Abundant numbers and the Riemann hypothesis. (English) Zbl 1149.11041

Briggs, Keith

Exp. Math. 15, No. 2, 251-256 (2006).

It has been shown by G. Robin [J. Math. Pures Appl. (9) 63, 187–213 (1984; Zbl 0516.10036)] that the Riemann hypothesis holds if and only if \[ \sigma(n)< e^\gamma n\log\log n\tag{\(*\)} \] for all \(n> 5040\). Moreover he showed that if the Riemann hypothesis is false, then the above inequality is violated by at least one “colossally abundant” integer \(n\), that is to say an integer which maximizes \(\sigma(n) n^{-\theta}\) for some \(\theta> 1\).
An algorithm to compute successive colossally abundant numbers \(n\) is given, with data on Robin’s inequality \((*)\) for such \(n\) up to \(10^{10^{10}}\). After a full discussion of the results it is conjectured that for colossally abundant numbers \(n\) the right-hand side of \((*)\) exceeds the left by \(n(\log\log n)^{1/2+o(1)}\).

Reviewer: Roger Heath-Brown (Oxford)

MSC:

11M26	Nonreal zeros of \(\zeta (s)\) and \(L(s, \chi)\); Riemann and other hypotheses
11N64	Other results on the distribution of values or the characterization of arithmetic functions
11Y55	Calculation of integer sequences

Keywords:

abundant numbers; Riemann hypothesis; computation

Citations:

Zbl 0516.10036

Software:

xrc