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Article Dans Une Revue Integers : Electronic Journal of Combinatorial Number Theory Année : 2011

The Robin Inequality for 7-Free Integers

Résumé

Recall that an integer is t-free if and only if it is not divisible by pt for some prime p. We give a method to check Robin’s inequality, !(n) < e!n log log n, for t-free integers n and apply it for t = 6, 7. We introduce !t, a generalization of the Dedekind ! function defined for any integer t ! 2, by !t(n) := n ! p|n " 1 + 1/p + · · · + 1/pt−1# . If n is t-free then the sum of divisor function !(n) is " !t(n). We characterize the champions for x #$ !t(x)/x, as primorial numbers. Define the ratio Rt(n) := !t(n) n log log n. We prove that, for all t, there exists an integer n1(t), such that we have Rt(Nn) < e! for n ! n1, where Nn = $n k=1 pk. Further, by combinatorial arguments, this can be extended to Rt(N) " e! for all N ! Nn, such that n ! n1(t). This yields Robin’s inequality for t = 6, 7. For t varying slowly with N, we also derive Rt(N) < e!.
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Dates et versions

hal-02300319 , version 1 (29-09-2019)

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  • HAL Id : hal-02300319 , version 1

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Patrick Solé, Michel Planat. The Robin Inequality for 7-Free Integers. Integers : Electronic Journal of Combinatorial Number Theory, 2011, 11, pp.A65 (8). ⟨hal-02300319⟩
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