--- title: Squarefree-Power-Like Functions url: https://www.emergentmind.com/topics/squarefree-power-like-functions type: topic --- # Squarefree-Power-Like Functions 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 [1012.4817]. 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 \(I(H)^{\{k\}}\) [2510.01366]. 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 [2502.10335]. ## 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 \(\psi(n)/n\) is unchanged when \(n\) is replaced by its squarefree kernel, and on squarefree integers one has the identity \(\psi(n)=\sigma(n)\) [1012.4817]. This is the basic “support-only” prototype. A second pattern consists of indicators defined by exclusion of prime powers. The squarefree indicator is \[ \mu^2(n)= \begin{cases} 1,& n\text{ squarefree},\\ 0,& \text{otherwise}, \end{cases} \] and more generally the \(s\)-free indicator is \[ E_s(n)=\sum_{j^s\mid n}\mu(j), \] so squarefreeness is the special case \(s=2\) [2502.10335] [1307.2066]. In the same direction, the largest squarefree divisor of an integer, \[ k(m)=\prod_{p\mid m}p, \] leads to the powered-number set \[ \mathcal A(\vartheta)=\{m\in\mathbb N:k(m)\le m^\vartheta\}, \] which measures how small the squarefree part of \(m\) is relative to \(m\) itself [2306.05981]. A third pattern is algebraic. A squarefree-power-like function in the formal sense is a map \((H,k)\mapsto \mathtt F(H,k)\) from hypergraphs and nonnegative integers to squarefree monomial ideals satisfying \[ \mathtt F(H,0)=S,\qquad \mathtt F(H,1)=I(H), \] the squarefree derivative condition \[ \partial^*\mathtt F(H,k)\subseteq \mathtt F(H,k-1), \] restriction to induced subhypergraphs, and the mixed-sum rule \[ \mathtt F(H_1+H_2,k)=\sum_{i=0}^k \mathtt F(H_1,i)\mathtt F(H_2,k-i) \] for disjoint unions [2510.01366]. The paper establishing this definition proves that both \(I(H)^{[k]}\) and \(I(H)^{\{k\}}\) fit this framework. ## 2. Support-type arithmetic functions The analytic prototype is Carella’s study of the Dedekind psi function. Because \[ \frac{\psi(n)}{n}=\prod_{p\mid n}\left(1+\frac1p\right), \] 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 \[ N_k=\prod_{j\le k}p_j, \] and establishes, for all sufficiently large \(k\), \[ \frac{\psi(N_k)}{N_k}>\frac{6}{\pi^2}e^\gamma \log\log N_k. \] It also derives \[ \frac{\psi(N_k)}{N_k}\sim \frac{6}{\pi^2}e^\gamma \log\log N_k \] from the prime product asymptotic \[ \prod_{p\le x}\left(1+\frac1p\right)=\frac{6}{\pi^2}e^\gamma\log x+O\!\left(\frac1{\log^2 x}\right), \] showing that prime-support products of the form \(f(n)=\prod_{p\mid n}F(p)\) are naturally optimized on primorials when \(F(p)>1\) decreases with \(p\) [1012.4817]. A complementary class is formed by multiplicative functions supported on the squarefree integers and taking values \(\pm1\) on primes. For such an \(f\), \[ F(s)=\sum_{n=1}^\infty \frac{f(n)}{n^s}=\prod_p\left(1+\frac{f(p)}{p^s}\right), \] and explicit constructions of the form \(f=\mu^2g\), where \(g\) is a completely multiplicative extension of a real non-principal Dirichlet character, yield summatory functions \[ M_f(x)=\sum_{n\le x}f(n) \] with strong cancellation. In particular, the paper gives deterministic examples with \(M_f(x)=o(\sqrt x)\), and under RH examples with \[ M_f(x)\ll x^{2/5+\varepsilon} \] for any \(\varepsilon>0\) [1908.11014]. This places squarefree-supported sign patterns into the same Euler-product regime as \(\mu\), but with analytically tunable prime signs. A related counting theory arises from the squarefree kernel. For \[ S_\vartheta(x)=\#\{m\le x:k(m)\le m^\vartheta\}, \] the asymptotic formula \[ S_\vartheta(x)=(1+o(1))x^\vartheta F((1-\vartheta)\log x) \] shows that the set of integers with unusually small squarefree kernel is only slightly denser than a pure power law \(x^\vartheta\) [2306.05981]. 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 \(s\)-free numbers, Brandes proves \[ \sum_{n\le x}E_s(n)E_s(n+1)=C_sx+O\!\left(x^{\frac{14}{7s+8}+\varepsilon}\right), \qquad C_s=\prod_p\left(1-\frac{2}{p^s}\right), \] generalizing Heath-Brown’s square-sieve method to arbitrary powers \(s\ge 2\) and improving the classical error term for consecutive \(s\)-free integers [1307.2066]. For arithmetic progressions, squarefree numbers are shown to equidistribute to smooth moduli far beyond earlier ranges. If \(q\) is squarefree, \(X^\eta\)-smooth, and \[ q\le X^{196/261-\varepsilon}, \] then the paper proves an asymptotic formula with power-saving error for \[ \sum_{\substack{n\le X\\ n\equiv a\!