Sign changes of error terms related to arithmetical functions

Paulo J. Almeida

doi:10.5802/jtnb.570

Paulo J. Almeida ¹

¹ Departamento de Matemática Universidade de Aveiro Campus Universitário de Santiago 3810-193 Aveiro, Portugal

Journal de théorie des nombres de Bordeaux, Tome 19 (2007) no. 1, pp. 1-25.

Résumé
Abstract

Soit $H (x) = \sum_{n \leq x} \frac{φ (n)}{n} - \frac{6}{π^{2}} x$ . Motivé par une conjecture de Erdös, Lau a développé une nouvelle méthode et il a démontré que $# {n \leq T : H (n) H (n + 1) < 0} ≫ T .$ Nous considérons des fonctions arithmétiques $f (n) = \sum_{d ∣ n} \frac{b_{d}}{d}$ dont l’addition peut être exprimée comme $\sum_{n \leq x} f (n) = α x + P (log (x)) + E (x)$ . Ici $P (x)$ est un polynôme, $E (x) = - \sum_{n \leq y (x)} \frac{b_{n}}{n} ψ (\frac{x}{n}) + o (1)$ 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 $H (x) = \sum_{n \leq x} \frac{φ (n)}{n} - \frac{6}{π^{2}} x$ . Motivated by a conjecture of Erdös, Lau developed a new method and proved that $# {n \leq T : H (n) H (n + 1) < 0} ≫ T .$ We consider arithmetical functions $f (n) = \sum_{d ∣ n} \frac{b_{d}}{d}$ whose summation can be expressed as $\sum_{n \leq x} f (n) = α x + P (log (x)) + E (x)$ , where $P (x)$ is a polynomial, $E (x) = - \sum_{n \leq y (x)} \frac{b_{n}}{n} ψ (\frac{x}{n}) + o (1)$ 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.

MR

DOI : 10.5802/jtnb.570

Affiliations des auteurs :

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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