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Robin’s theorem, primes, and a new elementary reformulation of the Riemann hypothesis. (English) Zbl 1235.11082
There are equivalent formulations of the Riemann hypothesis (RH) by G. Robin (1984, using Euler-constant \(\gamma\)) and J. C. Lagarias [Am. Math. Mon. 109, No. 6, 534–543 (2002; Zbl 1098.11005), using harmonic numbers). The authors give another elementary one, using Gronwall’s function \(G(n)= {\sigma(n)\over n\log\log n}\) \((n> 1)\): RH is true if and only if \(n= 4\) is the only composite number with the two properties:
(i) \(G(n)\geq G({n\over p})\) for every prime factor \(p\) of \(n\);
(ii) \(G(n)\geq G(an)\) for every positive integer \(a\).
The proof is elementary and uses the results of Gronwall and Robin.

MSC:
11M26 Nonreal zeros of \(\zeta (s)\) and \(L(s, \chi)\); Riemann and other hypotheses
11N64 Other results on the distribution of values or the characterization of arithmetic functions
11Y55 Calculation of integer sequences
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