Sign changes of error terms related to arithmetical functions

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 (2007)

  • Volume: 19, Issue: 1, page 1-25
  • ISSN: 1246-7405

Abstract

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Let H ( x ) = ∑ n ≤ x φ ( n ) n - 6 π 2 x . Motivated by a conjecture of Erdös, Lau developed a new method and proved that # { n ≤ T : H ( n ) H ( n + 1 ) < 0 } ≫ T . We consider arithmetical functions f ( n ) = ∑ d ∣ n b d d whose summation can be expressed as ∑ n ≤ x f ( n ) = α x + P ( log ( x ) ) + E ( x ) , where P ( x ) is a polynomial, E ( x ) = - ∑ n ≤ y ( x ) b n n ψ 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.

How to cite

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Almeida, 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 $\#\lbrace n\le T: H(n)H(n+1)&lt;0\rbrace \gg T.$ 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 $\#\lbrace n\le T: H(n)H(n+1)&lt;0\rbrace \gg T.$ 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 -

References

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  5. Y.-K. Lau, Sign changes of error terms related to the Euler function. Mathematika 46 (1999), 391–395. Zbl1033.11047MR1832629
  6. Y.-F. S. Pétermann, On the distribution of values of an error term related to the Euler function. Théorie des Nombres, (Quebec, PQ, 1987), 785–797. Zbl0685.10030MR1024603
  7. S. Ramanujan, Collected papers. Cambridge, 1927, 133–135. MR2280860
  8. R. Sitaramachandrarao, On an error term of Landau - II. Rocky Mount. J. of Math. 15 2 (1985), 579–588. Zbl0584.10027MR823269
  9. A. Walfisz, Teilerprobleme II. Math. Zeit. 34 (1931), 448–472. 

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