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

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 $$n(\log\log n)^{1/2+o(1)}$$.

### 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

Zbl 0516.10036

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