On effective mean-values of arithmetic functions

2025/07/14 by Gérald Tenenbaum, Tenenbaum, Gérald
Mathematics · #11N60 #11N64 #Advanced Mathematical Identities #FOS: Mathematics #Functional Equations Stability Results #Number Theory (math.NT) #primary 11N37 #secondary 11N25

paper · pdf · doi:10.48550/arxiv.2507.10483

openalex publication_date 2025/07/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let r, f be multiplicative functions with r\geqslant 0, f is complex valued, |f|\leqslant r, and r satisfies some standard growth hypotheses. Let x be large, and assume that, for some real number τ, the quantities r(p)-\Re\f(p)/p^iτ\ are small in various appropriate average senses over the set of prime numbers not exceeding x. We derive from recent effective mean-value estimates an effective comparison theorem between the mean-values of f and of r on the set of integers \leqslant x. We also provide effective estimates for certain weighted moments of additive functions and for sifted mean-values of non-negative multiplicative functions.

Citations

• Moyennes effectives de fonctions multiplicatives complexes
• Multiplicative Function Mean Values: Asymptotic Estimates
• Harmonic Analysis on the Positive Rationals I: Basic Results
• Harmonic Analysis on the Positive Rationals II: Multiplicative Functions and Maass Forms

Related

• Moyennes effectives de fonctions multiplicatives complexes
• Multiplicative Function Mean Values: Asymptotic Estimates
• Extremal estimates for strongly additive and strongly multiplicative arithmetic functions
• Moyennes de certaines fonctions multiplicatives sur les entiers friables, 3
• Decay of mean-values of multiplicative functions