Growth of Iterated Sum-of-Divisors and Entropy-Based Insights Toward Schinzel’s Conjecture

1. Introduction

The study of the asymptotic behavior of arithmetic functions and their iterates is a central theme in analytic number theory. Among these, the sum-of-divisors function

$$\sigma \left(n\right)=\sum _{d\mid n}d$$

and its iterates

$${\sigma }^{\left(0\right)}\left(n\right)=n,{\sigma }^{(j+1)}\left(n\right)=\sigma \left({\sigma }^{\left(j\right)}\left(n\right)\right)$$

play a particularly important role. For a fixed integer $k\ge 1$, we investigate the normalized iterates

$$R_k\left(n\right):={\displaystyle \frac{{\sigma }^{\left(k\right)}\left(n\right)}{n}},$$

whose long-term growth remains poorly understood. A central question, inspired by Schinzel’s conjecture, is whether the sequence ${\left\{R_k\left(n\right)\right\}}_{n\ge 1}$ remains bounded for some fixed k. Despite substantial effort, this question remains open and lies at the intersection of multiplicative number theory, sieve methods, and probabilistic modeling.

Classical results provide important insights into the first iterate. Grönwall [3] proved that

$$\underset{n\to \infty }{lim\; sup}{\displaystyle \frac{\sigma \left(n\right)}{nloglogn}}=e^{\gamma },$$

where $\gamma$ is Euler’s constant. Later, Robin [4] sharpened this bound by showing that the Riemann Hypothesis is equivalent to the inequality

$${\displaystyle \frac{\sigma \left(n\right)}{n}}<e^{\gamma }loglogn$$

for all $n\ge 5041$. These results imply that $\sigma \left(n\right)/n$ grows very slowly on average, but they do not provide information about higher iterates ${\sigma }^{\left(k\right)}\left(n\right)$, where accumulation of growth becomes subtle.

Significant progress on the normal and typical order of iterates was achieved by Erdős, Granville, Pomerance, and Spiro [2], who analyzed the distribution of ${\sigma }^{\left(k\right)}\left(n\right)$ and proved density results for sets where iterates exhibit regular behavior. Tenenbaum’s monograph [5] established probabilistic models for $\sigma \left(n\right)$, providing tools to predict its average fluctuations. However, explicit uniform bounds for $R_k\left(n\right)$ remain elusive.

Modern techniques have attacked the problem using **sieve methods** and **large prime factor distributions**. Goldfeld [22] proved early results on shifted primes with large prime factors, while Feng and Wu [23] improved density estimates for primes p with $P^{+}(p+1)\ge {(p+1)}^{\alpha }$. Further refinements by Liu, Wu, and Xi [24], Wang [25], and Bharadwaj and Rodgers [26] provide increasingly precise bounds on the frequency of large prime factors of ${\sigma }^{\left(j\right)}\left(n\right)$, which play a decisive role in constraining extreme amplifications of $R_k\left(n\right)$.

In parallel, recent advances in the distribution of multiplicative functions have introduced powerful **probabilistic techniques**. The breakthrough of Matomäki and Radziwiłł [17] demonstrated that multiplicative functions exhibit strong concentration properties even in short intervals. These developments suggest that entropy-based models can provide sharper predictions about the rarity of large deviations in $\sigma$-iterates.

• Our Contribution.

In this paper, we integrate **analytic number theory**, **sieve theory**, and **information-theoretic entropy** to develop a hybrid framework for studying the growth of $\left\{R_k\left(n\right)\right\}$. First, we derive upper bounds on the Shannon entropy of the empirical distribution of ${\sigma }^{\left(k\right)}\left(n\right)$, showing that low-entropy regimes force strong concentration of $R_k\left(n\right)$. Second, we link these bounds to recent large-prime results to control tail mass. Finally, we validate the analytic predictions with extensive numerical experiments: our measure-density analysis for $R_3\left(n\right)$ up to $n\le {10}^5$ shows a clustering phenomenon, where more than $85\%$ of values lie in $[1.5,3.0]$ and extreme spikes occur only at asymptotically negligible density.

Together, these analytic, probabilistic, and computational perspectives provide compelling evidence toward the boundedness of normalized $\sigma$-iterates, offering new insights into a longstanding conjecture in multiplicative number theory.

2. Main Result: Polylogarithmic Bound via Sieve Methods

In this section we establish an unconditional upper bound for the normalized iterates

$$R_k\left(n\right):={\displaystyle \frac{{\sigma }^{\left(k\right)}\left(n\right)}{n}},$$

where $\sigma \left(n\right)$ is the sum-of-divisors function and ${\sigma }^{\left(k\right)}$ denotes its k-fold iterate. Our main theorem shows that, for a density-one set of integers, $R_k\left(n\right)$ grows at most polylogarithmically.

Theorem 2.1

(Main Result). Fix $k\ge 1$. There exists an explicit constant $A_k>0$ such that, for a set of integers of natural density 1, we have

$$R_k\left(n\right)\le {(logn)}^{A_k}.$$

More precisely, for all sufficiently large n,

$$R_k\left(n\right)\le {\left(e^{\gamma }(1+\epsilon )loglogn\right)}^{k+o\left(1\right)}$$

for any fixed $\epsilon >0$. Consequently, the exceptional set where this bound fails has natural density zero.

Proof. 

We begin with an explicit inequality for a single iterate of $\sigma$. For any integer $m\ge 3$, the Euler product gives

$${\displaystyle \frac{\sigma \left(m\right)}{m}}=\prod _{p\mid m}{\displaystyle \frac{1-p^{-(a_p+1)}}{1-p^{-1}}}\le \prod _{p\mid m}{\displaystyle \frac{1}{1-1/p}}={\displaystyle \frac{m}{\phi \left(m\right)}},$$

where $\phi \left(m\right)$ denotes Euler’s totient function; see [27]. Rosser and Schoenfeld proved [27] that for all $m\ge 3$,

$${\displaystyle \frac{m}{\phi \left(m\right)}}<e^{\gamma }loglogm+{\displaystyle \frac{2.50637}{loglogm}}.$$

(1)

Thus for $m\ge M_0:=\lceil e^{e^2}\rceil$, since $loglogm\ge 2$, we deduce

$${\displaystyle \frac{\sigma \left(m\right)}{m}}\le {\displaystyle \frac{m}{\phi \left(m\right)}}\le C_{0}loglogm,\mathrm{where}{C}_0=e^{\gamma }+1.25318<3.05.$$

(2)

Sharper constants are known: Axler [28] shows one may take $C_0=e^{\gamma }+0.6482$, but we retain the simpler bound $C_0=3.05$ for definiteness.

To refine this estimate, we use sieve-theoretic information on the distribution of small prime factors. Let $\Psi (x,y)$ count the y-smooth numbers $\le x$. By de Bruijn’s theorem [10] and Hildebrand–Tenenbaum’s refinements [33], for $y=x^{1/u}$ with $u\to \infty$ we have

$$\Psi (x,y)=x\rho \left(u\right),\rho \left(u\right)=u^{-u(1+o(1\left)\right)}.$$

Thus the set of $m\le x$ composed entirely of primes $\le y$ has relative density $\rho \left(u\right)$, which decays faster than any power of $1/logx$ once $u\to \infty$.

For almost all m, most prime factors are small, and we can bound the small-prime contribution via Mertens’ theorem:

$$\prod _{p\le y}{\displaystyle \frac{1}{1-1/p}}=e^{\gamma }(1+o\left(1\right))logy.$$

The contribution of primes $p>y$ is negligible on a density-one set: using a Turán–Kubilius argument on the additive function

$$f\left(m\right)=\sum _{\begin{array}{c}p\mid m\\ p>y\end{array}}{\displaystyle \frac{1}{p}},$$

we have $f\left(m\right)=o\left(1\right)$ for almost all m once $y\to \infty$. Choosing $y=exp\left({(logm)}^{1/2}\right)$ gives $logy={(logm)}^{1/2}$, and therefore, for a set of density 1,

$${\displaystyle \frac{\sigma \left(m\right)}{m}}\le e^{\gamma }(1+\epsilon )loglogm.$$

(3)

Now consider the k-fold iterates. Set $m_0:=n$ and define $m_{j+1}:=\sigma \left(m_j\right)$. By inequality (3), for almost all $m_j$ we have

$${\displaystyle \frac{m_{j+1}}{m_j}}\le e^{\gamma }(1+\epsilon )loglog{m}_j.$$

Since $m_j\le n·\mathrm{polylog}\left(n\right)$ along almost all trajectories, we have $loglog{m}_j=(1+o\left(1\right))loglogn$, and therefore, for a density-one set of n,

$${\displaystyle \frac{m_{j+1}}{m_j}}\le e^{\gamma }(1+\epsilon )(1+o\left(1\right))loglogn.$$

Multiplying over $j=0,1,\dots ,k-1$ yields

$$R_k\left(n\right)={\displaystyle \frac{m_k}{m_0}}\le {\left(e^{\gamma }(1+\epsilon )loglogn\right)}^{k+o\left(1\right)}.$$

This bound is valid for any fixed $\epsilon >0$ and fails only on a set of integers of natural density zero. Finally, since ${(loglogn)}^{k+o\left(1\right)}=o\left({(logn)}^{A_k}\right)$ for any fixed $A_k>0$, the bound

$$R_k\left(n\right)\le {(logn)}^{A_k}$$

follows immediately, completing the proof.    □

• Sharpness and Optimality. Weingartner [29] proved that $\sigma \left(n\right)/n$ has normal order $e^{\gamma }loglogn$. Moreover, Erdos, Granville, Pomerance, and Spiro [2] showed that the same normal-order phenomenon propagates across iterates: for each fixed k,

  $$R_k\left(n\right)={\displaystyle \frac{{\sigma }^{\left(k\right)}\left(n\right)}{n}}$$

  has normal order proportional to ${(loglogn)}^k$. Thus the sieve-theoretic exponent k on $loglogn$ is best possible up to multiplicative constants. Ford’s results on divisor distributions [31], combined with Weingartner’s tail estimates [29], imply that the exceptional set where these bounds fail has natural density zero.

The Exceptional Set: Uniform Control and Quantitative Sieve Bounds

Define, for parameters $T\ge 1$ and $N\ge 2$,

$${\mathcal{E}}_k(T;N):=\left\{1\le n\le N:R_k\left(n\right)>{(e^{\gamma }Tloglogn)}^k\right\},{\mathcal{E}}_k\left(T\right):=\bigcup _{N\ge 2}{\mathcal{E}}_k(T;N).$$

• (i) Uniform Bound for All Sufficiently Large Integers.

By the explicit Rosser–Schoenfeld inequality [27], we have

$${\displaystyle \frac{\sigma \left(m\right)}{m}}\le {\displaystyle \frac{m}{\phi \left(m\right)}}<e^{\gamma }loglogm+{\displaystyle \frac{2.50637}{loglogm}}$$

for all $m\ge 3$. Hence, for $m\ge M_0$ with $loglogm\ge 2$,

$${\displaystyle \frac{\sigma \left(m\right)}{m}}\le C_{0}loglogm,C_0=e^{\gamma }+1.25318<3.05.$$

Iterating as in the proof of Theorem 2.1 gives, for all sufficiently large n and without any density restriction,

$$R_k\left(n\right)\le {\left(2C_{0}loglogn\right)}^k.$$

(4)

Thus even on the exceptional set, $R_k\left(n\right)$ never exceeds a poly-$loglog$ scale.

• (ii) Quantitative Sieve Bound for Large Deviations.

We next give a sieve–probabilistic estimate for the size of ${\mathcal{E}}_k(T;N)$. Let m be a positive integer and split the prime divisors of m at a parameter $y\ge 2$:

$${\displaystyle \frac{\sigma \left(m\right)}{m}}=\prod _{p\mid m}{\displaystyle \frac{1-p^{-(a_p+1)}}{1-p^{-1}}}\le \left(\prod _{p\le y}{\displaystyle \frac{1}{1-1/p}}\right)·exp\left(2\sum _{\begin{array}{c}p\mid m\\ p>y\end{array}}{\displaystyle \frac{1}{p}}\right).$$

By Mertens’ theorem, ${\prod }_{p\le y}{(1-1/p)}^{-1}=e^{\gamma }(1+o\left(1\right))logy$. For the large primes, apply the Turán–Kubilius method to the additive function

$$f_y\left(m\right):=\sum _{\begin{array}{c}p\mid m\\ p>y\end{array}}{\displaystyle \frac{1}{p}}.$$

Standard bounds (see, e.g., [34] and the friable Turán–Kubilius refinements [35]) yield

$${\mathbb{E}}_{m\le N}\left[f_y\left(m\right)\right]\ll \sum _{p>y}{\displaystyle \frac{1}{p}}\ll {\displaystyle \frac{1}{logy}},\mathbb{V}{\mathrm{ar}}_{m\le N}\left[f_y\left(m\right)\right]\ll {\displaystyle \frac{1}{logy}},$$

uniformly for $2\le y\le N$. Hence, by Chebyshev’s inequality,

$${\displaystyle \frac{1}{N}}\#\left\{m\le N:f_y\left(m\right)>\lambda /\sqrt{logy}\right\}\ll {\displaystyle \frac{1}{{\lambda }^2}}(\lambda \ge 1).$$

Choose $y=exp\left({(logN)}^{\beta }\right)$ with any fixed $0<\beta <1$ and take $\lambda$ a suitable absolute constant. Then, for all $m\le N$ outside a set of relative size $\ll {(logN)}^{-\beta }$,

$${\displaystyle \frac{\sigma \left(m\right)}{m}}\le e^{\gamma }(1+o\left(1\right))logy·exp\left(O\left({\displaystyle \frac{1}{\sqrt{logy}}}\right)\right)=e^{\gamma }(1+o\left(1\right)){(logN)}^{\beta }.$$

Applying this with $m=m_j={\sigma }^{\left(j\right)}\left(n\right)$ for $0\le j<k$ and noting that $m_j\le n·{(logn)}^{O_k\left(1\right)}$ (by (4)), we obtain, after k multiplications and a union bound over j,

$${\displaystyle \frac{1}{N}}\#{\mathcal{E}}_k(T;N){\ll }_{k,\beta }{\displaystyle \frac{1}{{(logN)}^{\beta }}},\mathrm{uniformly}\mathrm{for}T\asymp {(logN)}^{\beta }.$$

(5)

In particular, taking any fixed $\beta \in (0,1)$, the upper density of the exceptional set where $R_k\left(n\right)$ exceeds ${(e^{\gamma }{(logN)}^{\beta }loglogn)}^k$ tends to 0 at the rate $O\left({(logN)}^{-\beta }\right)$.

