A vanishing share of integers
Several combinations of three arithmetic functions produce exceptional sets that become negligible among the integers as the counting range grows, according to a theoretical analysis. For a fixed positive threshold, the study counts integers up to a limit x as x tends to infinity. It finds that the integers for which Euler’s totient function, applied to Dedekind’s function or to the sum-of-divisors function, produces a value at least a fixed positive multiple of the original integer number at most on the order of x divided by the square of the third iterated logarithm of x, with the implied constant allowed to depend on the threshold. In both cases, the count is negligible compared with x, meaning its share of all integers up to x tends to zero.
Here, an exceptional set is simply the collection of integers meeting one of the specified inequalities. This is not an empirical sample: the proofs range over the integers up to x and study how the counts behave as x grows without bound. In this setting, density zero means the proportion of qualifying integers among all integers up to x tends to zero. It is an asymptotic statement about the size of a set, not an exact count for a particular finite range.
The pattern repeats
A second pair of estimates turns the inequality around. For a fixed positive threshold, the paper counts integers for which two applications of Dedekind’s function, or Dedekind’s function applied after the sum-of-divisors function, produces a value no greater than the same fixed multiple of the original integer. Each exceptional set is bounded by x times a power of a logarithmic ratio, specifically the second iterated logarithm of x divided by the usual logarithm of x. The exponent is the reciprocal of a positive constant that depends only on the threshold, and both counts are negligible compared with x.
The analysis also lets the threshold grow with x. If a non-decreasing threshold function g grows more slowly than the third iterated logarithm, the associated variable-threshold set in the Dedekind-function case has size bounded by a constant multiple of one plus the square of g at the cutoff, multiplied by the cutoff and divided by the square of the third iterated logarithm. Its share among the integers still tends to zero. An analogous bound holds for a non-decreasing function h that also grows more slowly than the third iterated logarithm in the corresponding sum-of-divisors case. These results depend on the stated monotonicity and growth conditions.
The arithmetic mechanism
To obtain the estimates, the proof combines prime-product estimates, sieve estimates for integers avoiding prescribed prime divisors, small-prime divisibility results for Dedekind’s and the sum-of-divisors functions, and second-moment estimates for their normalized values. Normalization here means comparing a function’s value with the input integer. The argument therefore tracks both the prime factors appearing in the inner values and the average size of the normalized functions.
The authors describe a common arithmetic mechanism behind the different cases: small-prime divisibility in an inner value suppresses the normalized totient value while enhancing the normalized Dedekind value. That shared pattern appears across the large-value conditions involving the outer totient and the small-value conditions involving the outer Dedekind function, linking the different exceptional sets in the authors’ interpretation.
What the bounds leave open
A further result extends the density-zero conclusion to repeated use of Dedekind’s function. For every fixed number of iterations of at least 2 and every fixed positive threshold, the integers for which the resulting iterate is no more than that threshold times the original integer form a density-zero set. This remains an asymptotic conclusion, not an exact asymptotic formula or a pointwise statement about every integer.
The bounds come with important qualifications. They are upper bounds, not matching lower bounds or exact asymptotics, and the authors do not claim that the main order, x divided by the square of the third iterated logarithm, is optimal. Within the present approach, replacing the second moment with a higher fixed moment would not by itself improve that final order. The manuscript is a preprint, listed as arXiv:2608.19972v2 in math.NT and dated 6 September 2026.
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
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