Sign changes of error terms related to arithmetical functions
- [1] Departamento de Matemática Universidade de Aveiro Campus Universitário de Santiago 3810-193 Aveiro, Portugal
Journal de Théorie des Nombres de Bordeaux (2007)
- Volume: 19, Issue: 1, page 1-25
- ISSN: 1246-7405
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topAlmeida, Paulo J.. "Sign changes of error terms related to arithmetical functions." Journal de Théorie des Nombres de Bordeaux 19.1 (2007): 1-25. <http://eudml.org/doc/249961>.
@article{Almeida2007,
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.},
affiliation = {Departamento de Matemática Universidade de Aveiro Campus Universitário de Santiago 3810-193 Aveiro, Portugal},
author = {Almeida, Paulo J.},
journal = {Journal de Théorie des Nombres de Bordeaux},
keywords = {number of sign changes for error terms},
language = {eng},
number = {1},
pages = {1-25},
publisher = {Université Bordeaux 1},
title = {Sign changes of error terms related to arithmetical functions},
url = {http://eudml.org/doc/249961},
volume = {19},
year = {2007},
}
TY - JOUR
AU - Almeida, Paulo J.
TI - Sign changes of error terms related to arithmetical functions
JO - Journal de Théorie des Nombres de Bordeaux
PY - 2007
PB - Université Bordeaux 1
VL - 19
IS - 1
SP - 1
EP - 25
AB - 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.
LA - eng
KW - number of sign changes for error terms
UR - http://eudml.org/doc/249961
ER -
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