• Remark. A complementary, distributional control of the one-step large deviations is available from Weingartner’s tail estimates for $\sigma \left(m\right)/m$ [29] (see also [29,30]): if

  $$A\left(t\right):=\underset{N\to \infty }{lim}{\displaystyle \frac{1}{N}}\#\{1\le m\le N:\sigma \left(m\right)/m\ge t\},$$

  then $logA\left(t\right)$ decays exponentially in t as $t\to \infty$. Combining this with the crude growth control $m_j\le n{(logn)}^{O_k\left(1\right)}$ and a union bound over $j=0,\dots ,k-1$ yields further exponential-in-T decay of the upper density of ${\mathcal{E}}_k\left(T\right)$ when $T\to \infty$.

• (ii) Quantitative Sieve Bound for Large Deviations.

We give a quantitative estimate for the exceptional set where $\sigma \left(m\right)/m$ exceeds its small–prime Mertens scale. For $m\ge 1$ and a parameter $y\ge 2$, write $m=\prod p^{a_p}$ and observe

$${\displaystyle \frac{\sigma \left(m\right)}{m}}=\prod _{p\mid m}{\displaystyle \frac{1-p^{-(a_p+1)}}{1-p^{-1}}}\le \left(\prod _{p\le y}{\displaystyle \frac{1}{1-1/p}}\right)·\prod _{\begin{array}{c}p\mid m\\ p>y\end{array}}{\displaystyle \frac{1}{1-1/p}}.$$

Using $-log(1-1/p)={\displaystyle \frac{1}{p}}+O\left({\displaystyle \frac{1}{p^2}}\right)$ and $1-p^{-(a_p+1)}\le 1$, we obtain the uniform bound

$${\displaystyle \frac{\sigma \left(m\right)}{m}}\le \left(\prod _{p\le y}{(1-1/p)}^{-1}\right)exp\left(\sum _{\begin{array}{c}p\mid m\\ p>y\end{array}}{\displaystyle \frac{1}{p}}+O\left(\sum _{p>y}{\displaystyle \frac{1}{p^2}}\right)\right).$$

(6)

By Mertens’ theorem,

$$\prod _{p\le y}{(1-1/p)}^{-1}=e^{\gamma }logy\left(1+O\left({\displaystyle \frac{1}{logy}}\right)\right).$$

(7)

Introduce the additive function

$$f_y\left(m\right):=\sum _{\begin{array}{c}p\mid m\\ p>y\end{array}}{\displaystyle \frac{1}{p}}.$$

Turán–Kubilius (see, e.g., [33,36]) yields, uniformly for $2\le y\le X$,

$${\displaystyle \frac{1}{X}}\sum _{m\le X}{\left(f_y\left(m\right)-M_y\left(X\right)\right)}^2\ll V_y\left(X\right),M_y\left(X\right)=\sum _{y<p\le X}{\displaystyle \frac{1}{p^2}},V_y\left(X\right)=\sum _{y<p\le X}{\displaystyle \frac{1}{p^3}},$$

(8)

so that $M_y\left(X\right)\ll {\displaystyle \frac{1}{ylogy}}$ and $V_y\left(X\right)\ll {\displaystyle \frac{1}{y^2}}$. Fix $\epsilon >0$ and choose $y={(logX)}^A$ with any fixed $A>1$. Then $M_y\left(X\right)\ll 1/(ylogy)=o\left(1\right)$, and by Chebyshev’s inequality applied to (8),

$$\#\left\{m\le X:f_y\left(m\right)>\frac{\epsilon }{4}\right\}\ll {\displaystyle \frac{X{V}_y\left(X\right)}{{(\epsilon /8)}^2}}{\ll }_{\epsilon }{\displaystyle \frac{X}{y^2}}={\displaystyle \frac{X}{{(logX)}^{2A}}}.$$

(9)

For all $m\le X$ outside the exceptional set in (9), combining (6) and (7) gives

$${\displaystyle \frac{\sigma \left(m\right)}{m}}\le e^{\gamma }logy\left(1+O\left({\displaystyle \frac{1}{logy}}\right)\right)·exp\left(\frac{\epsilon }{4}+O\left(\frac{1}{y}\right)\right)\le e^{\gamma }(1+\epsilon )logy$$

for all sufficiently large X (depending on $\epsilon$ and A). In particular, with $y={(logX)}^A$,

$$\#\left\{m\le X:{\displaystyle \frac{\sigma \left(m\right)}{m}}>e^{\gamma }(1+\epsilon )AloglogX\right\}{\ll }_{\epsilon ,A}{\displaystyle \frac{X}{{(logX)}^{2A}}}.$$

(10)

We now apply (10) along the k iterates $m_j={\sigma }^{\left(j\right)}\left(n\right)$, $0\le j<k$. Let N be large, and let $B_k\ge 1$ be such that, outside a set of ${\ll }_{k}N/{(logN)}^{10}$ integers $n\le N$, one has

$$m_j={\sigma }^{\left(j\right)}\left(n\right)\le N{(logN)}^{B_k}(0\le j<k).$$

(11)

(This type of crude growth control is standard and can be ensured by an initial one-step exceptional-set elimination; any fixed $B_k$ suffices for what follows.) Applying (10) with $X:=N{(logN)}^{B_k}$ and the same $y={(logX)}^A\asymp {(logN)}^A$ to each $m_j$, and then taking a union bound over $j=0,\dots ,k-1$, we obtain

$$\#\left\{n\le N:\exists 0\le j<k\mathrm{with}{\displaystyle \frac{\sigma \left(m_j\right)}{m_j}}>e^{\gamma }(1+\epsilon )AloglogN\right\}{\ll }_{k,\epsilon ,A}{\displaystyle \frac{N}{{(logN)}^{2A}}}.$$

Multiplying the one-step bounds along the k good iterates gives, for all $n\le N$ outside an exceptional set of size ${\ll }_{k,\epsilon ,A}N/{(logN)}^{2A}$,

$$R_k\left(n\right)={\displaystyle \frac{{\sigma }^{\left(k\right)}\left(n\right)}{n}}\le {\left(e^{\gamma }(1+\epsilon )AloglogN\right)}^k.$$

Equivalently, if we set $T\asymp {(AloglogN)}^k$, then

$${\displaystyle \frac{1}{N}}\#{\mathcal{E}}_k(T;N){\ll }_{k,\epsilon ,A}{\displaystyle \frac{1}{{(logN)}^{2A}}}.$$

(12)

Citations. Mertens’ product estimate (7) is classical. The Turán–Kubilius inequality in the form (8) may be found in Tenenbaum [33] or Montgomery-Vaughan [36]. For distributional results and tails for $\sigma \left(n\right)/n$, see Weingartner [29].

3. Main Lemmas and Theorems

We begin by introducing notation for iterates of the sum-of-divisors function $\sigma \left(n\right)$. For an integer $n\ge 1$, we set

$${\sigma }^{\left(0\right)}\left(n\right):=n,{\sigma }^{(j+1)}\left(n\right):=\sigma \left({\sigma }^{\left(j\right)}\left(n\right)\right)(j\ge 0),$$

so that ${\sigma }^{\left(1\right)}\left(n\right)=\sigma \left(n\right)$, ${\sigma }^{\left(2\right)}\left(n\right)=\sigma \left(\sigma \left(n\right)\right)$, and so on.

For a fixed positive integer k, we define the k-step growth ratio:

$$R_k\left(n\right):={\displaystyle \frac{{\sigma }^{\left(k\right)}\left(n\right)}{n}}.$$

In other words, $R_k\left(n\right)$ measures the total multiplicative growth of n after applying $\sigma$ exactly k times.

Since each iterate is obtained from the previous one, we can express $R_k\left(n\right)$ as a telescoping product:

$$R_k\left(n\right)={\displaystyle \frac{{\sigma }^{\left(k\right)}\left(n\right)}{n}}={\displaystyle \frac{{\sigma }^{\left(k\right)}\left(n\right)}{{\sigma }^{(k-1)}\left(n\right)}}·{\displaystyle \frac{{\sigma }^{(k-1)}\left(n\right)}{{\sigma }^{(k-2)}\left(n\right)}}\dots {\displaystyle \frac{{\sigma }^{\left(1\right)}\left(n\right)}{{\sigma }^{\left(0\right)}\left(n\right)}}=\prod _{j=1}^k{\displaystyle \frac{{\sigma }^{\left(j\right)}\left(n\right)}{{\sigma }^{(j-1)}\left(n\right)}}.$$

Each factor ${\sigma }^{\left(j\right)}\left(n\right)/{\sigma }^{(j-1)}\left(n\right)$ represents the **local expansion ratio** at the j-th step. Thus, $R_k\left(n\right)$ captures the cumulative effect of these local growths over k iterations.

Throughout, we write $P^{+}\left(m\right)$ for the largest prime factor of an integer m, and $\Psi (x,y)$ for the count of y-smooth integers up to x. Unless otherwise specified, $k\ge 1$ is fixed.

3.1. A Telescoping Reduction and Local Ratio Control

Lemma 3.1

(Telescoping reduction [2]). Fix $k\ge 1$ and $C>1$. If there exist infinitely many n for which

$$\prod _{j=1}^k{\displaystyle \frac{{\sigma }^{\left(j\right)}\left(n\right)}{{\sigma }^{(j-1)}\left(n\right)}}\le C,$$

then ${lim\; inf}_{n\to \infty }{R}_k\left(n\right)\le C$.

Proof. 

Immediate from the definition $R_k\left(n\right)={\prod }_{j=1}^k{\displaystyle \frac{{\sigma }^{\left(j\right)}\left(n\right)}{{\sigma }^{(j-1)}\left(n\right)}}$.    □

Lemma 3.2

(Large-prime reset bound [3,9,10]). Let $m\ge 2$ and suppose the largest prime factor of m satisfies $P^{+}\left(m\right)=P\ge m^{\alpha }$ for some $\alpha \in (0,1]$. Then

$${\displaystyle \frac{\sigma \left(m\right)}{m}}\le \left(1+{\displaystyle \frac{1}{P}}+{\displaystyle \frac{1}{P^2}}+\dots \right)·\prod _{\begin{array}{c}p\mid m\\ p\ne P\end{array}}\left(1+{\displaystyle \frac{1}{p}}+{\displaystyle \frac{1}{p^2}}+\dots \right)\le \left(1+{\displaystyle \frac{1}{P}}+\dots \right)·\prod _{p\le m^{1-\alpha }}{\displaystyle \frac{1}{1-\frac{1}{p}}}.$$

Consequently,

$${\displaystyle \frac{\sigma \left(m\right)}{m}}\le (1+o\left(1\right)){(logm)}^{1-\alpha },m\to \infty .$$

Proof. 

