Typical Size of the Erdős–Hooley Function

Definitions and mathematical statement

For a positive integer $n$, a divisor means a positive integer $d$ satisfying $d\mid n$, and $\log$ denotes the natural logarithm. The supplied record asks for the typical asymptotic size of \[ \Delta(n)=\max_i\#\{d\in{\mathbb{N}}:d\mid n,\ e^i\leq d\leq e^{i+1}\}. \] Both endpoints are included, as in the source record. The range of $i$ is not specified. To expose this unresolved parameter, define, for $I\in\{{\mathbb{Z}},{\mathbb{R}}\}$, \[ \Delta_I(n)=\max_{i\in I}\#\{d\in{\mathbb{N}}:d\mid n,\ e^i\leq d\leq e^{i+1}\}. \] Here $I={\mathbb{Z}}$ uses fixed intervals, whereas $I={\mathbb{R}}$ allows the interval to slide. These are distinct definitions; the supplied record does not determine which is intended.

A set $A\subseteq{\mathbb{N}}$ has natural density one if \[ \lim_{x\to\infty}\frac{\#\{n\in A:n\leq x\}}{\lfloor x\rfloor}=1. \] This gives a conventional interpretation of "almost all integers." The record does not explicitly specify the probability or density convention underlying "typical."

Question, subject to these qualifications. Determine the asymptotic size of the source's $\Delta(n)$ for a typical positive integer $n$. The supplied curation additionally reports an almost-all lower bound \[ \Delta(n)\gg(\log\log n)^{\eta+o(1)}, \qquad \eta\approx0.353, \] and asks whether its exponent is optimal. No exact definition of $\eta$, range of the maximizing index, or precise quantification of the error term is supplied.

Under a natural-density interpretation, one possible precise reading of the reported lower bound is: for every $\varepsilon>0$, there exists $c_\varepsilon>0$ such that \[ \lim_{x\to\infty}\frac{1}{\lfloor x\rfloor} \#\left\{n\in{\mathbb{N}}:3\leq n\leq x,\ \Delta_I(n)\geq c_\varepsilon(\log\log n)^{\eta-\varepsilon}\right\}=1. \] This is an interpretation of the supplied claim, not an independently verified theorem. Under that interpretation, optimality could mean that no larger constant exponent satisfies the same almost-all lower-bound property. The stronger assertion that, for every $\varepsilon>0$, \[ \lim_{x\to\infty}\frac{1}{\lfloor x\rfloor} \#\left\{n\in{\mathbb{N}}:3\leq n\leq x,\ (\log\log n)^{\eta-\varepsilon}\leq\Delta_I(n) \leq(\log\log n)^{\eta+\varepsilon}\right\}=1 \] would specify a typical exponent. The record does not distinguish these targets. The intended index domain, exact exponent, and meaning of optimality require clarification before a unique fully quantified formulation can be fixed.

Short English statement

The problem asks how many positive divisors of a typical integer can lie in a multiplicative interval of ratio $e$, and whether the supplied almost-all lower-bound exponent $\eta\approx0.353$ is optimal. The interval endpoints are inclusive. The maximizing index domain, exact value of $\eta$, and intended meaning of optimality remain unresolved.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30004290$, title "Typical Size of the Erdős--Hooley Function," exact source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. The record lists proposed year 2019 and supplies no individual proposer.
• Supplied report citation: Analytic Number Theory (2020), Oberwolfach Reports, pp. 3141--3205, DOI 10.4171/owr/2019/50. Original report link: https://doi.org/10.4171/owr/2019/50. The supplied background describes Oberwolfach Reports as documenting mathematical workshops and their open-problem discussions.
• Supplied literature source: Ford--Green--Koukoulopoulos, "equal sums and divisor concentration" (descriptive label supplied by the registry): https://arxiv.org/abs/1908.00378.
• Supplied literature source: Koukoulopoulos--Tao, "upper mean-value bound" (descriptive label supplied by the registry): https://doi.org/10.1112/plms.12572.
• Literature assessment dated exactly 2026-08-22: the supplied curation labels the problem partially_solved and reports substantial progress on the general distribution question, sharp-looking lower bounds and extensive structural analysis of the Erdős--Hooley function, while stating that no complete description of the divisors of a typical integer is known. This is supplied dated curation derived from the OWR record, not an independently verified assessment; its time-sensitive status and cited claims require independent verification.
• The registry labels the statement corrected_verified and reports that the official Oberwolfach report was checked, the question was isolated from surrounding exposition, and notation was repaired. These are supplied provenance claims; no source was independently consulted for this draft.