2016/04/01 by Alexander P. Mangerel, Mangerel, Alexander P. · 2 citations
Mathematics · #Analytic Number Theory Research #FOS: Mathematics #Limits and Structures in Graph Theory #Mathematics and Applications #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.1604.00295
openalex publication_date 2016/04/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Given an arithmetic function g(n) write Mg(x) := ∑n ≤ x g(n). We extend and strengthen the results of a fundamental paper of Halász in several ways by proving upper bounds for the ratio of \frac|Mg(x)|M|g|(x), for any strongly multiplicative, complex-valued function g(n) under certain assumptions on the sequence \g(p)\p. We further prove an asymptotic formula for this ratio in the case that |arg(g(p))| is sufficiently small uniformly in p. In so doing, we recover a new proof of an explicit lower mean value estimate for Mf(x) for any non-negative, multiplicative function satisfying c1 ≤ |f(p)| ≤ c2 for c2 ≥ c1 > 0, by relating it to (x)/(log x)∏p ≤ x (1+(f(p))/(p)). As an application, we generalize our main theorem in such a way as to give explicit estimates for the ratio \frac|Mg(x)|Mf(x), whenever f: ℕ → (0,∞) and g: ℕ → ℂ are strongly multiplicative functions that are uniformly bounded on primes and satisfy |g(n)| ≤ f(n) for every n ∈ ℕ. This generalizes a theorem of Wirsing and extends recent work due to Elliott.