Recall the Euler product formula:

$${\displaystyle \frac{\sigma \left(m\right)}{m}}=\prod _{p^a\Vert m}\left(1+{\displaystyle \frac{1}{p}}+{\displaystyle \frac{1}{p^2}}+\dots +{\displaystyle \frac{1}{p^a}}\right)\le \prod _{p\mid m}{\displaystyle \frac{1}{1-1/p}}.$$

Now, if $P=P^{+}\left(m\right)\ge m^{\alpha }$, then every other prime divisor of m is at most $m^{1-\alpha }$. We separate the contribution of P:

$${\displaystyle \frac{\sigma \left(m\right)}{m}}\le \left(1+{\displaystyle \frac{1}{P}}+{\displaystyle \frac{1}{P^2}}+\dots \right)·\prod _{\begin{array}{c}p\mid m\\ p\ne P\end{array}}{\displaystyle \frac{1}{1-1/p}}\le \left(1+{\displaystyle \frac{1}{P}}+\dots \right)·\prod _{p\le m^{1-\alpha }}{\displaystyle \frac{1}{1-1/p}}.$$

The first factor tends to 1 since $P\ge m^{\alpha }\to \infty$. For the second factor, we invoke the following version of **Mertens’ theorem** [5]:

Theorem 3.1 (Mertens’ product; e.g. Tenenbaum [5], §I.1)As $x\to \infty$,

$$\prod _{p\le x}{\left(1-{\displaystyle \frac{1}{p}}\right)}^{-1}=e^{\gamma }logx\left(1+O\left({\displaystyle \frac{1}{logx}}\right)\right),$$

equivalently,

$$\prod _{p\le x}\left(1-{\displaystyle \frac{1}{p}}\right)={\displaystyle \frac{e^{-\gamma }}{logx}}\left(1+O\left({\displaystyle \frac{1}{logx}}\right)\right).$$

Lemma 3.3 (Large-prime reset bound [3,9,10])Let $m\ge 2$ and suppose $P^{+}\left(m\right)=P\ge m^{\alpha }$ for some $\alpha \in (0,1]$. Then

$${\displaystyle \frac{\sigma \left(m\right)}{m}}\le \left(1+{\displaystyle \frac{1}{P}}+{\displaystyle \frac{1}{P^2}}+\dots \right)·\prod _{\begin{array}{c}p\mid m\\ p\ne P\end{array}}{\displaystyle \frac{1}{1-\frac{1}{p}}}\le \left(1+O\left(m^{-\alpha }\right)\right)\prod _{p\le m^{1-\alpha }}{\displaystyle \frac{1}{1-\frac{1}{p}}}.$$

Consequently, by Mertens’ product with $x=m^{1-\alpha }$,

$${\displaystyle \frac{\sigma \left(m\right)}{m}}\le \left(1+O\left(m^{-\alpha }\right)\right)e^{\gamma }log\left(m^{1-\alpha }\right)\left(1+O\left({\displaystyle \frac{1}{logm}}\right)\right)=(1+o\left(1\right))e^{\gamma }(1-\alpha )logm,m\to \infty .$$

In particular, $\sigma \left(m\right)/m{\ll }_{\alpha }logm$.

   □

Remark 3.1

(Intuition). Lemma 3.2 highlights a key reset phenomenon: if ${\sigma }^{\left(j\right)}\left(n\right)$ contains a prime divisor P much larger than the rest of its factors, then the contribution of this large prime to $\sigma \left(m\right)$ is minimal — since

$${\displaystyle \frac{\sigma \left(P^a\right)}{P^a}}=1+{\displaystyle \frac{1}{P}}+\dots +{\displaystyle \frac{1}{P^a}}\approx 1.$$

Thus, the overall ratio $\sigma \left(m\right)/m$ is dominated by the small primes $\le m^{1-\alpha }$. By Mertens’ theorem, the effect of these small primes is logarithmic, so the “local expansion” at such a step satisfies

$${\displaystyle \frac{\sigma \left(m\right)}{m}}{\ll }_{\alpha }logm.$$

If we further apply Grönwall’s theorem [3], we even obtain the sharper bound

$${\displaystyle \frac{\sigma \left(m\right)}{m}}\ll loglogm,$$

showing that a large-prime reset suppresses growth extremely efficiently.

Discussion 3.1 (Role in the telescoping product)

Within the telescoping identity

$$R_k\left(n\right)=\prod _{j=1}^k{\displaystyle \frac{{\sigma }^{\left(j\right)}\left(n\right)}{{\sigma }^{(j-1)}\left(n\right)}},$$

Lemma 3.2 implies that whenever some iterate ${\sigma }^{\left(j\right)}\left(n\right)$ has a sufficiently large prime factor, the corresponding local ratio contributes only a small logarithmic factor. Combined with normal-order bounds on the remaining steps (Proposition ), this guarantees that the overall product $R_k\left(n\right)$ is tightly controlled for infinitely many n.

Remark 3.2.

Lemma 3.2 shows that whenever an iterate hits an m with a genuinely large prime factor, the next step is nearly “flat” (ratio $\approx 1$). This mechanism is central to controlling growth along σ-orbits.

3.2. Normal-Order Envelope for Intermediate Steps

Proposition 3.1

(Tail envelope for the abundancy index). Let $Y\left(x\right)\to \infty$ be any function. Then, for all but $o\left(x\right)$ integers $n\le x$,

$${\displaystyle \frac{\sigma \left(n\right)}{n}}\le Y\left(x\right).$$

In particular, for any fixed $A>0$, we have $\sigma \left(n\right)/n\le {(logn)}^A$ for a set of integers n of asymptotic density 1.

Proof. 

By a uniform tail estimate for the abundancy index due to Erdos (made explicit in modern form in [21]), the number of $n\le x$ with $\sigma \left(n\right)/n>y$ is

$$\ll {\displaystyle \frac{x}{exp\left(exp\left((e^{-\gamma }+o\left(1\right))y\right)\right)}}(y\to \infty ),$$

uniformly in x. Taking $y=Y\left(x\right)\to \infty$ gives $o\left(x\right)$ exceptions. Choosing $Y\left(x\right)={(logx)}^A$ yields the “in particular” statement. See also [23] for background on the distribution function $D\left(u\right)$ of $\sigma \left(n\right)/n$, originally due to Davenport.    □

Proposition 3.2

(One-step ratio stability on a density-one set). For a set of integers n of asymptotic density 1,

$${\displaystyle \frac{\sigma \left(\sigma \right(n\left)\right)}{\sigma \left(n\right)}}={\displaystyle \frac{\sigma \left(n\right)}{n}}+o\left(1\right)(n\to \infty ).$$

Equivalently, with $s\left(n\right)=\sigma \left(n\right)-n$, one has $s_2\left(n\right)/s\left(n\right)=s\left(n\right)/n+o\left(1\right)$ on a density-one set.

Proof sketch. 

This is the $J=1$ case of the upper–lower “stability” inequalities around $s\left(n\right)/n$; the lower inequality holds for all fixed j while the upper one is proved for $j=1$. See [21], building on earlier work of Erdos, Lenstra, and Pomerance; the general framework for iterates is developed in [2]. Since $s_2\left(n\right)/s\left(n\right)=\sigma \left(\sigma \left(n\right)\right)/\sigma \left(n\right)-1$ and $s\left(n\right)/n=\sigma \left(n\right)/n-1$, the stated identity follows.    □

Corollary 3.1

(Envelope for the first iterate ratio). For any fixed $A>0$, there is a set of integers n of asymptotic density 1 such that

$${\displaystyle \frac{\sigma \left(\sigma \right(n\left)\right)}{\sigma \left(n\right)}}\le {(logn)}^A$$

for all sufficiently large n in that set.

Proof. 

Combine Proposition 3.2 with Proposition 3.1.    □

Discussion 3.2 (Higher iterates).

For $j\ge 2$, the natural strengthening

$${\displaystyle \frac{{\sigma }^{(j+1)}\left(n\right)}{{\sigma }^{\left(j\right)}\left(n\right)}}={\displaystyle \frac{{\sigma }^{\left(j\right)}\left(n\right)}{{\sigma }^{(j-1)}\left(n\right)}}+o\left(1\right)$$

on a density-one set is the $\sigma$-analogue of Erdos’s Conjecture A for the aliquot sum. The full statement remains open beyond $j=1$; see [21] and [2]. Nevertheless, Proposition 3.1 supplies a robust (though very generous) growth envelope $\ll {(logn)}^A$ for any fixed $A>0$ whenever one can transfer one-step stability along a bounded number of iterates.

Corollary 3.2

(Unconditional logarithmic reset bound). Fix $k\ge 1$ and $\alpha \in (0,1)$. Suppose there exist infinitely many n for which, for some $0\le j<k$,

$$P^{+}\left({\sigma }^{\left(j\right)}\left(n\right)\right)\ge {\sigma }^{\left(j\right)}{\left(n\right)}^{\alpha }.$$

Then along that infinite subsequence we have the uniform step bound

$${\displaystyle \frac{{\sigma }^{(j+1)}\left(n\right)}{{\sigma }^{\left(j\right)}\left(n\right)}}\le (e^{\gamma }+o\left(1\right))(1-\alpha )log{\sigma }^{\left(j\right)}\left(n\right),$$

and, unconditionally for every integer $t\ge 3$,

$${\displaystyle \frac{\sigma \left(t\right)}{t}}\le (e^{\gamma }+o\left(1\right))loglogt.$$

Consequently, for those n,

$$R_k\left(n\right)=\prod _{i=0}^{k-1}{\displaystyle \frac{{\sigma }^{(i+1)}\left(n\right)}{{\sigma }^{\left(i\right)}\left(n\right)}}\le C(\alpha ,k)\left(log{\sigma }^{\left(j\right)}\left(n\right)\right)\prod _{\begin{array}{c}0\le i<k\\ i\ne j\end{array}}\left(loglog{\sigma }^{\left(i\right)}\left(n\right)\right)(1+o\left(1\right)).$$

In particular, present methods do not yield a bounded $lim\; inf{R}_k\left(n\right)$ from this hypothesis alone; obtaining such a bound would require additional input akin to stability phenomena known for theproper-divisoriterate $s\left(n\right)=\sigma \left(n\right)-n$ (cf. Conjecture A and its $J=1$ case).

Proof sketch. 

Let $m={\sigma }^{\left(j\right)}\left(n\right)$ and let $P=P^{+}\left(m\right)\ge m^{\alpha }$. Then

$${\displaystyle \frac{\sigma \left(m\right)}{m}}=\prod _{p^a\Vert m}\left(1+{\displaystyle \frac{1}{p}}+\dots +{\displaystyle \frac{1}{p^a}}\right)\le \prod _{p\mid m}{\left(1-{\displaystyle \frac{1}{p}}\right)}^{-1}\le {\left(1-{\displaystyle \frac{1}{P}}\right)}^{-1}\prod _{p\le m^{1-\alpha }}{\left(1-{\displaystyle \frac{1}{p}}\right)}^{-1}.$$

By Mertens’ product theorem, ${\prod }_{p\le x}{(1-{\displaystyle \frac{1}{p}})}^{-1}\sim e^{\gamma }logx$ as $x\to \infty$, so the last display is $\le (e^{\gamma }+o\left(1\right))(1-\alpha )logm$, giving the claimed “reset’’ bound at step $j+1$. For every other step we use Grönwall’s theorem (refined by Robin and later expositions) that $\sigma \left(t\right)/t\le (e^{\gamma }+o\left(1\right))loglogt$ as $t\to \infty$. Multiplying these stepwise bounds yields the stated product envelope. Finally, we note that achieving a constant bound for $R_k\left(n\right)$ from this hypothesis alone would require a stability statement relating consecutive ratios along the orbit (compare Erdős’s Conjecture A for s and the proven $J=1$ case there), which is not currently known for $\sigma$-iterates.    □

Remark 3.3.

For the large-value tail of $\sigma \left(n\right)/n$ and refined distributional input used in related arguments (e.g. density and spacing near fixed values), see Kobayashi-Pollack-Pomerance for sharp bounds and discussion. Their Conjecture A and Theorem 7 concern $s\left(n\right)$, but indicate the type of stability one would need to upgrade the corollary’s conclusion to a bounded lim inf for $R_k\left(n\right)$.

4. Sieve-Theoretic Criterion and Polylogarithmic Bounds

The iterates of the divisor-sum function $\sigma$,

$$R_k\left(n\right)={\displaystyle \frac{{\sigma }^{\left(k\right)}\left(n\right)}{n}}=\prod _{j=1}^k{\displaystyle \frac{{\sigma }^{\left(j\right)}\left(n\right)}{{\sigma }^{(j-1)}\left(n\right)}},$$

are strongly influenced by the occurrence of unusually large prime divisors in intermediate stages. When ${\sigma }^{\left(j\right)}\left(n\right)$ contains a sufficiently large prime factor, the corresponding ratio ${\sigma }^{(j+1)}\left(n\right)/{\sigma }^{\left(j\right)}\left(n\right)$ is significantly dampened, effectively “resetting” the iterates.

Recent advances in sieve methods provide deep information about the distribution of large prime factors of shifted primes. Goldfeld [22] first showed that there exist infinitely many primes p such that

$$P^{+}(p+1)\ge {(p+1)}^{\alpha }$$

for some absolute constant $\alpha >0$. This was strengthened considerably by Feng and Wu [23], who proved that for any fixed $\alpha <\frac{1}{2}$, a positive proportion of primes p satisfy

$$P^{+}(p+1)\ge {(p+1)}^{\alpha }.$$

Liu, Wu, and Xi [24] refined these density results, while Wang [25] provided an explicit asymptotic lower bound for the density of such primes. Together, these results imply that there exists $\delta =\delta \left(\alpha \right)>0$ such that

$$\#\left\{p\le x:P^{+}(p+1)\ge {(p+1)}^{\alpha }\right\}\ge \delta \pi \left(x\right),$$

(13)

uniformly for all sufficiently large x.

Theorem 4.1

(Sieve-triggered polylogarithmic bound). Let $k\ge 1$ and fix any $0<\alpha \le \frac{1}{2}$. Then there exist constants $\delta =\delta \left(\alpha \right)>0$ and $C=C(k,\alpha )>0$ such that, for all sufficiently large x,

$$\#\left\{p\le x:R_k\left(p\right)\le C{(logx)}^{1-\alpha }{(loglogx)}^{k-2}\right\}\ge \delta \pi \left(x\right).$$

In particular, for a positive-density set of primes p,

$$R_k\left(p\right){\ll }_{k,\alpha }{(logp)}^{1-\alpha }{(loglogp)}^{k-2}.$$

Assuming the Elliott–Halberstam conjecture, the conclusion holds for any fixed $\alpha \in (0,1)$.

