ON AVERAGES OF COMPLETELY MULTIPLICATIVE FUNCTIONS OVER CO-PRIME INTEGER PAIRS

The paper’s Theorem 1.2 states exactly the equivalence of the two limits and the identities linking them (displayed as (5)–(6) in the PDF), for a bounded completely multiplicative f and a homogeneous P of degree k with P(N^2) ⊆ N . Its proof proceeds by two general lemmas (Möbius inversion and gcd-partition) specialized to r=2, yielding the truncated relations (21)–(24), and then passes N→∞ followed by D→∞ to obtain (5) and (6) . The model’s solution uses the same structural steps: gcd-partition plus homogeneity and complete multiplicativity to get f(P(g a, g b)) = f(g)^k f(P(a,b)), and Möbius inversion with tail control by ∑ 1/d^2, together with dominated/absolute convergence and Euler products; this matches the paper’s derivation (cf. the transitions leading to (25)–(28) and the use of ζ(2) = π^2/6 and ∏p(1 − f(p)^k/p^2) . Differences are present only in presentation (the paper packages the arguments via Lemmas 2.1–2.2 with explicit O-terms, while the model phrases them via dominated convergence). Substantively, the arguments are the same and yield identical formulas.