an:01956329 Zbl 1098.11005 Lagarias, Jeffrey C. An elementary problem equivalent to the Riemann hypothesis. EN Am. Math. Mon. 109, No. 6, 534-543 (2002). 0002-9890 2002
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11A25 11M26 The sum-of-divisors function; Riemann Hypothesis 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 \textit{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 \textit{L. Alaoglu} and \textit{P. Erdős} [Trans. Am. Math. Soc. 56, 448--469 (1944; Zbl 0061.07903)] on these matters. Cem Yalçin Yildirim (Istanbul) 0516.10036; 0061.07903