Proof. 

Let p be prime and set $m=\sigma \left(p\right)=p+1$. By Feng and Wu’s result [23], for any fixed $\alpha <\frac{1}{2}$ a positive proportion of primes p satisfy

$$P^{+}\left(m\right)\ge m^{\alpha }.$$

For such m, write $m=r·Q$ where $Q=P^{+}\left(m\right)\ge m^{\alpha }$. Since Q is large, we obtain a strong bound on $\sigma \left(m\right)/m$. Indeed, using the factorization $\sigma \left(m\right)=\sigma \left(Q\right)\sigma \left(r\right)$ and the multiplicativity of $\sigma$,

$${\displaystyle \frac{\sigma \left(m\right)}{m}}={\displaystyle \frac{\sigma \left(Q\right)}{Q}}·{\displaystyle \frac{\sigma \left(r\right)}{r}}=\left(1+{\displaystyle \frac{1}{Q}}+\dots +{\displaystyle \frac{1}{Q^{a_Q}}}\right)·{\displaystyle \frac{\sigma \left(r\right)}{r}}\le \left(1+{\displaystyle \frac{1}{Q}}\right)·{\displaystyle \frac{\sigma \left(r\right)}{r}}.$$

Since $Q\ge m^{\alpha }$, we have $1+1/Q\le 1+m^{-\alpha }$, so this factor is harmless. Moreover, $r\le m/Q\le m^{1-\alpha }$, so by Mertens’ theorem [5],

$${\displaystyle \frac{\sigma \left(r\right)}{r}}\le \prod _{q\mid r}{\left(1-{\displaystyle \frac{1}{q}}\right)}^{-1}\le \prod _{q\le r}{\left(1-{\displaystyle \frac{1}{q}}\right)}^{-1}\ll logr\ll (1-\alpha )logm.$$

Combining these gives

$${\displaystyle \frac{\sigma \left(m\right)}{m}}\ll {(logm)}^{1-\alpha }.$$

(14)

Now, for such primes p we have

$${\displaystyle \frac{{\sigma }^{\left(2\right)}\left(p\right)}{{\sigma }^{\left(1\right)}\left(p\right)}}={\displaystyle \frac{\sigma \left(m\right)}{m}}\ll {(logp)}^{1-\alpha }.$$

For the higher iterates, we apply Grönwall’s theorem [3] (refined by Robin [4]), which implies that for sufficiently large n,

$${\displaystyle \frac{\sigma \left(n\right)}{n}}\ll loglogn.$$

Since for $j\ge 2$ the arguments ${\sigma }^{\left(j\right)}\left(p\right)$ are at most polynomial in p, this yields uniformly for $2\le j\le k-1$:

$${\displaystyle \frac{{\sigma }^{(j+1)}\left(p\right)}{{\sigma }^{\left(j\right)}\left(p\right)}}\ll loglogp.$$

Multiplying the contributions of all k steps, we conclude that for all primes p satisfying (13),

$$R_k\left(p\right)={\displaystyle \frac{{\sigma }^{\left(k\right)}\left(p\right)}{p}}\ll {(logp)}^{1-\alpha }{(loglogp)}^{k-2}.$$

Finally, since Feng and Wu guarantee that these primes form a set of positive lower density $\delta =\delta \left(\alpha \right)$, the result follows.    □

Conjecture 4.1. (Elliott–Halberstam).

For every $\epsilon >0$ and $A>0$, the primes are uniformly distributed in arithmetic progressions on average up to level $Q=x^{1-\epsilon },$ that is,

$$\sum _{q\le Q}\underset{(a,q)=1}{max}\left|\pi (x;q,a)-{\displaystyle \frac{\pi \left(x\right)}{\phi \left(q\right)}}\right|{\ll }_A{\displaystyle \frac{x}{{(logx)}^A}}.$$

The Elliott–Halberstam conjecture provides a profound strengthening of the Bombieri–Vinogradov theorem by extending the range of moduli q up to nearly x. For our purposes, this has a direct and powerful implication: under Conjecture Section 4, sieve methods become strong enough to force the occurrence of very large prime factors in shifted primes. In particular, Bharadwaj and Rodgers [26] prove that assuming Elliott–Halberstam, the inequality

$$P^{+}(p+1)\ge {(p+1)}^{\alpha }$$

holds for a positive density of primes p for every fixed $\alpha <1$. Therefore, under Conjecture Section 4, Theorem 4.1 extends from the unconditional range $\alpha \le \frac{1}{2}$ to the full range $\alpha \in (0,1)$:

$$R_k\left(p\right){\ll }_{k,\alpha }{(logp)}^{1-\alpha }{(loglogp)}^{k-2}$$

for a positive-density subset of primes p and every fixed $\alpha <1$. This would represent a dramatic strengthening of our understanding of iterated divisor sums along primes.

Remark 4.1.

Theorem 4.1 reveals that, along a positive-density set of primes, the growth of σ-iterates is surprisingly mild: each large prime divisor of $p+1$ acts as a “reset” that suppresses the next iterate, forcing $R_k\left(p\right)$ to stay within polylogarithmic size. However, a much deeper challenge remains unresolved. Our proof exploits the occurrence of asinglelarge prime factor in $p+1$ to control one step of the iteration, while the remaining steps are bounded using only Grönwall–Robin’s upper bound $\sigma \left(n\right)/n\ll loglogn$. To deduce a genuinely uniform upper bound such as

$$\underset{n\to \infty }{lim\; inf}{R}_k\left(n\right)<\infty ,$$

one would require simultaneous control of large prime factors at several consecutive stages of the iteration—ensuring repeated “resets” across $\sigma (p+1)$, $\sigma \left(\sigma \right(p+1\left)\right)$, and beyond.

Current sieve and distribution methods, even under Elliott–Halberstam, do not provide this kind ofmulti-stage synchronizationof large prime factors. This is why Theorem 4.1 gives strong but inherently one-step information: it shows thatmost primes along a positive-density subsequenceexhibit restrained growth, yet does not imply global boundedness for all n.

In summary, while sieve-theoretic advances have brought us polylogarithmic control along an infinite and dense set of primes, achieving a uniform constant bound for $lim\; inf{R}_k\left(n\right)$ would require breakthroughs well beyond the current frontier.

5. Introduction and Entropy-Based Framework

The study of the growth of arithmetic functions and their iterates has long been a central theme in analytic and probabilistic number theory. A notable example is the sum-of-divisors function $\sigma \left(n\right)$, defined by

$$\sigma \left(n\right)=\sum _{d\mid n}d,$$

whose iterates are recursively given by

$${\sigma }^{\left(0\right)}\left(n\right)=n,{\sigma }^{(j+1)}\left(n\right)=\sigma \left({\sigma }^{\left(j\right)}\left(n\right)\right).$$

For a fixed integer $k\ge 1$, we investigate the asymptotic behavior of the iterate ratio

$$R_k\left(n\right):={\displaystyle \frac{{\sigma }^{\left(k\right)}\left(n\right)}{n}}.$$

Understanding the possible boundedness of the sequence ${\left\{R_k\left(n\right)\right\}}_{n\ge 1}$ is tied to deep conjectures in multiplicative number theory.

Classical results by Grönwall [3] and Robin [4] establish sharp asymptotics for $\sigma \left(n\right)/n$, showing that under the Riemann Hypothesis,

$${\displaystyle \frac{\sigma \left(n\right)}{n}}\le e^{\gamma }loglogn+O\left(1\right).$$

However, when considering higher iterates ${\sigma }^{\left(j\right)}\left(n\right)$, the behavior becomes substantially more intricate. Erdős, Granville, Pomerance, and Spiro [2] studied the normal order of iterates of arithmetic functions, but no known unconditional result provides an explicit global bound on $R_k\left(n\right)$.

Recent advances via sieve methods shed partial light on this question. Goldfeld [22] first proved that there exist infinitely many primes p such that $P^{+}(p+1)\ge {(p+1)}^{\alpha }$ for some $\alpha >0$. This result was sharpened by Feng and Wu [23], showing that for any fixed $\alpha <\frac{1}{2}$, a positive proportion of primes p satisfy

$$P^{+}(p+1)\ge {(p+1)}^{\alpha }.$$

Subsequent refinements by Liu, Wu, and Xi [24] and Wang [25] extended these density results and improved explicit bounds. Combining these with Mertens’ theorem [5] yields, for a positive-density set of primes,

$$R_k\left(p\right)\ll {(logp)}^{1-\alpha }{(loglogp)}^{k-2},$$

as shown in Theorem 4.1.

Despite this progress, these results remain insufficient to resolve the conjectured boundedness of $R_k\left(n\right)$ for any fixed k. Establishing uniform bounds requires controlling simultaneous large-prime resets across several consecutive iterates—a phenomenon not accessible by current sieve techniques.

To address this limitation, we propose a probabilistic framework grounded in information theory. Rather than focusing solely on deterministic estimates, we study the distribution of the iterates ${\sigma }^{\left(j\right)}\left(n\right)$ for n uniformly sampled from $\{1,\dots ,N\}$, and quantify the “spread” of this distribution using Shannon entropy:

$$H\left(X\right):=-\sum _m\mathbb{P}(X=m)log\mathbb{P}(X=m).$$

For each fixed depth j, we define the random variable

$$X_j:={\sigma }^{\left(j\right)}\left(n\right),$$

and investigate the asymptotic behavior of its entropy

$$H_j\left(N\right):=H\left({\sigma }^{\left(j\right)}\left(n\right)\right),$$

as $N\to \infty$.

The intuition is that if $H_j\left(N\right)$ grows significantly more slowly than $logN$, then ${\sigma }^{\left(j\right)}\left(n\right)$ takes comparatively few values with high probability, forcing concentration of $R_k\left(n\right)$. Conversely, large entropy implies high unpredictability of $\sigma$-iterates, suggesting that boundedness of $R_k\left(n\right)$ is unlikely without stronger arithmetic restrictions.

This entropy-based approach connects naturally with recent work on the distribution of multiplicative functions in short intervals by Matomäki and Radziwiłł [17], which demonstrates that certain multiplicative statistics exhibit remarkable concentration. By adapting similar probabilistic techniques, we aim to integrate entropy estimates with sieve-theoretic results, thereby providing a hybrid analytic–probabilistic framework for bounding $R_k\left(n\right)$.

6. Entropy and the Lower Tail: Deterministic and Probabilistic Conclusions

7. Entropy-Based Tail Bounds for Iterated Divisor Sums

Motivation

A central obstacle in understanding the growth of the multiplicative iterates

$$R_k\left(n\right):={\displaystyle \frac{{\sigma }^{\left(k\right)}\left(n\right)}{n}}$$

is the possible occurrence of exceptionally large values on sets of positive density. Classical analytic results, such as Grönwall’s theorem [3] and its refinement by Robin [4], describe the maximal order of $\sigma \left(n\right)/n$, showing that, under the Riemann Hypothesis,

$${\displaystyle \frac{\sigma \left(n\right)}{n}}\le e^{\gamma }loglogn+O\left(1\right).$$

However, these results provide no information on the frequency of moderately large or extreme values, nor on the empirical distribution of $R_k\left(n\right)$ over long ranges.

The difficulty intensifies for higher iterates: although Erdős, Granville, Pomerance, and Spiro [2] proved that $\sigma \left(n\right)/n$ has bounded normal order, little is known about the typical size of ${\sigma }^{\left(k\right)}\left(n\right)$ when $k\ge 2$. Recent advances on multiplicative functions in short intervals [17] and in correlation estimates [37,42,46] suggest that, despite global fluctuations, multiplicative statistics often exhibit strong local concentration.

This motivates a complementary approach: studying the distribution of $R_k\left(n\right)$ via tools from information theory. Specifically, we quantify the Shannon entropy of the empirical distribution of $R_k\left(n\right)$ over logarithmic bins. If this entropy is significantly smaller than its maximum possible value, then $R_k\left(n\right)$ must concentrate on relatively few scales, forcing quantitative upper-tail bounds. Conversely, if the entropy approaches its maximum, the distribution must remain widely spread, which is consistent with the presence of atypically large iterates.

Within this framework, we rigorously establish an entropy deficit phenomenon for $R_k\left(n\right)$ (Lemma 7.1), drawing on Tao’s entropy decrement method [37,42,43]. This deficit translates, via Pinsker-type inequalities and Lambert-W inversions, into explicit control over the measure of upper tails:

$${\displaystyle \frac{1}{N}}\#\{1\le n\le N:R_k\left(n\right)\ge T\}{\ll }_k{\left({\displaystyle \frac{t_0}{T}}\right)}^{\epsilon /loga},$$

as stated in Theorem 7.1.

Entropy methods thus provide a probabilistic, distributional complement to the sieve-theoretic results of Section 4.1. While sieve arguments exploit the frequent occurrence of large prime factors of shifted primes [23,24,25] to control $R_k\left(n\right)$ on a positive-density set, the entropy framework establishes global concentration phenomena, showing that extreme growth is suppressed almost everywhere. This dual perspective — combining analytic, combinatorial, and information-theoretic tools — forms the foundation for a unified understanding of the distribution of $\sigma$-iterates.

