# zbMATH — the first resource for mathematics

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
##### Keywords:
The sum-of-divisors function; Riemann Hypothesis
Full Text: