Sign changes of error terms related to arithmetical functions

Almeida, Paulo J.

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

Let $H\left(x\right)={\sum }_{n\le x}\frac{\phi \left(n\right)}{n}-\frac{6}{{\pi }^2}x$. Motivated by a conjecture of Erdös, Lau developed a new method and proved that $\#\{n\le T:H(n\left)H\right(n+1)<0\}\gg T.$ We consider arithmetical functions $f\left(n\right)={\sum }_{d\mid n}\frac{b_d}{d}$ whose summation can be expressed as ${\sum }_{n\le x}f\left(n\right)=\alpha x+P(log\left(x\right))+E\left(x\right)$, where $P\left(x\right)$ is a polynomial, $E\left(x\right)=-{\sum }_{n\le y\left(x\right)}\frac{b_n}{n}\psi \left(\frac{x}{n}\right)+o\left(1\right)$ and $\psi \left(x\right)=x-\lfloor x\rfloor -1/2$. We generalize Lau’s method and prove results about the number of sign changes for these error terms.

Soit $H\left(x\right)={\sum }_{n\le x}\frac{\phi \left(n\right)}{n}-\frac{6}{{\pi }^2}x$. Motivé par une conjecture de Erdös, Lau a développé une nouvelle méthode et il a démontré que $\#\{n\le T:H(n\left)H\right(n+1)<0\}\gg T.$ Nous considérons des fonctions arithmétiques $f\left(n\right)={\sum }_{d\mid n}\frac{b_d}{d}$ dont l’addition peut être exprimée comme ${\sum }_{n\le x}f\left(n\right)=\alpha x+P(log\left(x\right))+E\left(x\right)$. Ici $P\left(x\right)$ est un polynôme, $E\left(x\right)=-{\sum }_{n\le y\left(x\right)}\frac{b_n}{n}\psi \left(\frac{x}{n}\right)+o\left(1\right)$ avec $\psi \left(x\right)=x-\lfloor x\rfloor -1/2$. Nous généralisons la méthode de Lau et démontrons des résultats sur le nombre de changements de signe pour ces termes d’erreur.

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