Arithmetical Functions I: Multiplicative Functions
نویسنده
چکیده
Truth be told, this definition is a bit embarrassing. It would mean that taking any function from calculus whose domain contains [1,+∞) and restricting it to positive integer values, we get an arithmetical function. For instance, e −3x cos2 x+(17 log(x+1)) is an arithmetical function according to this definition, although it is, at best, dubious whether this function holds any significance in number theory.
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On the binomial convolution of arithmetical functions
Let n = ∏ p p νp(n) denote the canonical factorization of n ∈ N. The binomial convolution of arithmetical functions f and g is defined as (f ◦g)(n) = ∑ d|n (∏ p (νp(n) νp(d) )) f(d)g(n/d), where ( a b ) is the binomial coefficient. We provide properties of the binomial convolution. We study the Calgebra (A,+, ◦,C), characterizations of completely multiplicative functions, Selberg multiplicative...
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