Squarefree-Power-Like Functions

• Squarefree-power-like functions are arithmetic and algebraic constructions that modify traditional prime exponent treatments by focusing solely on squarefree support.
• They bridge analytic number theory and combinatorial commutative algebra, exemplified by the normalized Dedekind psi function and hypergraph monomial ideals.
• Their applications span prime product asymptotics, graph matching invariants, and Hilbert function analysis, impacting regularity and additive representations.

Squarefree-power-like functions comprise a family of arithmetic and algebraic constructions in which higher prime exponents are either ignored, excluded, or replaced by squarefree-power operations. The collected works suggest two closely related usages. In analytic number theory, the model example is the normalized Dedekind psi function

$$\frac{\psi(n)}{n}=\prod_{p\mid n}\left(1+\frac1p\right),$$

whose value depends only on the support of the prime divisors of $n$ and not on their multiplicities (Carella, 2010). In combinatorial commutative algebra, the notion is formalized by an abstract function $\mathtt F(H,k)$ on hypergraphs, designed to unify the sequences of squarefree ordinary powers $I(H)^{[k]}$ and squarefree symbolic powers $I(H)^{\{k\}}$ (Chau et al., 1 Oct 2025). Around these two poles lie squarefree indicators such as $\mu^2$, $s$-free counting functions, squarefree kernels $k(m)$, squarefree powers of graph ideals, and squarefree-level $\eta$-quotients, all governed by local prime-support or squarefree combinatorics (Lowry-Duda, 14 Feb 2025).

1. Definitional patterns

Several recurrent definitions organize the subject. A first pattern consists of multiplicative functions determined only by the set of prime divisors. For the Dedekind psi function,

$\psi(n)=n\prod_{p\mid n}\left(1+\frac1p\right),$

the normalized quantity $n$0 is unchanged when $n$1 is replaced by its squarefree kernel, and on squarefree integers one has the identity $n$2 (Carella, 2010). This is the basic “support-only” prototype.

A second pattern consists of indicators defined by exclusion of prime powers. The squarefree indicator is

$n$3

and more generally the $n$4-free indicator is

$n$5

so squarefreeness is the special case $n$6 (Lowry-Duda, 14 Feb 2025, Brandes, 2013). In the same direction, the largest squarefree divisor of an integer,

$n$7

leads to the powered-number set

$n$8

which measures how small the squarefree part of $n$9 is relative to $\mathtt F(H,k)$0 itself (Brüdern et al., 2023).

A third pattern is algebraic. A squarefree-power-like function in the formal sense is a map $\mathtt F(H,k)$1 from hypergraphs and nonnegative integers to squarefree monomial ideals satisfying

$\mathtt F(H,k)$2

the squarefree derivative condition

$\mathtt F(H,k)$3

restriction to induced subhypergraphs, and the mixed-sum rule

$\mathtt F(H,k)$4

for disjoint unions (Chau et al., 1 Oct 2025). The paper establishing this definition proves that both $\mathtt F(H,k)$5 and $\mathtt F(H,k)$6 fit this framework.

2. Support-type arithmetic functions

The analytic prototype is Carella’s study of the Dedekind psi function. Because

$\mathtt F(H,k)$7

each prime contributes once, exponents are irrelevant, and squarefree integers become the natural arena. The paper proves that the extremal sequence is the primorial sequence

$\mathtt F(H,k)$8

and establishes, for all sufficiently large $\mathtt F(H,k)$9,

$I(H)^{[k]}$0

It also derives

$I(H)^{[k]}$1

from the prime product asymptotic

$I(H)^{[k]}$2

showing that prime-support products of the form $I(H)^{[k]}$3 are naturally optimized on primorials when $I(H)^{[k]}$4 decreases with $I(H)^{[k]}$5 (Carella, 2010).

A complementary class is formed by multiplicative functions supported on the squarefree integers and taking values $I(H)^{[k]}$6 on primes. For such an $I(H)^{[k]}$7,

$I(H)^{[k]}$8

and explicit constructions of the form $I(H)^{[k]}$9, where $I(H)^{\{k\}}$0 is a completely multiplicative extension of a real non-principal Dirichlet character, yield summatory functions

$I(H)^{\{k\}}$1

with strong cancellation. In particular, the paper gives deterministic examples with $I(H)^{\{k\}}$2, and under RH examples with

$I(H)^{\{k\}}$3

for any $I(H)^{\{k\}}$4 (Aymone, 2019). This places squarefree-supported sign patterns into the same Euler-product regime as $I(H)^{\{k\}}$5, but with analytically tunable prime signs.

