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On the distribution of $\phi(\psi(n))$ and $\psi(\psi(n))$

This paper investigates the distribution of the composite arithmetic functions $\phi(\psi(n))$ and $\psi(\psi(n))$ , providing quantitative bounds for the exceptional set of the former and proving that the latter has asymptotic density zero for any fixed positive constant.

Original authors: Aimin Guo

Published 2026-08-05

📖 4 min read🧠 Deep dive

CC BY 4.0

Original authors: Aimin Guo

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine a vast, invisible city where every building is a number, and the streets are paved with the rules of multiplication. In this city, mathematicians are like urban planners who study how these buildings interact when they are stacked or combined. Two famous "architects" in this city are Euler's totient function and Dedekind's arithmetic function. Think of them as special machines that take a number, look at its prime building blocks (the fundamental atoms of math), and spit out a new number based on a specific recipe. Euler's machine usually shrinks a number by removing its prime factors, while Dedekind's machine tends to expand it by adding a little extra weight to those factors.

For a long time, mathematicians have been fascinated by what happens when you run a number through one of these machines and then immediately feed the result into another. It's like taking a photo, running it through a filter, and then running that filtered photo through a second, different filter. The question is: do these double-processed numbers behave predictably, or do they go wild? Do they stay close to their original size, or do they shoot off to infinity or shrink to nothing? Understanding this helps us map the hidden landscape of numbers, revealing patterns that might otherwise stay invisible. It's not just about abstract puzzles; it's about understanding the fundamental rhythm of how numbers are built and how they change when we twist them.

In this new study, the author, Aimin Guo, dives deep into the behavior of two specific "double-filter" combinations: taking a number, running it through Dedekind's machine, and then feeding that result into Euler's machine ( $\phi(\psi(n))$ ), and the even more complex case of running a number through Dedekind's machine twice in a row ( $\psi(\psi(n))$ ). The paper tackles a question that was only vaguely answered before: just how rare are the numbers that break the expected rules?

Previously, researchers knew that for most numbers, the result of $\phi(\psi(n))$ is smaller than the original number, and the result of $\psi(\psi(n))$ is larger. But they didn't have a precise count of the "outliers"—the few numbers that disobey these trends. Guo's work provides a much sharper, quantitative map of these outliers. The paper proves that the set of numbers where $\phi(\psi(n))$ is unusually large (specifically, larger than a fixed fraction of the original number) is incredibly small. In fact, the author calculates an explicit upper bound for how many such "rebellious" numbers exist up to any given point $x$ . The formula shows that as you look at larger and larger ranges of numbers, the proportion of these outliers shrinks to almost nothing, vanishing faster than you might expect.

Furthermore, the paper investigates the second combination, $\psi(\psi(n))$ . It confirms that for any fixed small number $c$ , the set of integers $n$ where the double-Dedekind result is surprisingly small (less than $c$ times $n$ ) is so sparse that it effectively disappears in the grand scheme of things. The paper proves that the "density" of these exceptions is zero. This means that if you were to pick a number at random from a very large list, the chance of it being one of these rare exceptions is practically zero. The author also extends this finding to show that if you run a number through Dedekind's machine two or more times (for any fixed $k \ge 2$ ), the result will almost certainly be larger than any fixed fraction of the original number.

The study doesn't just say these exceptions are rare; it uses a mathematical technique called "sieve theory"—which is like using a fine mesh to filter out unwanted grains of sand—to count exactly how many grains remain. The author adapts methods used by other mathematicians to handle similar problems, refining the estimates to be much more precise than before. While the paper confirms that these weird behaviors are vanishingly rare, it also notes that finding the exact "normal" size for these functions and getting even tighter bounds on the exceptions remains a challenging open problem for future explorers. The work stands as a solid proof of just how orderly these chaotic-looking number combinations really are.

Technical Summary: On the Distribution of $\phi(\psi(n))$ and $\psi(\psi(n))$

Problem Statement
This paper investigates the distributional properties of compositions involving Dedekind's arithmetic function $\psi(n)$ and Euler's totient function $\phi(n)$ . Specifically, the author analyzes the frequency of integers $n \le x$ for which the composite functions $\phi(\psi(n))$ and $\psi(\psi(n))$ deviate from expected linear growth.

While $\phi(n)$ and $\psi(n)$ are multiplicative, their compositions generally lose this property, rendering classical multiplicative function theory inapplicable. Previous work by Sándor [8, 9] established qualitative density results, showing that for any fixed $c > 0$ , the sets $\{n \le x : \phi(\psi(n)) \ge cn\}$ and $\{n \le x : \psi(\psi(n)) \le cn\}$ have asymptotic density zero. However, these results were qualitative; quantitative estimates for the size of these "exceptional sets" remained limited. This paper aims to provide explicit upper bounds for these sets, refining Sándor's density results.

