Small values of the Euler function and the Riemann hypothesis

Jean-Louis Nicolas

Description

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.

• let ?


• function satisfying


• lower bounds


• euler's function


• riemann hypothesis


• chebichev functions


• hypothesis fails

Sujets

• Nicolas
• Jordan
• Euler function
• Riemann hypothesis

Informations

• Publié par: profil-urra-2012
• Publié le: 01 février 2012
• Nombre de lectures: 20
• Langue: English

Extrait

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 ( n n ) 6 e γ log log n + log 2 . l 5 o 1 g 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