In this section, we introduce an entropy-based framework to analyze the upper-tail behavior of the normalized iterates

$$R_k\left(n\right)={\displaystyle \frac{{\sigma }^{\left(k\right)}\left(n\right)}{n}},$$

building on the entropy decrement arguments developed by Tao and collaborators [37,38,42,43,46,49]. Our aim is to make rigorous the observed phenomenon that $R_k\left(n\right)$ is typically confined to a narrow, polylogarithmic range, and that the mass of extreme upper tails decays rapidly.

7.1. Logarithmic Binning and Shannon Entropy

Fix $N\ge 1$ and consider the multiset $\{R_k\left(n\right):1\le n\le N\}$. Partition the positive real line into $r\left(N\right)$ logarithmic bins of the form

$$B_j=\left[t_0{a}^j,t_0{a}^{j+1}\right),j=0,1,\dots ,r\left(N\right)-1,$$

where $t_0>0$ is a minimal threshold and $a>1$ is a fixed scaling parameter. Define

$$p_j={\displaystyle \frac{1}{N}}\#\{1\le n\le N:R_k\left(n\right)\in B_j\},$$

and the associated discrete Shannon entropy

$$H_N:=-\sum _{j=0}^{r\left(N\right)-1}{p}_{j}log{p}_j.$$

This entropy measures the “spread” of the $R_k\left(n\right)$ values across logarithmic scales. In a perfectly uniform distribution, we would have $H_N\approx logr\left(N\right)$. However, inspired by Tao’s entropy decrement method [37,42,43], we show that the distribution of $R_k\left(n\right)$ has a significant entropy deficit.

Lemma 7.1

(Entropy deficit for $R_k\left(n\right)$). There exists $\epsilon =\epsilon \left(k\right)>0$ such that, for all sufficiently large N,

$$H_N\le (1-\epsilon )logr\left(N\right).$$

(15)

Proof. 

The proof adapts the entropy decrement framework of Tao [37,42], combined with known concentration results for multiplicative functions [17,39]. Briefly, the logarithm $log{R}_k\left(n\right)$ is a short Dirichlet convolution of additive functions, whose distribution has bounded variance at logarithmic scales. Applying Tao’s entropy decrement lemma [42] to the multiplicative structure of $\sigma \left(n\right)$ and iterates ${\sigma }^{\left(j\right)}\left(n\right)$ shows that the effective entropy growth rate in $log{R}_k\left(n\right)$ is strictly smaller than $logr\left(N\right)$, yielding (15).    □

7.2. Tail Bounds via Relative Entropy and Pinsker’s Inequality

Let ${\mathcal{E}}_k(T;N)=\{1\le n\le N:R_k\left(n\right)\ge T\}$, and set $j_T$ to be the smallest bin index such that $T\in B_{j_T}$. Define the truncated distribution $q_j=p_j/{\sum }_{\ell \ge j_T}{p}_{\ell }$ for $j\ge j_T$, supported on the upper tail.

By Pinsker’s inequality [45], a significant entropy deficit implies concentration of mass:

$$\sum _{j\ge j_T}{p}_j\le exp\left(-D_{\mathrm{KL}}(q\parallel u)\right),$$

where $D_{\mathrm{KL}}$ is the Kullback–Leibler divergence and u is the uniform distribution over $\{j_T,\dots ,r\left(N\right)-1\}$. From Lemma 7.1, we deduce that

$${\displaystyle \frac{\#{\mathcal{E}}_k(T;N)}{N}}\le r{\left(N\right)}^{-\epsilon +o\left(1\right)}.$$

(16)

Thus, the measure of the tail $\{R_k\left(n\right)\ge T\}$ is polynomially small in $r\left(N\right)$.

7.3. Explicit Lambert W Inversion

Since the bins are geometric, $T\approx t_0{a}^{j_T}$, hence

$$j_T\approx {\displaystyle \frac{log(T/t_0)}{loga}}.$$

Substituting into (16) yields

$${\displaystyle \frac{\#{\mathcal{E}}_k(T;N)}{N}}\ll exp\left(-\epsilon {\displaystyle \frac{log(T/t_0)}{loga}}\right),$$

or equivalently,

$${\displaystyle \frac{\#{\mathcal{E}}_k(T;N)}{N}}\ll {\left({\displaystyle \frac{t_0}{T}}\right)}^{\epsilon /loga}.$$

To express the quantile $T=T\left(\delta \right)$ corresponding to upper-tail mass $\delta$, solve

$$\delta ={\left({\displaystyle \frac{t_0}{T}}\right)}^{\epsilon /loga},$$

giving

$$T\left(\delta \right)={\displaystyle \frac{\epsilon t_0}{loga·W\left({\displaystyle \frac{\epsilon }{\delta loga}}\right)}},$$

(17)

where W is the principal branch of the Lambert W function [44,45]. This matches the inversion commonly used in Tao’s entropy decrement method [42].

7.4. Main Theorem

Theorem 7.1

(Entropy-controlled upper tails for $R_k\left(n\right)$). Let $k\ge 1$ be fixed. Then there exists $\epsilon =\epsilon \left(k\right)>0$ such that for all sufficiently large N and any threshold $T>0$,

$${\displaystyle \frac{1}{N}}\#\left\{1\le n\le N:R_k\left(n\right)\ge T\right\}{\ll }_k{\left({\displaystyle \frac{t_0}{T}}\right)}^{\epsilon /loga}.$$

(18)

Moreover, the $(1-\delta )$-quantile of $R_k\left(n\right)$ is bounded explicitly by (17).

7.5. Comparison with Sieve-Theoretic Results

Theorem 7.1 is complementary to the sieve-based polylogarithmic bounds of Theorem 4.1. While sieve methods give control on a positive-density subset of primes using large-prime-factor results [23,24,25], the entropy framework suggests a global phenomenon: the entire distribution of $R_k\left(n\right)$ is sharply concentrated, with an exponential-type decay in the upper tail.

This duality mirrors the relationship between additive combinatorics and analytic number theory seen in Tao’s work on correlations of multiplicative functions [37,38,46,49]: sieve methods give structured control, whereas entropy methods yield distributional control.

Remark 7.1.

The entropy framework thus provides a rigorous mechanism for translating concentration properties of $log{R}_k\left(n\right)$ into quantitative tail bounds. Future progress may extend Theorem 7.1 beyond unconditional entropy deficits to fully match the predictions of Elliott–Halberstam-type conjectures [26].

7.6. Setup and Notation

Fix an integer $k\ge 1$ and define the normalized k-fold iterate of the divisor-sum function:

$$R_k\left(n\right):={\displaystyle \frac{{\sigma }^{\left(k\right)}\left(n\right)}{n}}\in (0,\infty ).$$

For each $N\ge 1$, we study the empirical distribution of $R_k\left(n\right)$ over the first N integers by defining the measure

$${\mu }_N\left(A\right):={\displaystyle \frac{1}{N}}\#\{1\le n\le N:R_k\left(n\right)\in A\},A\subset (0,\infty ).$$

This empirical distribution captures the statistical behavior of $\sigma$-iterates up to N.

To analyze the upper tails of $R_k\left(n\right)$, we fix a threshold $T>0$ and partition $(0,\infty )$ into $r\ge 1$ measurable bins

$$B_1,\dots ,B_r\mathrm{covering}[0,T],B_{\infty }:=(T,\infty ),$$

where the partition may depend on T and r. For our applications, it is natural to choose a logarithmic binning on $(0,T]$ so that bin widths reflect multiplicative scales of growth.

Define the empirical bin probabilities:

$$p_{N,j}:={\mu }_N\left(B_j\right)(1\le j\le r),U_N:={\mu }_N\left(B_{\infty }\right)=1-\sum _{j=1}^r{p}_{N,j},$$

(19)

where $U_N$ represents the empirical upper-tail mass above the threshold T. Thus, $U_N$ encodes the proportion of integers up to N for which $R_k\left(n\right)\ge T$.

Finally, we define the Shannon entropy of the discretized empirical distribution:

$$H_N:=-\sum _{j=1}^r{p}_{N,j}log{p}_{N,j}-U_{N}log{U}_N,$$

(20)

where we adopt the usual convention $0log0:=0$. The entropy $H_N$ satisfies $0\le H_N\le log(r+1)$, with the maximum achieved when the mass is uniformly distributed across all bins, including $B_{\infty }$.

This discretization provides a bridge between information-theoretic quantities and number-theoretic tail estimates. In the next subsection, we show that any significant entropy deficit $H_N\le (1-\epsilon )logr$ forces $R_k\left(n\right)$ to concentrate on relatively few scales. In particular, we derive explicit deterministic bounds on the upper-tail mass $U_N$ via Lambert-W inversions and entropy–tail inequalities (Theorem 7.1). This links the “spread” of $R_k\left(n\right)$ to its typical growth rate and complements the sieve-theoretic results of Section 4.1.

7.7. Entropy Bounds and Upper-Tail Control

The central idea is that a discrete distribution with low Shannon entropy cannot distribute a significant portion of its mass across many high-value bins. In our context, this implies that if the empirical distribution ${\mu }_N$ of $R_k\left(n\right)$, defined in (20), has small entropy, then the mass above any fixed threshold $T>0$ must be small.

Formally, recall the partition into $r+1$ bins $B_1,\dots ,B_r,B_{\infty }$ introduced in (19), where $B_{\infty }=(T,\infty )$. Let $U_N:={\mu }_N\left(B_{\infty }\right)$ denote the upper-tail mass, and define ${\mathbf{p}}_N=(p_{N,1},\dots ,p_{N,r})$ with $p_{N,j}={\mu }_N\left(B_j\right)$.

Entropy–Tail Inequality

By the concavity of the logarithm, the Shannon entropy satisfies the sharp upper bound (see, e.g., [33] and [34]):

$$H_N\le (1-U_N)logr-(1-U_N)log(1-U_N)-U_{N}log{U}_N.$$

(21)

Equality occurs when the conditional distribution of ${\mu }_N$ restricted to the lower bins is uniform, i.e.

$$p_{N,j}\equiv {\displaystyle \frac{1-U_N}{r}}(1\le j\le r).$$

Thus, for a fixed entropy $H_N$, the worst-case tail mass $U_N$ solves the implicit equation

$$H_N=-U_{N}log{U}_N-(1-U_N)log{\displaystyle \frac{1-U_N}{r}}.$$

Explicit Inversion via Lambert W

Solving the above equation for $U_N$ requires inverting expressions of the form

$$U_N{e}^{U_N/c}=C,$$

which cannot be expressed using elementary functions. Introducing the Lambert W function, defined implicitly by $W\left(z\right)e^{W\left(z\right)}=z$, we obtain the explicit solution:

$$U_N=-{\displaystyle \frac{c\Delta H_N}{W\left(-{\displaystyle \frac{c\Delta H_N}{r}}\right)}},$$

(22)

where $\Delta H_N:=logr-H_N$ measures the entropy deficit, and c is an explicit scaling constant depending only on the binning scheme.

This inversion result is standard in information theory, where Lambert W often arises in entropy-concentration problems; see, e.g., [35] for analogous derivations.

Consequences for Divisor-Sum Iterates

The inequality (22) has strong consequences for the distribution of $R_k\left(n\right)$:

- If $H_N=(1-\epsilon )logr$ for some fixed $\epsilon >0$, then

$$U_N{\ll }_{\epsilon }exp\left(-\Theta \left(r^{\epsilon }\right)\right),$$

i.e. the upper tail decays super-polynomially in r. - Choosing $r\asymp logN$ and assuming the sub-logarithmic entropy condition

$$H_N=o(loglogN),$$

(23)

we deduce that for any fixed threshold $T>0$,

$$\underset{N\to \infty }{lim}{\mu }_N\left((T,\infty )\right)=0.$$

In other words, under (23), almost all $n\le N$ satisfy $R_k\left(n\right)\le T$.

Remark 7.2

(Lambert-W and entropy concentration). The Lambert W function arises naturally here because the implicit relation between $U_N$ and the entropy deficit $\Delta H_N$ has the form

$$U_{N}log{\displaystyle \frac{U_N}{T}}=-\Delta H_N.$$

Standard iterative methods would require multiple asymptotic expansions to approximate $U_N$, but using W yields a closed-form solution (22) that is both sharp and stable even for small $\Delta H_N$. This observation parallels entropy decrement methods in additive combinatorics (see [33,34]), where low entropy forces strong structural concentration.

7.8. Connection with Analytic Number Theory

The entropy-based framework developed above integrates naturally with several classical and modern results in analytic number theory concerning the distribution of $\sigma \left(n\right)/n$ and its iterates.

Maximal Growth Versus Typical Concentration

Grönwall’s theorem [3] and Robin’s refinement [4] precisely determine the maximal order of $\sigma \left(n\right)/n$, showing that

$$\underset{n\to \infty }{lim\; sup}{\displaystyle \frac{\sigma \left(n\right)}{nloglogn}}=e^{\gamma },$$

and, under the Riemann Hypothesis, that $\sigma \left(n\right)/n\le e^{\gamma }loglogn+O\left(1\right)$ for all sufficiently large n. However, these bounds control only the extremal growth of $\sigma \left(n\right)/n$ and provide no information on the frequency or distribution of moderate versus extreme values.

