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An elementary problem equivalent to the Riemann hypothesis. (English) Zbl 1098.11005
Let \(\sigma(n)\) denote the sum-of-divisors function as usual, and write \(H_n = \sum_{j=1}^{n}{1\over j}\). The author proves that the problem of showing \(\sigma(n)\leq H_n + e^{H_n}\log H_n \), with equality only for \(n=1\), is equivalent to the Riemann Hypothesis (RH). The proof depends on two bounds for \(\sigma(n)\) given by G. Robin [J. Math. Pures Appl. (9) 63, 187–213 (1984; Zbl 0516.10036)].
The first bound is, assuming RH, \(\sigma(n) \leq e^{\gamma}n\log\log n \) for \(n \geq 5041\). Robin’s proof made use of delicate estimates involving prime-counting functions in terms of zeros of the Riemann zeta-function, and explicit error estimates for prime-counting functions. The second bound is, if RH is false, then there exist constants \(0<\beta < {1\over 2}\) and \(C>0\) such that \(\sigma(n) \geq e^{\gamma}n\log\log n + {Cn\log\log n\over (\log n)^{\beta}}\) holds for infinitely many \(n\).
The paper contains an exposition of highly composite numbers, superior highly composite numbers, superabundant numbers and collosally abundant numbers, and a brief account of the work of L. Alaoglu and P. Erdős [Trans. Am. Math. Soc. 56, 448–469 (1944; Zbl 0061.07903)] on these matters.

MSC:
11A25 Arithmetic functions; related numbers; inversion formulas
11M26 Nonreal zeros of \(\zeta (s)\) and \(L(s, \chi)\); Riemann and other hypotheses
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