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Semigroup forum

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Semigroup forum

### Sign changes of error terms related to arithmetical functions

Journal de Théorie des Nombres de Bordeaux

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 $#\left\{n\le T:H\left(n\right)H\left(n+1\right)<0\right\}\gg T.$ We consider arithmetical functions $f\left(n\right)={\sum }_{d\mid n}\frac{{b}_{d}}{d}$ whose summation can be expressed as ${\sum }_{n\le x}f\left(n\right)=\alpha x+P\left(log\left(x\right)\right)+E\left(x\right)$, 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 $\psi \left(x\right)=x-⌊x⌋-1/2$. We generalize Lau’s method and prove results about the number of sign changes for these error terms.

Semigroup forum

### The pseudovariety $J$ is hyperdecidable

RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications

### On some systems of equations with constraints in a free group. – Addenda. (Sur certains systèmes d'équations avec constraintes dans un groupe libre. – Addenda.)

Portugaliae Mathematica. Nova Série

### On some systems of equations with constraints in a free group. (Sur certains systèmes d'équations avec contraintes dans un groupe libre.)

Portugaliae Mathematica

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