On SA, CA, and GA numbers. (English) Zbl 1280.11049

Summary: Gronwall’s function \(G\) is defined for \(n>1\) by \(G(n)=\frac{\sigma(n)}{n \log\log n}\) where \(\sigma (n)\) is the sum of the divisors of \(n\). We call an integer \(N>1\) a GA1 number if \(N\) is composite and \(G(N)\geq G(N/p)\) for all prime factors \(p\) of \(N\). We say that \(N\) is a GA2 number if \(G(N)\geq G(aN)\) for all multiples \(aN\) of \(N\). In [Integers 11, No. 6, 753–763, A33 (2011; Zbl 1235.11082)], we used Robin’s and Gronwall’s theorems on \(G\) to prove that the Riemann Hypothesis (RH) is true if and only if 4 is the only number that is both GA1 and GA2. In the present paper, we study GA1 numbers and GA2 numbers separately. We compare them with superabundant (SA) and colossally abundant (CA) numbers (first studied by Ramanujan). We give algorithms for computing GA1 numbers; the smallest one with more than two prime factors is 183783600, while the smallest odd one is 1058462574572984015114271643676625. We find nineteen GA2 numbers \(\leq 5040\), and prove that a GA2 number \(N>5040\) exists if and only if RH is false, in which case \(N\) is even and \(>10^{8576}\).


11M26 Nonreal zeros of \(\zeta (s)\) and \(L(s, \chi)\); Riemann and other hypotheses
11A41 Primes
11Y55 Calculation of integer sequences


Zbl 1235.11082


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[1] Alaoglu, L., Erdos, P.: On highly composite and similar numbers. Trans. Am. Math. Soc. 56, 448–469 (1944) · Zbl 0061.07903
[2] Briggs, K.: Abundant numbers and the Riemann hypothesis. Exp. Math. 15, 251–256 (2006). http://www.expmath.org/expmath/volumes/15/15.2/Briggs.pdf (2006). Accessed 23 October 2011 · Zbl 1149.11041 · doi:10.1080/10586458.2006.10128957
[3] Caveney, G., Nicolas, J.-L., Sondow, J.: Robin’s theorem, primes, and a new elementary reformulation of the Riemann Hypothesis. Integers 11, A33 (2011). http://www.integers-ejcnt.org/l33/l33.pdf (2011). Accessed 23 October 2011 · Zbl 1235.11082 · doi:10.1515/INTEG.2011.057
[4] Choie, Y.-J., Lichiardopol, N., Moree, P., Sole, P.: On Robin’s criterion for the Riemann Hypothesis. J. Théor. Nombres Bordeaux 19, 351–366 (2007). http://arxiv.org/abs/math/0604314 (2006). Accessed 23 October 2011
[5] Dusart, P.: Estimates of some functions over primes without R.H. http://arxiv.org/abs/1002.0442v1 (2010). Accessed 23 October 2011 · Zbl 1426.11088
[6] Erdos, P., Nicolas, J.-L.: Répartition des nombres superabondants. Bull. Soc. Math. Fr. 103, 65–90 (1975). http://www.numdam.org/item?id=BSMF_1975_103_65_0 (1975). Accessed 23 October 2011
[7] Gronwall, T.H.: Some asymptotic expressions in the theory of numbers. Trans. Am. Math. Soc. 14, 113–122 (1913) · doi:10.1090/S0002-9947-1913-1500940-6
[8] Hardy, G.H., Wright, E.M.: An Introduction to the Theory of Numbers. In: Heath-Brown, D.R., Silverman, J.H. (eds.) 6th edn. Oxford University Press, Oxford (2008) · Zbl 1159.11001
[9] Lagarias, J.C.: An elementary problem equivalent to the Riemann hypothesis. Am. Math. Mon. 109, 534–543 (2002) · Zbl 1098.11005 · doi:10.2307/2695443
[10] Littlewood, J.E.: Sur la distribution des nombres premiers. C. R. Acad. Sci. Paris Sér. I Math. 158, 1869–1872 (1914)
[11] Nicolas, J.-L., Robin, G.: Majorations explicites pour le nombre de diviseurs de N. Can. Math. Bull. 26, 485–492 (1983) · doi:10.4153/CMB-1983-078-5
[12] Ramanujan, S.: Highly composite numbers. Proc. Lond. Math. Soc. 14, 347–400 (1915). Also In: Collected Papers, pp. 78–128. Cambridge University Press, Cambridge (1927) · doi:10.1112/plms/s2_14.1.347
[13] Ramanujan, S.: Highly composite numbers, annotated and with a foreword by J.-L. Nicolas and G. Robin. Ramanujan J. 1, 119–153 (1997) · Zbl 0917.11043 · doi:10.1023/A:1009764017495
[14] Robin, G.: Grandes valeurs de la fonction somme des diviseurs et hypothèse de Riemann. J. Math. Pures Appl. 63, 187–213 (1984) · Zbl 0516.10036
[15] Robin, G.: Sur l’ordre maximum de la fonction somme des diviseurs. In: Séminaire Delange-Pisot-Poitou Paris 1981–1982, pp. 233–242. Birkhäuser, Boston (1983)
[16] Schoenfeld, L.: Sharper bounds for the Chebyshev functions {\(\theta\)}(x) and {\(\psi\)}(x). II. Math. Comput. 30, 337–360 (1976) · Zbl 0326.10037
[17] Sloane, N.J.A.: The on-line encyclopedia of integer sequences. http://oeis.org (2011). Accessed 10 December 2011
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