SOME ERGODIC THEOREMS OVER SQUAREFREE NUMBERS

The paper’s Theorem 1.1 proves that for any bounded arithmetic function a(n) with Cesàro average invariant under left-multiplications, the average of a(n) along squarefree n avoiding a fixed finite set S of primes equals (α(S)/ζ(2))·A, with α(S)=∏_{p∈S} p/(p+1) (equation (7) in the paper) . The candidate solution proves the same statement via a clean Möbius/inclusion–exclusion expansion of μ^2(n)·1_{(n,M)=1}, a dominated convergence interchange justified by boundedness of a and a 1/[d^2,f] majorant, and an explicit prime-by-prime evaluation giving α(S)/ζ(2). The paper’s proof uses w_S-weighting (Lemma 3.1) and a truncated identity (Proposition 3.2) to pass to the limit and then evaluates the resulting Dirichlet series via identities (23)–(25), recovering the same constant (α(S)/ζ(2))A (equations (31)–(36)) . The two approaches are mathematically consistent, differ in technique, and lead to the same result.