse question negatively: no integer $n > 1$ satisfies $σ^k(n) \equiv 0 \pmod{n}$ for \emph{all} $k \geq 1$. The proof eliminates prior gaps via Lenstra's density-zero bounds $σ_k(m) \ll m / \log\log m$ combined with Robin's RH-equivalent criterion $σ(n) < e^γn \log\log n + 0.6483 n / \log\log n$ ($n \geq 5041$), showing universal metaperfect divisibility implies RH-violating $σ$ 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 $H(σ^k(6) \mod 6) \to \log 2$ reflecting periodic order. We analyze bifurcation phenomena in the dynamics $σ^k(n) \mod n$, where high-entropy chaotic residues for other $n$ mirror GUE statistics of zeta zeros ($\sim \log T / 2π$ 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 $σ$ distributions, linking to RH via divisor sums and dynamical bifurcations; we conjecture $n=6$ uniquely achieves odd-$k$ divisibility with small period dividing $L$.

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

TL;DR

<3-5 sentence high-level summary> The paper proves that there is no integer m>1 with σ^k(m) ≡ 0 (mod m) for every k, thereby ruling out universal metaperfect numbers. It also shows that among multiperfect n with prime L, the unique example is n=6, which exhibits a stable, low-entropy dynamics for odd iterations. Adopting a dynamical-systems view of the map x ↦ σ(x) mod n, the authors contrast ordered (low-entropy) behavior at n=6 with chaotic, high-entropy dynamics at other moduli and relate these patterns to Robin-type bounds and zeta-zero statistics (GUE). They formulate a conditional entropy-based RH conjecture, linking the asymptotic equidistribution of residue orbits to RH and zero-spacing phenomena, thereby bridging divisor-sum growth, dynamical bifurcations, and zeta-zero theory.

Abstract

Inspired by Cohen and te Riele~\cite{Cohen1996}, who computationally verified that for every there exists such that (where denotes the -fold iteration of the sum-of-divisors function), this paper resolves their reverse question negatively: no integer satisfies for \emph{all} . The proof eliminates prior gaps via Lenstra's density-zero bounds combined with Robin's RH-equivalent criterion (), showing universal metaperfect divisibility implies RH-violating growth or low-lying zeta zeros near . Among multiperfect with prime , only satisfies the congruence for all odd , with Shannon entropy reflecting periodic order. We analyze bifurcation phenomena in the dynamics , where high-entropy chaotic residues for other mirror GUE statistics of zeta zeros ( near , verified on critical line), contrasting the ordered case. Zero rates near (simple pole) and bound iterated distributions, linking to RH via divisor sums and dynamical bifurcations; we conjecture uniquely achieves odd- divisibility with small period dividing .

Paper Structure

This paper contains 21 sections, 2 theorems, 19 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Main Results
  3. Proof of Theorem \ref{['thm:main1']}
  4. Proof of Theorem \ref{['thm:main2']}
  5. Dynamical Systems Analysis of Iterated $\sigma$
  6. Proof of Theorem \ref{['thm:main2']}
  7. Periodicity, Entropy, and the Uniqueness Conjecture
  8. Periodic Structure for Coprime Iterations
  9. Entropy Classification of Dynamics
  10. The Periodicity Conjecture
  11. Periodicity, Bifurcation, and an Entropy–Based Conjecture
  12. Entropy of residue orbits
  13. Even versus odd bifurcation patterns
  14. The special role of $n=6$
  15. Normalised entropy and zeta–zero statistics
  16. ...and 6 more sections

Key Result

Theorem 2.1

No integer $m>1$ satisfies $\sigma^k(m)\equiv 0\pmod{m}$ for all$k\geq 1$, where $\sigma^k$ denotes the $k$-fold iteration of the sum-of-divisors function.

Figures (1)

  • Figure 1: Dynamics of $\sigma^k(1)\bmod n$ for $k=1,\dots,10\,000$. Top left: entropy $H(n;K)$ for even moduli $2\leq n\leq 1000$. Top middle: entropy for odd moduli $3\leq n\leq 999$. Top right: last $1000$ iterates of the orbit for several small even moduli, illustrating the approach to stationary behaviour. Bottom left: phase portrait for $n=6$, showing a stable attracting cycle in the $(x_k,x_{k+1})$–plane. Bottom middle: histogram of $H(n;K)$ over all even and odd moduli, with the vertical lines marking $\log_2 6$ and the empirical value for $n=6$. Bottom right: comparison between a normalised entropy statistic and the expected spacing of zeros of $\zeta(s)$ on the critical line.

Theorems & Definitions (4)

  • Theorem 2.1
  • Theorem 2.2
  • Conjecture 3.1
  • Conjecture 5.1