A new mathematics preprint puts quantitative limits on several exceptional patterns produced by composing Dedekind’s function ψ, Euler’s totient function ϕ and the sum-of-divisors function σ. The paper counts integers n up to a cutoff x that meet inequalities such as ϕ(ψ(n)) ≥ cn or ψ(ψ(n)) ≤ cn, with c fixed and positive.
This is a theoretical result, not an analysis of observed data. Its basic question is how large these sets of integers can be as x grows without bound. Here, an exceptional set is simply the collection of n≤x satisfying one of the paper’s composition inequalities.
The main numerical bounds
The first main result concerns the set of integers for which ϕ(ψ(n)) is at least cn. For every fixed c>0, the paper gives the upper bound (15/(π²c) + o(1)) x/log₃ x. In the paper’s notation, log₃ x is the third iterated logarithm, while o(1) marks a term that tends to zero in the asymptotic limit.
A parallel estimate applies when σ replaces ψ inside the totient. The count of n≤x with ϕ(σ(n))≥cn is bounded by (π²/(6c) + o(1)) x/log₃ x. The paper says this improves the order of the earlier upper bound from x/log₄ x to x/log₃ x. The stated gain concerns the scale of the bound as x becomes large, rather than an exact count.
The authors describe the first estimate as a quantitative refinement of a density result by Sándor and the second as an improvement on work by Dixit and Bhattacharjee. They also explicitly say the main upper bounds are not claimed to be optimal, leaving their sharpness unresolved.
What the formulas measure
The formulas are asymptotic, which means they describe behavior in the limit x→∞ rather than promising a fixed result for a particular finite cutoff. The o(1) term is part of that statement: it becomes negligible only in the limit. Readers should therefore treat x/log₃ x as a long-run upper-bound scale, not as a ready-made percentage for a finite list of integers.
The paper then lets the threshold vary. For non-decreasing functions g and h that grow more slowly than log₃ x—written g(x)=o(log₃ x) and h(x)=o(log₃ x)—the corresponding sets G_g(x) and H_h(x) are each o(x). Their displayed bounds scale as g(x)x/log₃ x and h(x)x/log₃ x. Saying a set is o(x) means its count divided by x tends to zero as x grows.
This variable-threshold result has a clear condition attached: g and h must be non-decreasing, and the growth restrictions must hold. The conclusion is not stated outside those assumptions. It extends the fixed-threshold picture without claiming that arbitrary threshold functions behave in the same way.
A wider family of exceptions
The other main estimates turn to outcomes that fall at or below the threshold after composition. For each fixed c>0, the set with ψ(ψ(n))≤cn is bounded by a c-dependent multiple of x times (log₂ x/log x)^(1/z_c), and is o(x). The same form of bound holds for the set with ψ(σ(n))≤cn. Because both are o(x), the paper describes these exceptional sets as having density zero.
The symbol z_c is specified only as a positive constant that may depend on c; no numerical value is given in the supplied analysis. That matters for interpretation: the result gives a quantitative rate, but the rate is c-dependent rather than a single universal numerical percentage. The conclusion remains asymptotic, with x tending to infinity.
A corollary broadens the repeated-function statement. For every fixed k≥2 and c>0, the set of n≤x satisfying ψ^(k)(n)≤cn has density zero. Here ψ^(k) means applying ψ k times, so the statement covers every fixed iterate order starting at the second, although the supplied result does not give a sharper general rate for all such k.
A proof-driven study
Behind the conclusions is a chain of analytic estimates rather than a data set. The proofs combine products over primes, sieve estimates for integers that avoid prescribed prime divisors, and results about small prime divisors of ψ(n) and σ(n).
One key step is an estimate for T_p(x), the number of integers n≤x for which p does not divide ψ(n). The analysis identifies that estimate as the key ingredient in proving Theorem 1.3. This sieve perspective connects the final exceptional-set bounds to restrictions on which primes can divide the intermediate values.
The paper does not use a sampled population: its objects are integers n≤x and the inequalities that select them. The result is therefore a mathematical statement about counts and limiting behavior, not empirical evidence in humans, animals or laboratory systems.
What remains open
The main caution is straightforward: the upper bounds in Theorem 1.1 are not claimed to be optimal. The estimates are asymptotic, so they do not establish how well the formulas perform over any particular finite range of x. The variable-threshold conclusions likewise depend on their monotonicity and growth conditions.
The document is an arXiv version 1 preprint dated 20 Aug 2026. Its contribution is a set of upper bounds and density-zero conclusions for specified exceptional sets, with the main rates explicitly left open to possible sharpening.
Paper data and sources
Original title: Exceptional sets for compositions involving Euler's function,the divisor-sum function and Dedekind's function
Authors: Aimin Guo
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-20
DOI: Not available
Original paper · Full text