迭代求和幂因数函数与黎曼假说的新等价关系
Iterating sum of power divisor function and New equivalence to the Riemann hypothesis
AI总结：
本文研究了迭代求和幂因数函数的动力学，证明了不存在满足特定条件的整数，并建立了与黎曼假说的新等价关系。
摘要：
本文研究了迭代求和因数函数σ_k(m)的动力学及其模m的行为，受完美数和多重完美数的经典问题以及σ_k(m)≡0 mod m的同余式启发。完美数和多重完美数仍然极为稀少；奇完美数仍未知且必须极其庞大。此处重点在于迭代和统计结构，而非孤立例子。主要得出三个结果。首先，证明没有整数m>1能对所有k≥0满足σ_k(m)≡0 mod m，从而排除了“多重完美”数的存在，并表明σ迭代无法永久停留在模m的余数类0中。其次，对于某些明确的整数如m=6,12,24，序列σ_k(m) mod m严格周期性，周期大小除以L=lcm(e_i+1)，其中e_i是m的素数指数。分叉图和分布分析揭示了随着m增加，从刚性的双循环结构向更复杂的余数动态转变。第三，建立了一个与黎曼假说的新等价关系：若且仅若对于每个偶非平方自由的m≥5041且包含一个五次方素数，有σ_k(m)/(σ_{k-1}(m)loglogσ_{k-1}(m)) ≤e^γ，并且序列σ_k(m) mod m最终周期性，均匀于k≥0，则黎曼假说成立。广泛计算支持这些周期性现象，得出非正态离散分布模型，且暗示与新提出的Schrodinger型“Caceres”算子的联系，该算子的谱数值重现黎曼ζ函数非平凡零点的关键统计特征。
英文摘要：
This paper investigates the dynamics of the iterated sum-of-divisors function $σ_k(m)$ and its behaviour modulo $m$, motivated by classical questions on perfect and multiperfect numbers and by the congruences $σ_k(m) \equiv 0 \pmod m$. 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 $σ_k(m) \equiv 0 \pmod m$ for all $k \ge 0$, thereby ruling out the existence of "metaperfect" numbers and showing that the iteration of $σ$ cannot remain permanently trapped in the residue class $0$ modulo $m$. Second, for certain explicit integers such as $m=6,12,24$, the sequence $σ_k(m) \bmod m$ 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, \[ \frac{σ_k(m)}{σ_{k-1}(m)\log\logσ_{k-1}(m)} \le e^γ, \] and the sequence $σ_k(m) \bmod m$ 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.
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