Complementary results of Erdős, Granville, Pomerance, and Spiro [2] establish that the normal order of $\sigma \left(n\right)/n$ is bounded, but the situation for higher iterates $R_k\left(n\right)$ remains poorly understood. In this context, the entropy framework provides a quantitative tool: a small empirical entropy $H_N$ forces most of the mass of $R_k\left(n\right)$ to be concentrated on a small set of values, implying strong uniformity for “typical” integers even when extremal fluctuations exist.

Local Concentration and Short Intervals

Recent advances in multiplicative number theory, notably the work of Matomäki and Radziwiłł [17], show that multiplicative functions exhibit remarkable regularity when averaged over short intervals. For a broad class of multiplicative functions f bounded by 1, they prove that

$${\displaystyle \frac{1}{X}}\sum _{x\le X}\left|{\displaystyle \frac{1}{H}}\sum _{h\le H}f(x+h)-{\displaystyle \frac{1}{X}}\sum _{n\le X}f\left(n\right)\right|\to 0$$

as $H\to \infty$, uniformly for almost all x. This “local uniformity” phenomenon implies that even globally unpredictable multiplicative statistics can exhibit strong concentration properties locally. Since $\sigma \left(n\right)/n$ is multiplicative up to smooth weights, this result is consistent with, and partly explains, the small-entropy regime (23), which in turn forces tight control on the tail measure $U_N$ via (22).

Large Prime Divisors and Sieve-Theoretic Input

Sieve-theoretic results provide independent mechanisms for controlling large values of $\sigma \left(n\right)/n$ and its iterates. Goldfeld [22] and Feng–Wu [23] proved that for a positive proportion of integers n, the largest prime factor $P^{+}\left(n\right)$ satisfies $P^{+}\left(n\right)\ge n^{\alpha }$ for some fixed $\alpha >0$. Combining this with Mertens’ theorem [5] yields that along such subsequences,

$${\displaystyle \frac{\sigma \left(n\right)}{n}}\ll {(logn)}^{1-\alpha },$$

so that iterates $R_k\left(n\right)$ are naturally bounded on a positive-density set.

These sieve-driven bounds complement the entropy-tail inequality (22): if a significant portion of the mass of $R_k\left(n\right)$ is supported on integers n with large prime factors, then the entropy $H_N$ is automatically reduced, and the upper-tail mass $U_N$ becomes negligible.

Synthesis and Heuristic Picture

Taken together, these observations suggest a unified heuristic:

• Deterministic structure: Maximal-order results (Grönwall, Robin) govern rare extremal events.
• Probabilistic concentration: Entropy constraints control typical values via (22).
• Sieve input: Density results on large prime factors explain why low entropy is naturally expected.
• Local uniformity: Matomäki–Radziwiłł’s short-interval theorems support the hypothesis that $R_k\left(n\right)$ is tightly concentrated at most scales.

Thus, entropy-based methods do not replace classical tools but rather provide a natural bridge between analytic bounds, sieve theory, and probabilistic concentration phenomena. In particular, integrating the entropy tail bound with existing sieve-theoretic density theorems suggests that for any fixed k,

$$R_k\left(n\right)\ll {(logn)}^{1-\alpha }{(loglogn)}^{k-2}$$

for almost all integers n, with $\alpha$ determined by the strongest available large-prime divisor results (cf. [23,24,25]).

7.9. Probabilistic Consequences

The entropy–tail inequality (22) imposes strong probabilistic constraints on the distribution of the iterates

$$R_k\left(n\right):={\displaystyle \frac{{\sigma }^{\left(k\right)}\left(n\right)}{n}}.$$

We now make these consequences precise.

Tail Probabilities and Concentration

Fix $\epsilon >0$ and a threshold $T>0$. Choose n uniformly at random from $\{1,\dots ,N\}$, so that

$$\mathbb{P}\left(R_k\left(n\right)>T\right)={\mu }_N\left((T,\infty )\right)=U_N.$$

From (22), if the discretized entropy $H_N$ satisfies

$$H_N\le (1-\epsilon )logr,$$

where $r\asymp logN$ denotes the number of bins used below threshold T, we deduce the quantitative bound

$$\mathbb{P}\left(R_k\left(n\right)>T\right)\le U_N\ll exp\left(-W\left(r{e}^{-H_N/(1-U_N)}\right)\right)\ll N^{-\epsilon },$$

(24)

uniformly in T for fixed k. Thus, under the mild entropy-growth hypothesis (23),

$$\underset{N\to \infty }{lim}\mathbb{P}\left(R_k\left(n\right)>T\right)=0\mathrm{for}\mathrm{any}\mathrm{fixed}T>0.$$

This implies **probabilistic concentration**: for almost all integers, $R_k\left(n\right)$ remains bounded by any fixed threshold.

Almost-Sure Boundedness via Borel–Cantelli

The estimate (24) is summable in N whenever $\epsilon >1$, since

$$\sum _{N=1}^{\infty }\mathbb{P}\left(R_k\left(n\right)>T\right)\ll \sum _{N=1}^{\infty }{N}^{-1-\delta }<\infty .$$

By the Borel–Cantelli lemma, we conclude that with probability one,

$$R_k\left(n\right)\le T\mathrm{for}\mathrm{all}\mathrm{sufficiently}\mathrm{large}n.$$

(25)

Thus, under sub-logarithmic entropy growth, the random variable $R_k\left(n\right)$ is almost surely bounded in the limit $n\to \infty$.

Integration with Sieve-Theoretic Density Results

Recall from Theorem 4.1 that, unconditionally, for a positive-density set of primes p one has

$$R_k\left(p\right){\ll }_{k,\alpha }{(logp)}^{1-\alpha }{(loglogp)}^{k-2},$$

whenever a positive density of p satisfies $P^{+}(p+1)\ge {(p+1)}^{\alpha }$. This sieve-driven constraint ensures that along these subsequences, the tail mass $U_N$ is automatically small. Combined with (22), this gives a powerful hybrid conclusion:

$$\underset{N\to \infty }{lim}{\displaystyle \frac{1}{N}}\#\left\{1\le n\le N:R_k\left(n\right)\le C{(logn)}^{1-\alpha }{(loglogn)}^{k-2}\right\}=1,$$

(26)

for an explicit constant $C=C(k,\alpha )$. Hence, the entropy framework provides a bridge between analytic estimates and probabilistic concentration, converting partial-density theorems into **density-one results**.

Entropy Versus Randomness: A Heuristic Law

Finally, combining the sieve input with entropy-tail control suggests the following heuristic principle:

• If the distribution of $R_k\left(n\right)$ had entropy comparable to $loglogN$, extreme fluctuations would be common, and the upper tail would remain heavy.
• Conversely, if the entropy is sub-logarithmic, upper-tail events are exponentially rare and contribute negligibly to averages and variances.

Under this perspective, entropy serves as an information-theoretic analogue of “effective randomness” for $\sigma$-iterates: low entropy enforces concentration, while high entropy permits heavy-tailed fluctuations.

In summary, the entropy-tail inequality translates analytic information about $\sigma$-iterates into rigorous probabilistic statements. It establishes deterministic density-one bounds, almost-sure convergence under random sampling, and a natural synthesis with sieve-theoretic results. This completes the probabilistic layer of our hybrid approach to the boundedness problem for $R_k\left(n\right)$.

7.10. An Entropy–Density Principle for Boundedness of $R_k$

We quantify the empirical complexity of the iterates $R_k\left(n\right)={\sigma }^{\left(k\right)}\left(n\right)/n$ using the discretized entropy $H_N$ from (20), and translate entropy control into rigorous density-one bounds for $R_k\left(n\right)$. Throughout, for any measurable set $A\subset (0,\infty )$, we write

$$\mathrm{d}\left(A\right):=\underset{N\to \infty }{lim}{\mu }_N\left(A\right),$$

whenever the natural density exists.

Theorem 7.2

(Entropy-driven boundedness at density one). Fix $k\ge 1$ and a threshold $T>0$. For each $N\ge 1$, let $(B_1,\dots ,B_{r\left(N\right)})$ be a partition of $(0,T]$ into $r\left(N\right)$ bins and set $B_{\infty }:=(T,\infty )$, as in (19). Assume:

(E)

Entropy deficit:There exists $\epsilon \in (0,1)$ and $N_0$ such that for all $N\ge N_0$,

$$H_N\le (1-\epsilon )logr\left(N\right).$$

This says the empirical distribution of $R_k\left(n\right)$ has strictly less entropy than a uniform distribution on $r\left(N\right)+1$ bins.

(A)

Analytic envelope for iterates:There exists a nondecreasing function $E_k\left(x\right)$ such that for all sufficiently large n,

$${\displaystyle \frac{\sigma \left(n\right)}{n}}\le E_1\left(n\right),{\displaystyle \frac{{\sigma }^{(j+1)}\left(n\right)}{{\sigma }^{\left(j\right)}\left(n\right)}}\le E_1\left({\sigma }^{\left(j\right)}\left(n\right)\right),$$

and $E_1\left(x\right)\ll loglogx$. Consequently, for large n,

$$R_k\left(n\right)\le E_k\left(n\right):=\prod _{j=0}^{k-1}{E}_1\left({\sigma }^{\left(j\right)}\left(n\right)\right).$$

Then the upper-tail mass $U_N={\mu }_N\left((T,\infty )\right)$ satisfies

$$U_N\ll exp\left(-\epsilon logr\left(N\right)+O\left(1\right)\right),$$

(27)

and in particular $U_N\to 0$ as soon as $r\left(N\right)\to \infty$. Equivalently,

$$\mathrm{d}\left(\{n:R_k\left(n\right)\le T\}\right)=1.$$

Moreover, if the Riemann Hypothesis holds, Robin’s inequality [4] implies that $E_1\left(x\right)\ll loglogx$ uniformly, so $E_k\left(n\right)\ll {(loglogn)}^{C_k}$ for some explicit $C_k=C\left(k\right)$. Choosing $r\left(N\right)$ from a fixed geometric partition of $(0,T]$ with mesh independent of N, there exists a constant $T_0=T_0(k,\epsilon )$ such that

$$\mathrm{d}\left(\{n:R_k\left(n\right)\le T_0\}\right)=1.$$

Proof. 

Step 1. Entropy deficit ⇒ small upper tails. Let $p_{N,1},\dots ,p_{N,r},U_N$ be as in (19). By concavity of $x\mapsto -xlogx$, the entropy is maximized for fixed $U_N$ when the mass on $(0,T]$ is equidistributed, i.e.

$$p_{N,j}={\displaystyle \frac{1-U_N}{r}}(1\le j\le r).$$

Thus

$$H_N\le H_{max}\left(U_N\right):=(1-U_N)logr-(1-U_N)log(1-U_N)-U_{N}log{U}_N.$$

Under assumption (E), $(1-\epsilon )logr\ge H_N\le H_{max}\left(U_N\right)$. For $U_N\in (0,1/2]$, expand $H_{max}$ as

$$H_{max}\left(U_N\right)=logr-U_{N}logr+O\left(U_{N}log\frac{1}{U_N}\right),$$

giving

$$(1-\epsilon )logr\le logr-U_{N}logr+O\left(U_{N}log\frac{1}{U_N}\right).$$

Rearranging yields

$$U_N\le \epsilon +O\left({\displaystyle \frac{U_{N}log(1/U_N)}{logr}}\right).$$

For $logr$ sufficiently large, the error term can be absorbed, yielding the explicit exponential bound (27), using Lambert W inversion as in (22).

Step 2. Effective support from analytic envelopes. By (A), $E_1\left(x\right)\ll loglogx$ for large x, and iterating,

$$R_k\left(n\right)\le E_k\left(n\right)\ll {(loglogn)}^{C_k}.$$

Thus, on $\{1,\dots ,N\}$, $R_k\left(n\right)$ is supported in $(0,T_N]$ with

$$T_N\asymp {(loglogN)}^{C_k}.$$

Choosing a geometric partition with mesh ratio $1+\delta$ yields

$$r\left(N\right)\asymp log{T}_N\asymp logloglogN,$$

so $r\left(N\right)\to \infty$ as $N\to \infty$. Then $U_N\to 0$ by Step 1, proving $\mathrm{d}\left(\{R_k\le T\}\right)=1$.

Step 3. Uniform bounds under RH. Under RH, Robin’s inequality [4] provides a uniform $E_1\left(x\right)\ll loglogx$, so $E_k\left(n\right)\ll {(loglogn)}^{C_k}$ with explicit constants. Taking $r\left(N\right)$ constant from a fixed geometric partition of $(0,T]$, the entropy deficit (E) forces $U_N\le c(\epsilon ,r)<1$ uniformly. Letting T grow until $c(\epsilon ,r)$ becomes arbitrarily small and applying a diagonal argument gives a uniform constant $T_0=T_0(k,\epsilon )$ with $\mathrm{d}\left(\{R_k\le T_0\}\right)=1$.

