# Oscillations d'un terme d'erreur lié à la fonction totient de Jordan

• Volume: 3, Issue: 2, page 311-335
• ISSN: 1246-7405

top

## Abstract

top
Let ${J}_{k}\left(n\right):={n}^{k}{\prod }_{p\mid n}\left(1-{p}^{-k}\right)$ (the $k$-th Jordan totient function, and for $k=1$ the Euler phi function), and consider the associated error term${E}_{k}\left(x\right):=\sum _{n\le x}\phantom{\rule{4pt}{0ex}}{J}_{k}\left(n\right)-\frac{{x}^{k+1}}{\left(k+1\right)\zeta \left(k+1\right)}.$When $k\ge 2$, both ${i}_{k}:={E}_{k}\left(x\right){x}^{-k}$ and ${s}_{k}:=lim sup{E}_{k}\left(x\right){x}^{-k}$ are finite, and we are interested in estimating these quantities. We may consider instead$Ik:=\underset{n\in ℕ,n\to \infty }{lim inf}$d 1 (d)dk ( 12 - { nd} ), since from [AS] ${i}_{k}={I}_{k}-{\left(\zeta \left(k+1\right)\right)}^{-}1$ and from the present paper ${s}_{k}=-{i}_{k}$. We show that ${I}_{k}$ belongs to an interval of the form$\left(\frac{1}{2\zeta \left(k\right)}-\frac{1}{\left(k-1\right){N}^{k-1}},\frac{1}{2\zeta \left(k\right)}\right),$where $N=N\left(k\right)\to \infty$ as $k\to \infty$. From a more practical point of view we describe an algorithm capable of yielding arbitrary good approximations of ${I}_{k}$. We apply this algorithm to the small values of $k$ and obtain $.29783<I-2<.29877,.415891<{I}_{3}<.415923,$ and $.46196896<{I}_{4}<.46196916$.

## How to cite

top

Pétermann, Y.-F. S.. "Oscillations d'un terme d'erreur lié à la fonction totient de Jordan." Journal de théorie des nombres de Bordeaux 3.2 (1991): 311-335. <http://eudml.org/doc/93542>.

@article{Pétermann1991,
author = {Pétermann, Y.-F. S.},
journal = {Journal de théorie des nombres de Bordeaux},
keywords = {error term estimates; Jordan totient function; algorithm; upper and lower bounds},
language = {fre},
number = {2},
pages = {311-335},
publisher = {Université Bordeaux I},
title = {Oscillations d'un terme d'erreur lié à la fonction totient de Jordan},
url = {http://eudml.org/doc/93542},
volume = {3},
year = {1991},
}

TY - JOUR
AU - Pétermann, Y.-F. S.
TI - Oscillations d'un terme d'erreur lié à la fonction totient de Jordan
JO - Journal de théorie des nombres de Bordeaux
PY - 1991
PB - Université Bordeaux I
VL - 3
IS - 2
SP - 311
EP - 335
LA - fre
KW - error term estimates; Jordan totient function; algorithm; upper and lower bounds
UR - http://eudml.org/doc/93542
ER -

## References

top
1. [AS] S.D. Adhikari and A. Sankaranarayanan, On an error term related to the Jordan totient function Jk(n), J. Number Theory34 (1990), 178-188. Zbl0694.10041MR1042491
2. [ES] P. Erdös and H.N. Shapiro, The existence of a distribution function for an error term related to the Euler function, Canad. J. Math.7 (1955), 63-75. Zbl0067.27601MR65580
3. [M] H.L. Montgomery, Fluctuations in the mean of Euler's phi function, Proc. Indian Acad. Sci. (Math.Sci.)97 (1987), 239-245. Zbl0656.10042MR983617
4. [P1] Y.-F.S. Pétermann, Existence of all the asymptotic λ-th means for certain arithmetical convolutions, Tsukuba J. Math.12 (1988), 241-248. Zbl0661.10056
5. [P2] Y.-F.S. Pétermann, On the distribution of values of an error term related to the Euler function, Proc. Conf. Théorie des nombres Univ. Laval juillet 1987, 785-797, Walter de Gruyter, Berlin (1989). Zbl0685.10030MR1024603
6. [P3] Y.-F.S. Pétermann, On the average behaviour of the largest divisor of n which is prime to a fixed integer k, prépublication.
7. [W] A. Walfisz, Weylsche Exponentialsummen in der neueren Zahlentheorie, VEB Deutscher Verlag der Wissenschaften, Berlin (1963). Zbl0146.06003MR220685

top

## 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.