Summary
Denote by P(x) the number of integers n≤x satisfying σ(n)≡0 (mod n), and by P 2 (x) the number of integers n ≤ x satisfying σ(n)=2n. The author proves that P(x)<x3/4+ɛ and P 2 (x)<x(1−c)/2 for a certain c>0.
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Literatur
Dickson,History of the Theorie des Numbers, Vol. 1, Chapter 1.
« Journal für die reine und angew. Math. », 194 (1955), 218–220.
« Archiv der Math. » 6 (1955), 442–443.
Landau,Verteilung der Primzahlen, Vol. 1, p. 217.Landau states his result forEuler's ϕ function, but the result for σ(n) follows immediately.
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Erdös, P. On perfect and multiply perfect numbers. Annali di Matematica 42, 253–258 (1956). https://doi.org/10.1007/BF02411879
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DOI: https://doi.org/10.1007/BF02411879