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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 {\it G. Robin} (1984, using Euler-constant $\gamma$) and {\it 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) $G(n)\ge G({n\over p})$ for every prime factor $p$ of $n$; (ii) $G(n)\ge G(an)$ for every positive integer $a$. The proof is elementary and uses the results of Gronwall and Robin.

11M26Nonreal zeros of $\zeta (s)$ and $L(s, \chi)$; Riemann and other hypotheses
11N64Characterization of arithmetic functions
11Y55Calculation of integer sequences
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