Ruzsa 1977 general multiplicative functions

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Ruzsa, I. Z., General multiplicative functions. Acta Arith. 32 (1977), 313-347. The image-only scan shows no copyright or license line on its rendered first or last page; the journal's record offers the PDF under the download link "Pobierz zgodnie z CC-BY", rendered "Free download under CC-BY license" on the English site, and names no version or URL for it (https://www.impan.pl/get/doi/10.4064/aa-32-4-313-347, read 2026-10-02): the Creative Commons Attribution license, with no version stated.

Ruzsa generalizes multiplicative and additive arithmetic functions to G-multiplicative functions, meaning f: N -> G into a groupoid G with f(nm) = f(n) o f(m) for coprime n, m (Definitions 1.1-1.3), and proves local limit theorems on the counting of n <= x with f(n) = g. Theorem 1.4 states that if G is an Abelian group and f is G-multiplicative then the sequence f^{-1}(g) has an asymptotic density for every g in G; Corollary 1.6 deduces that the solution set of an infinite system f_k(n) = g_k, with complex-valued multiplicative f_k that never take the value 0, has an asymptotic density, and for functions taking values in {1,-1} Theorem 1.4 is equivalent to existence of the mean value M(f) conjectured by Erdős and proved by Wirsing. Statement 1.8 identifies the number-theoretic content: S contained in N arises as f^{-1}(g) for a strongly G-multiplicative f exactly when S = qG_0 intersect N for a subgroup G_0 of the multiplicative group of the rationals and a nonzero rational q, equivalently when the divisibility-closure condition (1.10) holds. The framework rests on density sets, density and uniform density classes, concentration complexes and factor-functions (Sections 3, 7-9), and Sections 7-8 show that a set T of prime powers with sum_{t in T} 1/t < infinity is always negligible. The paper is cited for problem 121, on the largest subsets of {1,...,N} in which no k distinct elements multiply to a square.

Source: https://eudml.org/doc/205553.

Bears on. #121

Results to transcribe.

• Theorem 1.4: If G is an Abelian group and f is G-multiplicative, the sequence f^{-1}(g) has an asymptotic density for every g in G. Corollary 1.6 fails if infinitely many of the f_k may take the value 0, which the paper reads as showing that G must be a group.
• Corollary 1.6: For complex-valued multiplicative functions f_k with f_k(n) != 0 for all n and arbitrary g_k in C, the set of solutions of the infinite system f_k(n) = g_k has an asymptotic density.
• Statement 1.8: For S contained in N the following are equivalent: S = f^{-1}(g) for some Abelian group G, g in G and strongly G-multiplicative f; S = qG_0 intersect N for a subgroup G_0 of the multiplicative rationals and nonzero rational q; and the multiplicative closure condition (1.10) holds.
• Problem 1.7: Poses the number-theoretic question of characterizing which sequences S contained in N are of the form f^{-1}(g) for an Abelian group G, a G-multiplicative f and some g in G.
• Negligibility of thin prime-power sets (Sections 7-8): A set T of prime powers with sum_{t in T} 1/t < infinity, for instance all p^i with i >= 2, is always negligible for these density questions.