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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.doi:10.4064/aa155-3-7 fatcat:qc267tqjanfnfcrfynyfonkeu4