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Paulo J. Almeida
Sign changes of error terms related to arithmetical functions
Journal de théorie des nombres de Bordeaux, 19 no. 1 (2007), p. 1-25, doi: 10.5802/jtnb.570
Article PDF | Reviews MR 2332050 | Zbl pre05186971

Résumé - Abstract

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

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