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

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.

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.

Received: 2006-01-08
Published online: 2008-12-03
DOI: https://doi.org/10.5802/jtnb.570
@article{JTNB_2007__19_1_1_0,
     author = {Paulo J. Almeida},
     title = {Sign changes of error terms related to arithmetical functions},
     journal = {Journal de Th\'eorie des Nombres de Bordeaux},
     publisher = {Universit\'e Bordeaux 1},
     volume = {19},
     number = {1},
     year = {2007},
     pages = {1-25},
     doi = {10.5802/jtnb.570},
     zbl = {pre05186971},
     mrnumber = {2332050},
     language = {en},
     url={jtnb.centre-mersenne.org/item/JTNB_2007__19_1_1_0/}
}
Almeida, Paulo J. 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. https://jtnb.centre-mersenne.org/item/JTNB_2007__19_1_1_0/

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