The conjectured Robin inequality for an integer n > 7! is (n) < en log log n, where denotes the Euler constant, and (n) = P d|n d. Robin proved that this conjecture is equivalent to the Riemann hypothesis (RH). Writing D(n) = en log log n(n), and d(n) = D(n) n , we prove unconditionally that lim infn!1 d(n) = 0. The main ingredients of the proof are an estimate for the Chebyshev summatory function, and an e?ective version of Mertens’ third theorem due to Rosser and Schoenfeld. A new criterion for RH depending solely on lim infn!1 D(n) is derived.
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