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
j
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