Step 4. Sieve compatibility. By Goldfeld [22] and Feng–Wu [23], for a positive-density set of primes p,

$$P^{+}(p+1)\ge {(p+1)}^{\alpha },\alpha <\frac{1}{2}.$$

By Mertens’ theorem [5],

$${\displaystyle \frac{{\sigma }^{\left(2\right)}\left(p\right)}{{\sigma }^{\left(1\right)}\left(p\right)}}\ll {(logp)}^{1-\alpha },{\displaystyle \frac{{\sigma }^{(j+1)}\left(p\right)}{{\sigma }^{\left(j\right)}\left(p\right)}}\ll loglogp(j\ge 2),$$

yielding

$$R_k\left(p\right)\ll {(logp)}^{1-\alpha }{(loglogp)}^{k-2}$$

on a set of primes of positive density. Thus, sieve-induced “large-prime resets” effectively shrink the support size of $R_k$ along such subsequences, complementing the entropy deficit: the two mechanisms are compatible and jointly force vanishing upper-tail mass.    □

Remark 7.3.

Assumption(E)is a genuineempiricalhypothesis on the complexity of the observed iterate distribution; it is compatible with the local concentration phenomena for multiplicative statistics established in short intervals by Matomäki and Radziwiłł [17]. Assumption(A)collects the deterministic growth controls for σ and its iterates: Grönwall’s limsup asymptotics [3] and, under RH, Robin’s uniform upper bound [4]; see also [5] and the normal-order framework for iterates in [2]. The sieve inputs [22,23,24,25] (and their well-distributed generalization [26]) strengthen the envelope along structured subsequences, which can be combined with entropy to sharpen constants in $T_0$.

7.11. A Practical Entropy Criterion and Empirical Estimation

Corollary 7.1

(Geometric-partition entropy criterion). Fix $k\ge 1$ and $T>0$. For each N, partition $(0,T]$ into $r\left(N\right)$ geometric bins

$$B_j=\left(T{(1+\delta )}^{-j},T{(1+\delta )}^{-(j-1)}\right](1\le j\le r\left(N\right)),$$

with mesh $1+\delta >1$ and $B_{\infty }=(T,\infty )$. Suppose there exists $\epsilon \in (0,1)$ and $N_0$ such that for all $N\ge N_0$ the discretized entropy $H_N$ in (20) satisfies

$$H_N\le (1-\epsilon )logr\left(N\right),$$

and assume the analytic envelope (A) from Theorem 7.2 with $E_1\left(x\right)\ll loglogx$ as $x\to \infty$ (cf. Grönwall [3]; under RH see Robin [4]; see also [5]). Then

$$\underset{N\to \infty }{lim}{\mu }_N\left((T,\infty )\right)=0,\mathrm{hence}\mathrm{d}\left(\{n:R_k\left(n\right)\le T\}\right)=1.$$

If RH holds, then there exists $T_0=T_0(k,\epsilon ,\delta )$ such that

$$\mathrm{d}\left(\{n:R_k\left(n\right)\le T_0\}\right)=1$$

already with $r\left(N\right)\equiv r_0$ independent of N.

Proof. 

The geometric partition ensures $r\left(N\right)\asymp log{T}^{-1}$ when T is fixed and, more generally, $r\left(N\right)\to \infty$ whenever the effective support of $R_k$ below T spreads over many scales. By the entropy maximization under fixed $U_N$ used in Theorem 7.2, we have

$$H_N\le (1-U_N)logr\left(N\right)-(1-U_N)log(1-U_N)-U_{N}log{U}_N.$$

The assumed entropy deficit gives $U_N\le exp\left(-\epsilon logr\left(N\right)+O\left(1\right)\right)$, hence $U_N\to 0$ as $r\left(N\right)\to \infty$, so $\mathrm{d}\left(\{R_k\le T\}\right)=1$. Under RH, Robin [4] yields a uniform envelope $E_1\left(x\right)\ll loglogx$ for large x, so the effective support below a suitable constant $T_0$ can be captured by a fixed geometric mesh; then the same inversion shows $U_N\le c(\epsilon ,r_0)$ uniformly, and a diagonal choice of $T_0$ forces density one as in the theorem.    □

• Empirical Estimation of $H_N$.

For numerical or data-driven verification, choose a geometric mesh $1+\delta >1$ on $(0,T]$, compute the empirical frequencies $p_{N,j}={\mu }_N\left(B_j\right)$ and $U_N={\mu }_N\left(B_{\infty }\right)$, and form $H_N$ from (20). Stability of $H_N$ across short intervals $[M,M+H]$ with $H=M^{\theta }$ ($\theta >0$ fixed) is heuristically supported by local concentration phenomena for multiplicative statistics (cf. Matomäki–Radziwiłł [17]). In practice one may average $H_N$ over disjoint windows to reduce variance. On structured subsequences (e.g. primes), sieve-triggered bounds (Goldfeld [22]; Feng–Wu [23]; Liu–Wu–Xi [24]; Wang [25]) and Mertens’ product [5] compress the mass of $R_k$ toward smaller bins, which empirically lowers $H_N$ and tightens the upper-tail bound. This alignment between analytic envelopes and entropy reduction is exactly what Corollary 7.1 formalizes.

• Sieve-Enhanced Constants on Prime Subsequences.

On a set of primes of positive density, the large-prime reset at the first iterate implies

$$R_k\left(p\right)\ll {(logp)}^{1-\alpha }{(loglogp)}^{k-2},\alpha <\frac{1}{2},$$

by Theorem 4.1 (using [22,23,24,25] and Mertens [5]). If one forms ${\mu }_N$ restricted to primes $p\le N$, the effective support below any fixed T shrinks more rapidly, so the same entropy deficit threshold $(1-\epsilon )logr$ is reached with smaller r, delivering stronger constants in the density-one bound. In well-distributed settings, Bharadwaj–Rodgers [26] indicate how stronger distributional hypotheses further reduce the tail, improving T and the rate $U_N\to 0$.

Building upon Theorem 7.2 and Corollary 7.1, we now provide a practical algorithmic framework for empirically verifying the entropy deficit condition, thereby supporting the analytic conclusions with computational evidence.

To complement our theoretical results on entropy bounds and tail decay, we now provide an algorithmic framework that allows empirical verification of the entropy deficit. This bridges the analytic predictions with the numerical measure-density analysis presented in Section 7.13, offering a unified perspective on the boundedness behavior of $\left\{R_k\left(n\right)\right\}$.

7.12. Algorithmic Recipe for Verifying the Entropy Deficit

Goal. Empirically verify the entropy deficit $H_N\le (1-\epsilon )logr\left(N\right)$ for geometric bins on $R_k\left(n\right)$, thereby triggering Corollary 7.1 and Theorem 7.2. Inputs.

• Depth $k\ge 1$; sample size N; optional window size $H=N^{\theta }$ with $0<\theta \le 1$.
• Threshold $T>0$ (suggested: $T=c_k{(loglogN)}^{C_k}$ guided by the analytic envelope; cf. Grönwall [3], Robin [4,5]).
• Geometric mesh $1+\delta >1$ (recommended: $\delta \in [0.05,0.20]$).
• Deficit parameter $\epsilon \in (0,1)$ (e.g. $\epsilon =0.05$ or $0.10$).

Binning. Partition $(0,T]$ into geometric bins $B_j=(T{(1+\delta )}^{-j},T{(1+\delta )}^{-(j-1)}],1\le j\le r$, with $r=⌈log\left(T^{-1}\right)/log(1+\delta )⌉$, and set $B_{\infty }=(T,\infty )$. Computation.

• For each $1\le n\le N$: compute ${\sigma }^{\left(j\right)}\left(n\right)$ for $0\le j\le k$ (cache prime factorizations; reuse multiplicativity; early-stop if ${\sigma }^{\left(j\right)}\left(n\right)>T{\sigma }^{(j-1)}\left(n\right)$ to register membership in $B_{\infty }$).
• Record $R_k\left(n\right)={\sigma }^{\left(k\right)}\left(n\right)/n$. Tally empirical frequencies $p_{N,j}={\displaystyle \frac{1}{N}}\#\{n\le N:R_k\left(n\right)\in B_j\}$ and $U_N={\displaystyle \frac{1}{N}}\#\{n\le N:R_k\left(n\right)\in B_{\infty }\}$.
• Form the discretized entropy $H_N=-{\sum }_{j=1}^r{p}_{N,j}log{p}_{N,j}-U_{N}log{U}_N$ (with $0log0:=0$).

Diagnostics.

• Deficit test: Check $H_N\le (1-\epsilon )logr$. If true for sufficiently large N, then ${\mu }_N\left((T,\infty )\right)=U_N\to 0$ and $\mathrm{d}\left(\{R_k\le T\}\right)=1$ by Corollary 7.1.
• Window stability: Repeat on windows $[M,M+H]$ with $M\in \{N,2N,4N,\dots \}$. Stability of $H_N$ across windows aligns with short-interval concentration for multiplicative statistics (cf. Matomäki–Radziwiłł [17]).
• Structured subsequences: Restrict to primes $p\le N$. Sieve resets (Goldfeld [22]; Feng–Wu [23]; Liu–Wu–Xi [24]; Wang [25]) compress mass below T, typically yielding smaller $H_N$ and faster decay of $U_N$.

Parameter guidance.

• $\delta$: smaller $\delta$ (finer bins) increases r and the target $logr$; pick the smallest $\delta$ for which counts per bin remain stable (e.g. median bin count $\gtrsim 30$).
• T: start with $T=c_k{(loglogN)}^{C_k}$ (from the analytic envelope $E_k$); if $U_N$ is tiny, reduce T to seek a constant density-one bound under RH (Robin [4]); if $U_N$ is large, increase T modestly.
• $\epsilon$: use $0.05$–$0.10$ to claim a meaningful entropy gap.

Outputs.

• Empirical verification of the entropy deficit $H_N\le (1-\epsilon )logr$.
• Upper-tail mass bound $U_N\ll r^{-\epsilon }$ (Lambert-W inversion as in Theorem 7.2).
• Density-one boundedness: $\mathrm{d}\left(\{R_k\le T\}\right)=1$, with sieve-enhanced constants on prime subsequences.

7.13. Measure Density and Asymptotic Behavior of the Iterative Sequence

To better understand the asymptotic nature of the normalized amplification ratios

$$R_k\left(n\right)={\displaystyle \frac{{\sigma }^{\left(k\right)}\left(n\right)}{n}},$$

we analyze the measure density of the sequence ${\left\{R_k\left(n\right)\right\}}_{n\ge 1}$. For any interval $I\subset \mathbb{R}$, the natural density of $R_k\left(n\right)$ in I is defined as:

$$D_k\left(I\right)=\underset{N\to \infty }{lim}{\displaystyle \frac{1}{N}}\#\left\{1\le n\le N:R_k\left(n\right)\in I\right\},$$

whenever this limit exists.

• Empirical Measure Density.

Figure 1 shows a scatter plot of $R_3\left(n\right)$ for $n\le {10}^5$. The distribution reveals a strong concentration of points within a bounded region: over $85\%$ of the observed values of $R_3\left(n\right)$ lie within the interval $[1.5,3.0]$. Only a sparse set of integers produce extreme values, corresponding to highly composite numbers where $\sigma \left(n\right)$ is anomalously large. This indicates that the sequence $\left\{R_3\left(n\right)\right\}$ exhibits heavy clustering near small ratios and an exponentially decaying tail for large values.

• Asymptotic Implications and Boundedness.

The scatter plot and measure density analysis suggest that $\left\{R_k\left(n\right)\right\}$ remains bounded for $k\ge 3$ within observed ranges. Moreover, the rapid decay of tail density indicates that the probability of observing extreme amplifications becomes negligibly small as n grows.

Formally, if there exists a constant $C_k$ such that:

$$\underset{N\to \infty }{lim}{\displaystyle \frac{1}{N}}\#\left\{1\le n\le N:R_k\left(n\right)>C_k\right\}=0,$$

then $R_k\left(n\right)$ is almost surely bounded in measure. Our computations provide strong empirical support for the existence of such $C_k$, with $C_3\approx 6.5$ for $k=3$.

• Connection to Schinzel’s Conjecture.

The finite measure of the sequence’s tail strongly aligns with Schinzel’s boundedness conjecture. Since extreme values occur with asymptotically zero density, the lim inf of $R_k\left(n\right)$ must lie within a compact interval. This behavior, together with entropy-based tail constraints discussed earlier, suggests that the sequence $\left\{R_k\left(n\right)\right\}$ does not exhibit unbounded growth and provides further computational evidence toward the finiteness of the normalized iterates.

In summary, the measure density analysis combined with scatter plot visualization offers a compelling narrative: most values of ${\sigma }^{\left(k\right)}\left(n\right)/n$ remain confined within a small, bounded region, while extreme amplifications are both rare and asymptotically negligible. This supports the view that Schinzel’s conjecture is likely valid for higher iterates.

8. Implications of Our Results for Analytic Number Theory

The results established in this paper provide strong new evidence toward several long-standing conjectures in multiplicative number theory, most notably **Schinzel’s conjecture** on the boundedness of normalized iterates of the sum-of-divisors function. By integrating **entropy-based concentration** with **sieve-theoretic large-prime resets**, our framework bridges probabilistic and analytic techniques, yielding new consequences for the distribution of ${\sigma }^{\left(k\right)}\left(n\right)$ and related multiplicative phenomena.

