Small values of the Euler function and the Riemann hypothesis
12
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Documents
2012
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12
pages
English
Documents
2012
Obtenez un accès à la bibliothèque pour le consulter en ligne En savoir plus
Small values of the Euler function and the Riemann hypothesis Jean-Louis Nicolas ? 2 février 2012 À André Schinzel pour son 75ème anniversaire, en très amical hommage. Abstract Let ? be Euler's function, ? be Euler's constant and Nk be the product of the first k primes. In this article, we consider the function c(n) = (n/?(n)?e? log logn) √ log n. Under Riemann's hypothesis, it is proved that c(Nk) is bounded and explicit bounds are given while, if Riemann's hypothesis fails, c(Nk) is not bounded above or below. Keywords : Euler's function, Riemann hypothesis, Explicit formula. 2010 Mathematics Subject Classification : 11N37, 11M26, 11N56. 1 Introduction Let ? be the Euler function. In 1903, it was proved by E. Landau (cf. [5, 59] and [4, Theorem 328]) that lim sup n?∞ n ?(n) log log n = e? = 1.7810724179 . . . where ? = 0.5772156649 . . . is Euler's constant. In 1962, J. B. Rosser and L. Schoenfeld proved (cf. [9, Theorem 15]) (1.1) n ?(n) 6 e? log log n+ 2.51 log log n for n > 3 and asked if there exists an infinite number of n such that n/?(n) > e? log log n.
Voir
- let ?
- function satisfying
- lower bounds
- euler's function
- riemann hypothesis
- chebichev functions
- hypothesis fails
Small values of the Euler function and the Riemann hypothesis Jean-Louis Nicolas ∗ 2 février 2012
À André Schinzel pour son 75ème anniversaire, en très amical hommage.
Abstract Let ϕ be Euler’s function, γ be Euler’s constant and N k be the product of the first k primes. In this article, we consider the function c ( n ) = ( n/ϕ ( n ) − e γ log log n ) √ log n . Under Riemann’s hypothesis, it is proved that c ( N k ) is bounded and explicit bounds are given while, if Riemann’s hypothesis fails, c ( N k ) is not bounded above or below. Keywords : Euler’s function, Riemann hypothesis, Explicit formula. 2010 Mathematics Subject Classification : 11N37, 11M26, 11N56.
1 Introduction Let ϕ be the Euler function. In 1903, it was proved by E. Landau (cf. [5, §59] and [4, Theorem 328]) that li n m → s ∞ up ϕ ( n ) lo n g log n = e γ = 1 . 7810724179 . . . where γ = 0 . 5772156649 . . . is Euler’s constant. In 1962, J. B. Rosser and L. Schoenfeld proved (cf. [9, Theorem 15]) (1.1) ϕ ( nn ) 6 e γ log log n +log2 . l5o1g n for n > 3 and asked if there exists an infinite number of n such that n/ϕ ( n ) > e γ log log n . In [6], (cf. also [7]), I answer this question in the affirmative. Soon after, A. Schinzel told me that he had worked unsuccess-fully on this question, which made me very proud to have solved it. ∗ Research partially supported by CNRS, Institut Camille Jordan, UMR 5208. 1