A related counting theory arises from the squarefree kernel. For

$I(H)^{\{k\}}$6

the asymptotic formula

$I(H)^{\{k\}}$7

shows that the set of integers with unusually small squarefree kernel is only slightly denser than a pure power law $I(H)^{\{k\}}$8 (Brüdern et al., 2023). This suggests a bridge between exact powers and multiplicatively smoothed “powered numbers”.

3. Distribution, values, and additive representations

The squarefree indicator and its higher-power analogues appear in several distribution problems. For $I(H)^{\{k\}}$9-free numbers, Brandes proves

$\mu^2$0

generalizing Heath-Brown’s square-sieve method to arbitrary powers $\mu^2$1 and improving the classical error term for consecutive $\mu^2$2-free integers (Brandes, 2013).

For arithmetic progressions, squarefree numbers are shown to equidistribute to smooth moduli far beyond earlier ranges. If $\mu^2$3 is squarefree, $\mu^2$4-smooth, and

$\mu^2$5

then the paper proves an asymptotic formula with power-saving error for

$\mu^2$6

thereby pushing squarefree distribution past the $\mu^2$7-barrier for a positive-density family of moduli (Mangerel, 2020).

Squarefree-value problems for sparse polynomials fit the same pattern. For fixed nonzero integers $\mu^2$8 with $\mu^2$9 squarefree,

$s$0

satisfies

$s$1

where

$s$2

This gives the conjectural local-density constant together with a power-saving error term for the sparse family $s$3 (Sanjaya et al., 2021).

Additive representation by a squarefree number and a power of two yields another squarefree-power-like problem. For odd integers, the paper verifies computationally that every

$s$4

can be represented as

$s$5

with $s$6 squarefree and in fact $s$7, extending earlier verification of Erdős’s conjecture by more than $s$8 in range (Hercher, 2024).

A more classical squarefree product phenomenon occurs in Faulhaber denominators. If $s$9 is the denominator of the power-sum polynomial and $k(m)$0, then

$k(m)$1

so $k(m)$2 is squarefree and the full denominator has the form

$k(m)$3

The same paper derives squarefree product formulas for Bernoulli polynomial denominators (Kellner et al., 2017).

4. Abstract squarefree powers and admissible sets

The formal theory developed for hypergraphs packages squarefree ordinary and symbolic powers into a single algebraic object. For a squarefree-power-like function $k(m)$4, the squarefree derivative condition forces eventual vanishing, so there is a largest $k(m)$5 with $k(m)$6; this is the $k(m)$7-number $k(m)$8. The framework is rigid on complete intersections: if $k(m)$9 is a disjoint union of edges, then every squarefree-power-like function satisfies

$\eta$0

It is also additive on disjoint unions: $\eta$1 (Chau et al., 1 Oct 2025).

The central combinatorial invariant is the $\eta$2-admissible $\eta$3-set. If

$\eta$4

is such a set, then each part $\eta$5 contains an edge, the induced hypergraph $\eta$6 is the disjoint union of the $\eta$7, one has the budget condition

$\eta$8

and the top piece on each part is principal: $\eta$9 The associated invariant

$\psi(n)=n\prod_{p\mid n}\left(1+\frac1p\right),$0

gives the universal regularity lower bound

$\psi(n)=n\prod_{p\mid n}\left(1+\frac1p\right),$1

and this bound is sharp when $\psi(n)=n\prod_{p\mid n}\left(1+\frac1p\right),$2 is a complete intersection (Chau et al., 1 Oct 2025).

Specializing to squarefree symbolic powers, the same framework yields the first general combinatorial lower bound

$\psi(n)=n\prod_{p\mid n}\left(1+\frac1p\right),$3

where $\psi(n)=n\prod_{p\mid n}\left(1+\frac1p\right),$4 is the $\psi(n)=n\prod_{p\mid n}\left(1+\frac1p\right),$5-admissible independence number. In two notable classes this becomes exact: for every block graph $\psi(n)=n\prod_{p\mid n}\left(1+\frac1p\right),$6,

$\psi(n)=n\prod_{p\mid n}\left(1+\frac1p\right),$7

for all squarefree symbolic powers, and for every Cohen–Macaulay chordal graph $\psi(n)=n\prod_{p\mid n}\left(1+\frac1p\right),$8 with $\psi(n)=n\prod_{p\mid n}\left(1+\frac1p\right),$9,

$n$00

(Chau et al., 1 Oct 2025).