\!\!\pmod q}}\mu^2(n), \] thereby pushing squarefree distribution past the \(X^{3/4}\)-barrier for a positive-density family of moduli [2008.11163]. Squarefree-value problems for sparse polynomials fit the same pattern. For fixed nonzero integers \(\alpha,\beta\) with \(\gcd(\alpha,\beta)\) squarefree, \[ N(X;\alpha,\beta)=\#\{(a,b)\in\mathbb Z^2:\max\{|a|^{1/3},|b|^{1/4}\}\nu(G)\), and it turns regularity into a matching-theoretic problem. The paper proves the lower bound \[ \operatorname{reg}(I(G)^{[k]})\ge 2k+\nu_1(G)-k \qquad (k\le \nu_1(G)), \] establishes \[ \operatorname{reg}(I(G)^{[2]})\le 2+\nu(G), \] and shows that the top squarefree power \(I(G)^{[\nu(G)]}\) has linear quotients [1909.11420]. The first-syzygy threshold has a sharp combinatorial description. For \(2\le p\le \nu(G)\), \[ I(G)^{[p]}\text{ is linearly related} \] if and only if there is no induced subgraph \(H\subseteq G\) on \(2p+2\) vertices such that \(H\) is disconnected and \(\nu(H)=p+1\). In the equivalent language of a \(1\)-dimensional flag simplicial complex \(\Delta\), the obstruction becomes the presence of an induced complete bipartite graph \[ K_{2+i,\,2p-i} \] with even \(i\), and linear resolution further requires the absence of the crown graph \(Cr(2p+1)\) [2607.00838]. The same paper gives the explicit counting formula \[ \beta_{1,2p+2}(I_\Delta^{[p]}) = \sum_{\substack{0\le i\le p\\ i\text{ even}}} \#\{\text{induced }K_{2+i,\,2p-i}\subseteq \Delta\}, \] 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 \(T\), the highest non-vanishing squarefree power \[ NI(T)^{[\nu(N(T))]} \] is componentwise linear if and only if \(T\) 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 [2603.15229]. The same paper shows that \[ \operatorname{reg}(NI(G)^{[\nu]})-\deg(NI(G)^{[\nu]}) \] can be any prescribed positive integer on suitable trees, while for caterpillar graphs one has the exact formula \[ \operatorname{reg}\!\left(\frac{S}{NI(G)^{[\nu]}}\right)=\deg(NI(G)^{[\nu]})-1 \] [2603.15229]. Depth supplies another homological normalization. For a squarefree monomial ideal \(I\), the normalized depth function \[ g_I(k)=\operatorname{depth}(S/I^{[k]})-(d_k-1) \] satisfies \(g_I(k)\ge 0\), and the paper conjectures that \(g_I(k)\) is nonincreasing. In the graph case, if \(G^c\) is disconnected then \[ g_{I(G)}(k)=0 \] for all \(1\le k\le \nu(G)\), and for every graph without isolated vertices one always has \[ g_{I(G)}(\nu(G))=0 \] [2209.07847]. ## 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 \(I\), the \(a_i\)-invariants of \(S/I^n\) for \(i\ge 1\) and the geometric regularity are linear functions of \(n\) from \(n=2\), and one has the exact bridge \[ \mathrm{g\text{-}reg}(S/I^n)=\operatorname{reg}(S/I^{(n)}) \qquad (n\ge 1) \] [1808.07266]. 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 \[ \frac{t^\ell}{(1-t)^\ell},\qquad \ell=0,\dots,n. \] 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 [1201.0896]. This recasts squarefree Hilbert functions as support-controlled growth laws. Squarefree indexing also governs a modular-form problem. For squarefree level \(N\), \(\eta\)-quotients \[ f(\tau)=\prod_{\delta\mid N}\eta(\delta\tau)^{r_\delta} \] are indexed by squarefree divisors, the cusps are indexed by the same divisor set, and the cusp-order matrix \[ A_N=\left(\frac{N\gcd(d_i,d_j)^2}{d_id_j}\right) \] 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 \[ h_N\mid k, \] together with a weight-\(2\) obstruction on an explicit exceptional set \(S\) [1911.00096]. 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 \(\mu^2\), but the paper shows that they are not reconstructing \(n\) or testing \(p^2\mid n\) directly. Instead, they exploit conditional squarefree probabilities attached to small-prime divisibility patterns; indeed, using only \[ (n\bmod 2,\ n\bmod 3) \] already gives \(70.1\%\) accuracy for \(\mu^2(n)\), close to the \(70.64\%\) accuracy of the full 100-prime CRT input [2502.10335]. 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. Source: https://www.emergentmind.com/topics/squarefree-power-like-functions