Extrapolating the mean-values of multiplicative functions
نویسندگان
چکیده
منابع مشابه
Mean values of multiplicative functions
Let f(n) be a totally multiplicative function such that |f(n)| ≤ 1 for all n, and let F (s) = ∑∞ n=1 f(n)n−s be the associated Dirichlet series. A variant of Halász’s method is developed, by means of which estimates for ∑N n=1 f(n)/n are obtained in terms of the size of |F (s)| for s near 1 with 1. The result obtained has a number of consequences, particularly concerning the zeros of the p...
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p 1−f(p) p diverges then the limit in (1.1) exists, and equals 0 = Θ(f,∞). Wirsing’s result settled an old conjecture of P. Erdős and Wintner that every multiplicative function f with −1 ≤ f(n) ≤ 1 had a mean-value. The situation for complex valued multiplicative functions is more delicate. For example, the function f(n) = n (0 6= α ∈ R) does not have a mean-value because 1 x ∑ n≤x n iα ∼ x 1+i...
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exists . The question of the existence of the mean value M(g) has been much studied, but it has been solved only for certain subclasses . One of the definite results is the following theorem, due to H . Delange [2] (throughout the paper, p denotes a prime, and E and n denote a sum ana a product, respectively, taken over all primes) : p . p If g(n) is a strongly multiplicative number-theoretical...
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Let f : Nn → C be an arithmetic function of n variables, where n ≥ 2. We study the mean-value M(f) of f that is defined to be lim x1,...,xn→∞ 1 x1 · · ·xn ∑ m1≤x1, ... , mn≤xn f(m1, . . . , mn), if this limit exists. We first generalize the Wintner theorem and then consider the multiplicative case by expressing the mean-value as an infinite product over all prime numbers. In addition, we study ...
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ژورنال
عنوان ژورنال: Indagationes Mathematicae (Proceedings)
سال: 1989
ISSN: 1385-7258
DOI: 10.1016/1385-7258(89)90004-8