Briggs, Keith Abundant numbers and the Riemann hypothesis. (English) Zbl 1149.11041 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) Cited in 1 ReviewCited in 12 Documents 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 × Cite Format Result Cite Review PDF Full Text: DOI Euclid EuDML Online Encyclopedia of Integer Sequences: Superabundant [or super-abundant] numbers: n such that sigma(n)/n > sigma(m)/m for all m < n, sigma(n) being A000203(n), the sum of the divisors of n. Colossally abundant numbers: m for which there is a positive exponent epsilon such that sigma(m)/m^{1 + epsilon} >= sigma(k)/k^{1 + epsilon} for all k > 1, so that m attains the maximum value of sigma(m)/m^{1 + epsilon}. Prime numbers that when multiplied in order yield the sequence of colossally abundant numbers A004490. Integer part of sigma(m)/phi(m) for colossally abundant numbers m. Integer part of sigma(m)*phi(m)/m for colossally abundant numbers m. a(n) = floor( exp(gamma) k log log k ) - sigma(k), where gamma is Euler’s constant (A001620) and sigma(k) is sum of divisors of k (A000203), the n-th colossally abundant number (A004490). Numbers k > 1 with sigma(k)/(k * log(log(k))) > sigma(m)/(m * log(log(m))) for all m > k, sigma(k) being A000203(k), the sum of the divisors of k.