5. Edge ideals, graph powers, and homological thresholds

For edge ideals, squarefree powers are controlled exactly by graph matchings: $n$01 This gives immediate nilpotence at $n$02, and it turns regularity into a matching-theoretic problem. The paper proves the lower bound

$n$03

establishes

$n$04

and shows that the top squarefree power $n$05 has linear quotients (Erey et al., 2019).

The first-syzygy threshold has a sharp combinatorial description. For $n$06,

$n$07

if and only if there is no induced subgraph $n$08 on $n$09 vertices such that $n$10 is disconnected and $n$11. In the equivalent language of a $n$12-dimensional flag simplicial complex $n$13, the obstruction becomes the presence of an induced complete bipartite graph

$n$14

with even $n$15, and linear resolution further requires the absence of the crown graph $n$16 (Navarra et al., 1 Jul 2026). The same paper gives the explicit counting formula

$n$17

so failure of linear relatedness is measured exactly by induced complete bipartite obstructions.

Closed neighborhood ideals exhibit a different squarefree-power behavior. For a tree $n$18, the highest non-vanishing squarefree power

$n$19

is componentwise linear if and only if $n$20 satisfies the two forbidden-configuration conditions (C1) and (C2). When either fails, the paper constructs disconnected generator graphs and proves non-componentwise-linearity; when both hold, it proves linear quotients for the top squarefree power (Nambi et al., 16 Mar 2026). The same paper shows that

$n$21

can be any prescribed positive integer on suitable trees, while for caterpillar graphs one has the exact formula

$n$22

(Nambi et al., 16 Mar 2026).

Depth supplies another homological normalization. For a squarefree monomial ideal $n$23, the normalized depth function

$n$24

satisfies $n$25, and the paper conjectures that $n$26 is nonincreasing. In the graph case, if $n$27 is disconnected then

$n$28

for all $n$29, and for every graph without isolated vertices one always has

$n$30

(Erey et al., 2022).

6. Extended algebraic and computational frameworks

Squarefree-power-like behavior also appears in asymptotic regularity of ordinary powers. For a two-dimensional squarefree monomial ideal $n$31, the $n$32-invariants of $n$33 for $n$34 and the geometric regularity are linear functions of $n$35 from $n$36, and one has the exact bridge

$n$37

(Lu, 2018). This identifies a setting where ordinary powers of squarefree ideals already display strongly controlled “power-like” behavior.

At the level of numerical invariants, the cone of Hilbert functions of squarefree modules is simplicial, with extremal rays

$n$38

For squarefree modules generated in degree zero, the cone coincides with the cone generated by Stanley–Reisner rings, and its defining linear inequalities can be compared asymptotically with the nonlinear Kruskal–Katona bound (Bertone et al., 2012). This recasts squarefree Hilbert functions as support-controlled growth laws.

Squarefree indexing also governs a modular-form problem. For squarefree level $n$39, $n$40-quotients

$n$41

are indexed by squarefree divisors, the cusps are indexed by the same divisor set, and the cusp-order matrix

$n$42

has the property that every divisor appears exactly once in each row and column. The paper proves a broad existence theorem based on the condition

$n$43

together with a weight-$n$44 obstruction on an explicit exceptional set $n$45 (Allen, 2019). This suggests a squarefree divisor-lattice version of “power-like” exponent data.

A computational perspective comes from machine learning. Small transformer models trained on CRT encodings of integers achieve nontrivial performance on $n$46, but the paper shows that they are not reconstructing $n$47 or testing $n$48 directly. Instead, they exploit conditional squarefree probabilities attached to small-prime divisibility patterns; indeed, using only

$n$49

already gives $n$50 accuracy for $n$51, close to the $n$52 accuracy of the full 100-prime CRT input (Lowry-Duda, 14 Feb 2025). This suggests that even in learned approximations, squarefree-power-like behavior is dominated by local multiplicative biases rather than full arithmetic reconstruction.

Taken together, these results suggest that squarefree-power-like phenomena are governed by a common principle: arithmetic or algebraic data are organized by squarefree support, by exclusion of higher prime powers, or by squarefree power operations, and the resulting extremal, homological, and distributional behavior is then controlled by prime products, matching and cover invariants, or squarefree divisor lattices.