Squarefree Kernel Counts Outside the Established Range

Definitions and mathematical statement

Under the usual interpretation of squarefree kernel as the product of the distinct prime divisors, put \[ k(n)=\prod_{p\mid n}p\quad(n\in{\mathbb{Z}}_{\geq1}),\qquad k(1)=1, \] where the product is over primes. Define \[ N(x,y)=\#\{n\in{\mathbb{Z}}_{\geq1}:n\leq x,\ k(n)\leq y\}. \] The positive-integer counting domain and the product-of-distinct-primes convention make explicit the customary interpretation; the supplied record does not expressly define either convention.

For sufficiently large real $x$, write the source's region as \[ H_{\varepsilon}=\left\{(x,y):\exp\left((\log\log x)^{\varepsilon}\right)\leq y\leq x\right\}, \] where $\log$ denotes the natural logarithm. The source asks for the asymptotic behavior of $N(x,y)$ outside $H_{\varepsilon}$, apart from a known small-$y$ range.

This is an incompletely specified asymptotic request. The record supplies neither the admissible values and quantification of $\varepsilon$, nor the ambient parameter domain whose complement is intended, nor the boundaries of the excluded small-$y$ range. It also leaves unspecified how $y$ may depend on $x$, the required uniformity, and the intended form of an asymptotic estimate. Consequently, the displayed request cannot be made into a uniquely quantified asymptotic assertion from this record alone. The descriptions of established and known ranges are supplied source qualifications, not independently verified results.

Short English statement

Count positive integers up to $x$ whose product of distinct prime divisors is at most $y$. The source asks for asymptotics beyond the displayed established region, excluding a known small-$y$ range. The omitted parameter conditions and exclusion boundaries require clarification before a precise asymptotic target can be stated.

Sources

• UnsolvedMath local registry, supplied record with permanent ID $30002432$, title "Squarefree Kernel Counts Outside the Established Range," exact source code catalogue entry; collection: Oberwolfach Reports Open Problems. No individual proposer is supplied; the proposed year is recorded as 2013.
• Supplied report citation: Analytic Number Theory (2014), Oberwolfach Reports, pp. 2963--3037, DOI 10.4171/owr/2013/51. Supplied source URL: https://doi.org/10.4171/owr/2013/51. The literature-sources entry labeled "original OWR report" supplies the same link: https://doi.org/10.4171/owr/2013/51. No more specific problem location is supplied.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem open and states that the checked literature provides no general asymptotic outside the established domain. This is supplied dated curation, not independently verified literature assessment. The background expressly requests independent verification of time-sensitive status and cited claims.
• The registry labels the statement corrected_verified and says that notation, typography, and wording were normalized without changing the mathematical content. This records the supplied verification claim; it does not constitute mathematical review of this draft.