Claims

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2001_07_01_grytczuk_luca_wojtowicz: Grytczuk, Luca and Wójtowicz prove the totient of n minus its totient exceeds the totient of n infinitely often, with a growing gap, and the reverse on a set of lower density at least 0.54; refereed and credited by the site.

2002_01_01_luca_pomerance: Luca and Pomerance prove that the totient of n exceeds the totient of n minus its totient for almost all n, by a margin of any order below n, and remark without proof that the reverse holds infinitely often by any factor.

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