Sign changes of error terms related to arithmetical functions

Sign changes of error terms related to arithmetical functions

Let H(x) = ∑ n≤x φ(n) n − 6 π2x. Motivated by a conjecture of Erdös, Lau developed a new method and proved that #{n ≤ T : H(n)H(n + 1) < 0} T. We consider arithmetical functions f(n) = ∑ d|n bd d whose summation can be expressed as ∑ n≤x f(n) = αx+P (log(x))+E(x), where P (x) is a polynomial, E(x) = − ∑ n≤y(x) bn n ψ ( x n ) + o(1) and ψ(x) = x− bxc − 1/2. We generalize Lau’s method and prove r...

Sign changes of error terms related to arithmetical functions par Paulo

Résumé. Soit H(x) = ∑ n≤x φ(n) n − 6 π2x. Motivé par une conjecture de Erdös, Lau a développé une nouvelle méthode et il a démontré que #{n ≤ T : H(n)H(n + 1) < 0} T. Nous considérons des fonctions arithmétiques f(n) = ∑ d|n bd d dont l’addition peut être exprimée comme ∑ n≤x f(n) = αx+ P (log(x)) + E(x). Ici P (x) est un polynôme, E(x) = − ∑ n≤y(x) bn n ψ ( x n ) + o(1) avec ψ(x) = x − bxc − 1...

Arithmetical Functions I: Multiplicative Functions

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 n...

Arithmetical Semigroups Related to Trees and Polyhedra

Fourier Expansions of Arithmetical Functions