Sign changes of error terms related to arithmetical functions
Paulo J. Almeida1
1Departamento de Matemática Universidade de Aveiro Campus Universitário de Santiago 3810-193 Aveiro, Portugal
Journal de théorie des nombres de Bordeaux, Volume 19 (2007) no. 1, pp. 1-25

Let H(x)=∑ n≤x φ(n) n-6 π 2 x. 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 b d 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) 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.

Soit H(x)=∑ n≤x φ(n) n-6 π 2 x. 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 b d 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) 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.

Received:
Published online:
DOI: 10.5802/jtnb.570

Paulo J. Almeida  1

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