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Sur la répartition des valeurs de la fonction d‘Euler

Published online by Cambridge University Press:  04 December 2007

MICHEL BALAZARD  and
GÉRALD TENENBAUM
MICHEL BALAZARD
Affiliation:
Cnrs Umr 9936, Algorithmique arthmétique, Université Bordeaux 1, 351, cours de la Libération, 33405 Talence, France
GÉRALD TENENBAUM
Affiliation:
Institut Élie Cartan, Université de Nancy 1, BP 239, 54506 Vandœuvre Cedex, France
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Abstract

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Let Φ(x) denote the number of those integers n with φ(n)[les ] x, where φ denotes the Euler function. Improving on a well-known estimate of Bateman (1972), we show that Φ(x)-Ax [Lt] R(x), where A=ζ(2)ζ(3)/ζ(6) and R(x) is essentially of the size of the best available estimate for the remainder term in the prime number theorem.

Keywords

Euler functionexponential sumsVinogradov methodprime number theoremshifted primesVaughan‘s identity
Type
Research Article
Information
Compositio Mathematica , Volume 110 , Issue 2 , January 1998 , pp. 239 - 250
Copyright
© 1998 Kluwer Academic Publishers