Methodology
The author adapts sieve-theoretic methods previously utilized by Dixit and Bhattacharjee [2] in their study of $\phi(\sigma(n))$ . The core strategy involves analyzing integers $n$ for which the value of the inner function (specifically $\psi(n)$ ) omits certain small prime divisors.

Key technical components include:

Sieve Estimates: The proof relies on bounding the number of integers $n \le x$ such that a specific prime $p$ does not divide $\psi(n)$ . This is denoted as $T_p(x) = \#\{n \le x : p \nmid \psi(n)\}$ .
Lemma 2.4: A critical lemma establishes that for any prime $p$ and sufficiently large $x$ , $T_p(x) \ll x \left(\frac{\log_2 x}{\log x}\right)^{\frac{1}{p-1}}$ . This bound is derived by observing that if $q \equiv -1 \pmod p$ and $q \mid n$ , then $p \mid \psi(n)$ . Consequently, if $p \nmid \psi(n)$ , $n$ must be free of prime factors $q \equiv -1 \pmod p$ . The Siegel-Walfisz theorem is used to estimate the density of such primes.
Product of Primes: The author defines $P(z) = \prod_{p \le z} p$ . They utilize the fact that if $P(z) \mid \psi(n)$ , then $\phi(\psi(n))$ is significantly smaller than $\psi(n)$ (due to the product of $(1-1/p)$ terms), while $\psi(\psi(n))$ is significantly larger (due to the product of $(1+1/p)$ terms).
Partial Summation: The sum $\sum_{n \le x} \psi(n)/n$ is evaluated using known asymptotics for $\sum \psi(n)$ to bound the proportion of $n$ where $\psi(n)$ is "large" or "small."

Key Contributions and Results

Quantitative Bound for $\phi(\psi(n))$ (Theorem 1.1):
The paper provides an explicit upper bound for the exceptional set where $\phi(\psi(n)) \ge cn$ . For every $c > 0$ :
$\#\{n \le x : \phi(\psi(n)) \ge cn\} \le \frac{15x}{\pi^2 c \log_4 x} + O\left(\frac{x \log_3 x}{(\log x)^{1/(\log_3 x \log_4 x)}}\right)$
This refines Sándor's qualitative result by showing the exceptional set is not just $o(x)$ , but decays at a rate roughly proportional to $x/\log_4 x$ .
Refinement with Slowly Varying Thresholds (Theorem 1.2):
The author extends the result to non-decreasing functions $g(x)$ . If $g(x) = o(\log_4 x)$ , the set $G_g(x) = \{n \le x : \phi(\psi(n)) \ge n/g(n)\}$ satisfies:
$|G_g(x)| = O\left(\frac{xg(x)}{\log_4 x} + \frac{x \log_3 x}{(\log x)^{1/(\log_3 x \log_4 x)}}\right) = o(x)$
Distribution of $\psi(\psi(n))$ (Theorem 1.3):
The paper proves that for any fixed $c > 0$ , the set of integers where $\psi(\psi(n)) \le cn$ has asymptotic density zero. The proof demonstrates that if $\psi(\psi(n)) \le cn$ , then $\psi(n)$ must omit a product of small primes $P(z)$ (where $z$ depends on $c$ ). Using the sieve estimates, the size of this set is shown to be $o(x)$ .
Generalization to Higher Iterates (Corollary 1.1):
It is shown that for any fixed $k \ge 2$ , the set $\{n \le x : \psi^{(k)}(n) \le cn\}$ also has asymptotic density zero, as $\psi^{(k)}(n) \ge \psi(\psi(n))$ .

Significance and Claims
The paper provides quantitative estimates for the exceptional sets associated with the compositions $\phi(\psi(n))$ and $\psi(\psi(n))$ , refining earlier qualitative density results. By moving from qualitative density ( $o(x)$ ) to explicit upper bounds involving iterated logarithms, the work offers a more precise understanding of the distribution of these arithmetic functions.

The author notes that their quantitative bounds depend essentially on the sieve estimate for $T_p(x)$ . They suggest that any substantial improvement in this specific sieve estimate would directly lead to sharper bounds for the exceptional sets. While the paper mentions in the remarks that similar density-zero results can be derived for the composition $\psi(\sigma(n))$ , it explicitly confines its main quantitative analysis and scope to $\phi(\psi(n))$ and $\psi(\psi(n))$ . The author concludes that determining the normal orders of these compositions and obtaining even sharper estimates for the exceptional sets remain interesting open problems.

Technical Summary: On the Distribution of ϕ(ψ(n))\phi(\psi(n))ϕ(ψ(n)) and ψ(ψ(n))\psi(\psi(n))ψ(ψ(n))

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Technical Summary: On the Distribution of $\phi(\psi(n))$ and $\psi(\psi(n))$