A Strengthening of Theorems of Halász and Wirsing

الأصل · EN

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 |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/ 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 |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.