Sign changes of error terms related to arithmetical functions
Journal de théorie des nombres de Bordeaux, Tome 19 (2007) no. 1, pp. 1-25.

Soit H(x)= nx φ(n) n-6 π 2 x. Motivé par une conjecture de Erdös, Lau a développé une nouvelle méthode et il a démontré que #{nT:H(n)H(n+1)<0}T. Nous considérons des fonctions arithmétiques f(n)= dn b d d dont l’addition peut être exprimée comme nx f(n)=αx+P(log(x))+E(x). Ici P(x) est un polynôme, E(x)=- ny(x) b n nψx n+o(1) avec ψ(x)=x-x-1/2. Nous généralisons la méthode de Lau et démontrons des résultats sur le nombre de changements de signe pour ces termes d’erreur.

Let H(x)= nx φ(n) n-6 π 2 x. Motivated by a conjecture of Erdös, Lau developed a new method and proved that #{nT:H(n)H(n+1)<0}T. We consider arithmetical functions f(n)= dn b d d whose summation can be expressed as nx f(n)=αx+P(log(x))+E(x), where P(x) is a polynomial, E(x)=- ny(x) b n nψx n+o(1) and ψ(x)=x-x-1/2. We generalize Lau’s method and prove results about the number of sign changes for these error terms.

DOI : 10.5802/jtnb.570
Paulo J. Almeida 1

1 Departamento de Matemática Universidade de Aveiro Campus Universitário de Santiago 3810-193 Aveiro, Portugal
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Paulo J. Almeida. Sign changes of error terms related to arithmetical functions. Journal de théorie des nombres de Bordeaux, Tome 19 (2007) no. 1, pp. 1-25. doi : 10.5802/jtnb.570. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.570/

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