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

top
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

top

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

top
  1. A. Baker, Linear forms in the logarithms of algebraic numbers (III). Mathematika 14 (1967), 220–228. Zbl0161.05301MR220680
  2. U. Balakrishnan, Y.-F. S. Pétermann, The Dirichlet series of ζ ( s ) ζ α ( s + 1 ) f ( s + 1 ) : On an error term associated with its coefficients. Acta Arithmetica 75 (1996), 39–69. Zbl0846.11054MR1379390
  3. S. Chowla, Contributions to the analytic theory of numbers. Math. Zeit. 35 (1932), 279–299. Zbl0004.10202MR1545300
  4. G. H. Hardy, E. M. Wright, An Introduction to the Theory of Numbers. 5th ed. Oxford 1979. Zbl0423.10001MR67125
  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. 

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.