Sur les entiers inférieurs à x ayant plus de log ( x ) diviseurs

Marc Deléglise; Jean-Louis Nicolas

Journal de théorie des nombres de Bordeaux (1994)

  • Volume: 6, Issue: 2, page 327-357
  • ISSN: 1246-7405

Abstract

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Let τ ( n ) be the number of divisors of n ; let us define S λ ( x ) = C a r d n ≤ x ; τ ( n ) ≥ ( log x ) λ log 2 if λ ≥ 1 C a r d n ≤ x ; τ ( n ) < ( log x ) λ log 2 if λ < 1 It has been shown that, if we set f ( λ , x ) = x ( log x ) λ log λ - λ + 1 log log x the quotient S λ ( x ) / f ( λ , x ) is bounded for λ fixed. The aim of this paper is to give an explicit value for the inferior and superior limits of this quotient when λ ≥ 2 . For instance, when λ = 1 / log 2 , we prove lim inf S λ ( x ) f ( λ , x ) = 0 . 938278681143 ⋯ and lim inf S λ ( x ) f ( λ , x ) = 1 . 148126773469 ⋯

How to cite

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Deléglise, Marc, and Nicolas, Jean-Louis. "Sur les entiers inférieurs à $x$ ayant plus de $\log (x)$ diviseurs." Journal de théorie des nombres de Bordeaux 6.2 (1994): 327-357. <http://eudml.org/doc/93607>.

@article{Deléglise1994,
author = {Deléglise, Marc, Nicolas, Jean-Louis},
journal = {Journal de théorie des nombres de Bordeaux},
keywords = {integers with more than prime factors; asymptotic behavior; divisor function; numerical computations},
language = {fre},
number = {2},
pages = {327-357},
publisher = {Université Bordeaux I},
title = {Sur les entiers inférieurs à $x$ ayant plus de $\log (x)$ diviseurs},
url = {http://eudml.org/doc/93607},
volume = {6},
year = {1994},
}

TY - JOUR
AU - Deléglise, Marc
AU - Nicolas, Jean-Louis
TI - Sur les entiers inférieurs à $x$ ayant plus de $\log (x)$ diviseurs
JO - Journal de théorie des nombres de Bordeaux
PY - 1994
PB - Université Bordeaux I
VL - 6
IS - 2
SP - 327
EP - 357
LA - fre
KW - integers with more than prime factors; asymptotic behavior; divisor function; numerical computations
UR - http://eudml.org/doc/93607
ER -

References

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  1. [1] M. Balazard, J.-L. Nicolas, C. Pomerance, G. Tenenbaum, Grandes déviations pour certaines fonctions arithmétiques, J. Number Theory40 (1992), 146-164. Zbl0745.11041MR1149734
  2. [2] L. Comtet, Analyse combinatoire, Tomes 1 et 2. Presses universitaires de France, 1970. Zbl0221.05002MR262087
  3. [3] H.H. Crapo, Permanent by Möbius inversion, Journal of Combinatorial Theory4 (1968), 198-200. Zbl0162.03201MR220613
  4. [4] H.T. Davis, Table of the higher mathematical functions, The principia Press, Bloomington, Indiana, 1935, vol. 2. Zbl0013.21603
  5. [5] M. Deléglise, Applications des ordinateurs à la théorie des nombres, Thèse Université de Lyon1, 1991. 
  6. [6] P.D.T.A. Elliot, Probabilistic number theory, vol I and II, Grundlehren der Mathematischen Wissenschaften239-240, Springer-Verlag, 1979. Zbl0431.10029MR551361
  7. [7] P. Flageolet, I. Vardi, Numerical evaluation of Euler products, Prepublication. 
  8. [8] W.L. Glaisher, On the sums of the inverse powers of the prime numbers, Quartely Journal of Math25 (1891), 347-362. Zbl23.0275.02JFM23.0275.02
  9. [9] G.H. Hardy and E.M. Wright, An introduction to the theory of numbers, Oxford at the Clarendon Press1962. Zbl0086.25803MR568909
  10. [10] K.K. Norton, On the number of restricted prime factors of an integer, Illinois J. Math20 (1976), 681-705. Zbl0329.10035MR419382
  11. [11] H. Riesel, Prime numbers and computer methods for factorization, Birkhaüser, 1985. Zbl0582.10001MR897531

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