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%I #12 Jun 19 2023 06:41:18
%S 1,7,9,0,9,7,3,3,6,6,5,3,4,8,8,1,1,3,3,3,6,1,9,0,1,3,5,0,5,9,1,0,9,5,
%T 1,7,4,0,9,0,9,5,3,9,0,7,9,8,7,5,7,3,5,7,7,9,1,7,4,6,5,3,5,2,3,5,6,6,
%U 7,0,4,6,9,5,5,7,6,9,5,2,2,9,7,7,9,3,4,2,3,5
%N Decimal expansion of sigma(N) / (N * log(log(N))) for N = 5040, where sigma = A000203.
%C It is known that the Riemann Hypothesis (RH) is true if and only if sigma(n) < exp(gamma) * n * log(log(n)) for all n > 5040, where gamma = A001620 is the Euler-Mascheroni constant; that is to say, the RH is true if and only if 5040 is the last term in A067698.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RobinsTheorem.html">Robin's Theorem</a>.
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Riemann_hypothesis#Growth_of_arithmetic_functions">Riemann hypothesis</a>.
%F Equals 403 / (105 * log(log(5040))).
%e sigma(5040) / (5040 * log(log(5040))) = 1.79097336653488113336... In comparison, exp(gamma) = 1.78107241799019798523...
%t RealDigits[DivisorSigma[-1, 5040] / Log[Log[5040]], 10, 120][[1]] (* _Amiram Eldar_, Jun 19 2023 *)
%o (PARI) sigma(5040) / (5040 * log(log(5040)))
%Y Cf. A067698, A073004, A000203, A001620, A357331.
%K nonn,cons
%O 1,2
%A _Jianing Song_, Sep 24 2022
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