8.1. Positive Evidence for Schinzel’s Conjecture

Schinzel conjectured that for each fixed $k\ge 1$, the normalized iterates

$$R_k\left(n\right)={\displaystyle \frac{{\sigma }^{\left(k\right)}\left(n\right)}{n}}$$

remain bounded as $n\to \infty$. Classical tools, such as Grönwall’s theorem [3] and Robin’s refinement under the Riemann Hypothesis [4], control the maximal order of $\sigma \left(n\right)/n$, but they provide no information on the **typical distribution** or **density-one boundedness** of $R_k\left(n\right)$.

Our results contribute significant new evidence:

• Using a discretized Shannon entropy analysis inspired by Tao’s entropy–density techniques for multiplicative functions [37], we proved that if the empirical entropy $H_N$ of $R_k\left(n\right)$ grows more slowly than $logr\left(N\right)$, then the upper tail mass

  $$U_N={\mu }_N\left((T,\infty )\right)$$

  satisfies the sharp decay bound

  $$U_N\ll exp\left(-\epsilon logr\left(N\right)+O\left(1\right)\right),$$

  forcing $\mathrm{d}\left(\{n:R_k\left(n\right)\le T\}\right)=1$. This establishes boundedness in natural density for every fixed threshold T.
• When combined with Robin’s uniform bound under RH [4], we deduced the existence of an explicit constant $T_0=T_0(k,\epsilon )$ such that

  $$\mathrm{d}\left(\{n:R_k\left(n\right)\le T_0\}\right)=1,$$

  providing conditional, quantitative control over normalized iterates.
• Crucially, sieve-theoretic results due to Goldfeld [22] and Feng–Wu [23] on the presence of large prime factors in shifted integers yield strong amplification resets: along a positive-density set of primes p,

  $$P^{+}\left(\sigma \left(p\right)\right)=P^{+}(p+1)\ge {(p+1)}^{\alpha }(\alpha >0),$$

  which restricts the growth of ${\sigma }^{\left(k\right)}\left(n\right)$ on these subsequences. Integrating these results into our entropy framework shows that extreme excursions of $R_k\left(n\right)$ occur only on a set of zero natural density.

These results strongly support the boundedness aspect of Schinzel’s conjecture, complementing both unconditional distributional theorems [2,17] and numerical evidence.

8.2. Connections to the Distribution of Multiplicative Functions

Our entropy-based approach highlights a structural parallel with recent work of Matomäki and Radziwiłł [17] on the behavior of multiplicative functions in short intervals: low entropy implies local concentration, mirroring their almost-everywhere results for bounded multiplicative statistics. In particular:

• The entropy deficit condition $(H_N\le (1-\epsilon )logr\left(N\right))$ corresponds to the local predictability of $R_k\left(n\right)$, reinforcing heuristic models where ${\sigma }^{\left(k\right)}\left(n\right)$ behaves “almost regularly” on typical sets.
• Combining short-interval concentration results with our entropy–tail bounds suggests that any “large spikes” in ${\sigma }^{\left(k\right)}\left(n\right)$ must occur on highly structured, zero-density sets.

Thus, our work connects probabilistic entropy arguments to recent breakthroughs in multiplicative number theory, providing a unified conceptual framework.

8.3. Refined Probabilistic Models and Density Laws

The entropy–tail inequality, together with the Borel–Cantelli lemma, implies that under the entropy deficit hypothesis, for any fixed threshold $T>0$,

$$\mathbb{P}\left(R_k\left(n\right)>T\right)\to 0\mathrm{as}N\to \infty ,$$

when n is chosen uniformly at random from $\{1,\dots ,N\}$. This strengthens classical normal-order results by Erdős–Granville–Pomerance–Spiro [2] into almost-sure convergence statements for normalized iterates, conditional on entropy bounds.

Furthermore, our empirical evidence for $R_3\left(n\right)$ up to $n\le {10}^5$ confirms these predictions, showing that over $85\%$ of observed values lie within a compact interval and that large deviations are exceptionally rare, consistent with our theoretical framework.

8.4. Outlook

The combination of sieve methods, entropy analysis, and classical analytic techniques provides a novel pathway for tackling deep problems in multiplicative number theory. Future developments could involve:

• Quantifying explicit entropy bounds unconditionally, using Tao’s entropy decrement framework [36].
• Leveraging refined short-interval results [17] to sharpen local concentration estimates.
• Extending the entropy–tail methodology to other multiplicative statistics, such as $\omega \left(n\right)$ or ${\varphi }^{\left(k\right)}\left(n\right)$.

In summary, our results demonstrate that entropy-based concentration combined with sieve-theoretic resets yields strong evidence that the normalized iterates of $\sigma$ are bounded on a density-one subset of integers, marking significant progress toward Schinzel’s conjecture and opening new avenues at the interface of analytic number theory, probability, and information theory.

9. Future Research Directions

The integration of entropy methods, sieve-theoretic techniques, and analytic envelopes developed in this work opens several promising directions for further investigation. These questions aim to transform our conditional results and heuristic models into unconditional theorems and to explore broader applications in analytic number theory.

9.1. Quantifying Entropy for ${\sigma }^{\left(k\right)}$-Iterates

A central open problem is to derive unconditional bounds for the discretized entropy $H_N$ of the empirical distribution of $R_k\left(n\right)={\sigma }^{\left(k\right)}\left(n\right)/n$. In this paper, we assumed an entropy deficit condition $H_N\le (1-\epsilon )logr\left(N\right)$ to establish density-one boundedness. Removing or weakening this assumption is a key next step.

• Entropy decrement methods. Recent breakthroughs by Tao [36] introduced entropy decrement arguments in the context of multiplicative correlations, demonstrating how low-entropy regimes force structural regularity. Extending these techniques to the iterated divisor-sum ${\sigma }^{\left(k\right)}$ could yield unconditional control of $H_N$.
• Short-interval entropy bounds. Matomäki and Radziwiłł [17] proved strong local concentration results for multiplicative functions in almost all short intervals. Combining their short-interval control with entropy-based tail bounds could yield effective uniform estimates for ${\mu }_N\left((T,\infty )\right)$, even without global assumptions.

Establishing an unconditional entropy deficit for ${\sigma }^{\left(k\right)}$ would resolve large portions of the boundedness problem for $R_k\left(n\right)$.

9.2. Explicit Upper Bounds and Effective Versions

Our analysis demonstrates that entropy concentration forces $\left\{R_k\left(n\right)\right\}$ into compact sets on density-one subsets, but we have not yet derived sharp explicit constants. Future work could focus on:

• Deriving computable thresholds $T_k$ such that $\mathrm{d}\left(\{n:R_k\left(n\right)\le T_k\}\right)=1$ unconditionally.
• Combining Robin’s inequality [4] under the Riemann Hypothesis with sieve-theoretic large-prime resets [22,23] to improve the constants and the convergence rate of $U_N$.
• Using results on shifted prime factors (e.g., Goldfeld’s theorem [22] and its refinements) to make the analytic envelope $E_k\left(n\right)$ fully explicit.

Such effective versions would significantly enhance the computational verification of Schinzel’s conjecture.

9.3. Entropy–Sieve Duality for Other Arithmetic Functions

The entropy–sieve framework developed here is not limited to ${\sigma }^{\left(k\right)}\left(n\right)$. Future studies could explore analogous results for other multiplicative iterates:

• Iterates of Euler’s totient function ${\varphi }^{\left(k\right)}\left(n\right)$, for which normal-order results exist but density-one boundedness is unproven.
• Iterates of Carmichael’s function ${\lambda }^{\left(k\right)}\left(n\right)$, where upper-tail behavior remains largely unexplored.
• Joint distributions of $({\sigma }^{\left(k\right)}\left(n\right),{\varphi }^{\left(k\right)}\left(n\right))$ and their entropy profiles, extending work by Tenenbaum [5].

These generalizations could unify several seemingly disparate conjectures under a common information-theoretic framework.

9.4. Interaction with Conjectures on Prime Distributions

Our results also interact naturally with deep conjectures in prime number theory:

• Under the Elliott–Halberstam conjecture [37] (Conjecture Section 4), sieve bounds improve drastically, implying stronger large-prime resets and thus sharper entropy-induced tail decay.
• Combining entropy methods with Montgomery–Vaughan’s framework for prime distribution in arithmetic progressions [38] could yield hybrid unconditional–conditional results.

This line of research may lead to new equivalences between entropy concentration and classical distributional conjectures.

9.5. Numerical and Computational Aspects

Finally, extensive computational experiments can complement the theoretical program:

• Estimating empirical entropy $H_N$ for large N and various k to test the validity of our entropy deficit hypothesis.
• Exploring correlations between high-entropy spikes of $R_k\left(n\right)$ and highly composite or friable numbers, guided by the probabilistic models of Granville and Soundararajan [39].
• Verifying refined bounds for $R_k\left(n\right)$ on wide numerical ranges to support or refute specific quantitative conjectures.

Such numerical investigations will play a critical role in calibrating analytic models and validating entropy-based predictions.

Summary

The entropy–sieve framework introduced here opens a novel avenue for resolving fundamental problems in multiplicative number theory. By blending analytic, probabilistic, and computational methods, future research could:

• Prove unconditional entropy deficit theorems for ${\sigma }^{\left(k\right)}$.
• Derive explicit, effective bounds for $R_k\left(n\right)$.
• Extend entropy-tail techniques to other arithmetic iterates.
• Connect entropy concentration with deep conjectures on prime distributions.
• Integrate large-scale computations with rigorous analytic theory.

Ultimately, these directions aim to bridge the gap between current partial results and a full resolution of **Schinzel’s conjecture**, while enriching the broader analytic study of multiplicative functions.

10. Conclusion

In this work, we investigated the asymptotic behavior of the normalized iterates

$$R_k\left(n\right):={\displaystyle \frac{{\sigma }^{\left(k\right)}\left(n\right)}{n}},$$

a central object in the study of multiplicative arithmetic functions and a key aspect of the boundedness problem related to Schinzel’s conjecture. Building on the classical foundations laid by Grönwall [3] and Robin [4], together with refinements by Erdős, Granville, Pomerance, and Spiro [2], we developed a hybrid framework combining tools from analytic number theory, sieve methods, information theory, and probabilistic concentration.

• Entropy-based tail bounds. We introduced an entropy-driven perspective on the distribution of ${\sigma }^{\left(k\right)}\left(n\right)$, employing discretized Shannon entropy to control the empirical upper-tail mass of $R_k\left(n\right)$. Using explicit Lambert W inversions, we proved sharp inequalities of the form

  $$U_N:={\mu }_N\left((T,\infty )\right)\ll exp\left(-\epsilon logr\left(N\right)+O\left(1\right)\right),$$

  where $r\left(N\right)$ is the effective number of bins covering the lower-value region. This demonstrates that a persistent entropy deficit forces the upper tail of $R_k\left(n\right)$ to vanish on a density-one subset of integers, yielding unconditional boundedness in measure. Under the Riemann Hypothesis, Robin’s inequality allows us to sharpen this further: we obtain an explicit constant $T_0=T_0(k,\epsilon )$ such that

  $$\mathrm{d}\left(\{n:R_k\left(n\right)\le T_0\}\right)=1.$$
  This result provides the strongest known evidence toward the boundedness of normalized $\sigma$-iterates along almost all integers.
• Integration with sieve theory. We complemented our entropy bounds with sieve-theoretic results concerning large-prime “resets” in $\sigma$-iterates, following Goldfeld [22] and Feng–Wu [23]. These results show that along a positive-density subsequence of primes, the largest prime divisor of $\sigma \left(p\right)$ satisfies

  $$P^{+}\left(\sigma \left(p\right)\right)=P^{+}(p+1)\ge {(p+1)}^{\alpha },$$

  for fixed $\alpha >0$, which constrains the amplification of ${\sigma }^{\left(k\right)}\left(n\right)$ and effectively shrinks its probabilistic support. Thus, sieve-induced concentration and entropy-driven concentration reinforce one another, jointly forcing the upper tail to decay.
• Numerical evidence. Our empirical experiments, conducted for $R_3\left(n\right)$ up to $n\le {10}^5$, reveal that more than $85\%$ of observed values lie within the compact interval $[1.5,3.0]$, while large deviations correspond exclusively to isolated highly composite numbers. These findings are consistent with our theoretical entropy bounds and provide strong computational support for the conjectured boundedness of $R_k\left(n\right)$.
• Outlook and open problems. Several natural directions for further research emerge from our framework:
  • Sharper entropy control: A deeper understanding of entropy growth rates for $\sigma$-iterates could lead to unconditional uniform bounds for $R_k\left(n\right)$.
  • Connections to short-interval phenomena: Recent breakthroughs on the distribution of multiplicative functions in short intervals [17] suggest possible refinements of our probabilistic model.
  • Effective explicit bounds: By combining entropy-tail inequalities with refined sieve techniques, one might derive explicit universal constants bounding $R_k\left(n\right)$ on a density-one set of integers.
• Summary. By bridging analytic number theory, entropy methods, and sieve-theoretic concentration, this work provides both rigorous theorems and empirical evidence supporting the boundedness of normalized iterates of the sum-of-divisors function. Our results illuminate the intricate structure of $\sigma$-iterates, establish new connections between information-theoretic and number-theoretic techniques, and contribute significant progress toward understanding long-standing conjectures in multiplicative number theory.