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

Y.-F. S. Pétermann

Journal de théorie des nombres de Bordeaux (1991)

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

Abstract

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Let J k ( n ) : = n k ∏ p ∣ n ( 1 - p - k ) (the k -th Jordan totient function, and for k = 1 the Euler phi function), and consider the associated error term E k ( x ) : = ∑ n ≤ x J k ( n ) - x k + 1 ( k + 1 ) ζ ( k + 1 ) . When k ≥ 2 , both i k : = E k ( x ) x - k and s k : = lim sup E k ( x ) x - k are finite, and we are interested in estimating these quantities. We may consider instead I k : = lim inf n ∈ ℕ , n → ∞ d 1 (d)dk ( 12 - { nd} ), since from [AS] i k = I k - ( ζ ( k + 1 ) ) - 1 and from the present paper s k = - i k . We show that I k belongs to an interval of the form 1 2 ζ ( k ) - 1 ( k - 1 ) N k - 1 , 1 2 ζ ( k ) , where N = N ( k ) → ∞ as k → ∞ . 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

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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

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  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

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