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A357331
Decimal expansion of sigma(N) / (exp(gamma) * N * log(log(N))) for N = 5040, where sigma = A000203 and gamma = A001620 is the Euler-Mascheroni constant.
1
1, 0, 0, 5, 5, 5, 8, 9, 8, 1, 4, 5, 6, 7, 2, 0, 1, 0, 3, 6, 4, 2, 4, 7, 0, 7, 6, 7, 7, 8, 1, 5, 5, 4, 4, 3, 1, 6, 9, 8, 4, 4, 3, 0, 1, 4, 6, 7, 4, 1, 5, 2, 7, 9, 7, 3, 6, 8, 0, 2, 5, 8, 3, 2, 2, 5, 7, 4, 6, 5, 9, 5, 4, 9, 5, 5, 8, 5, 2, 2, 7, 8, 7, 7, 1, 4, 6, 2, 3, 9
OFFSET
1,4
COMMENTS
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.
LINKS
Eric Weisstein's World of Mathematics, Robin's Theorem.
Wikipedia, Riemann hypothesis.
FORMULA
Equals 403 / (exp(gamma) * 105 * log(log(5040))).
EXAMPLE
sigma(5040) / (exp(gamma) * 5040 * log(log(5040))) = 1.00555898145672010364... > 1.
MATHEMATICA
RealDigits[DivisorSigma[-1, 5040] / (Exp[EulerGamma] * Log[Log[5040]]), 10, 120][[1]] (* Amiram Eldar, Jun 19 2023 *)
PROG
(PARI) sigma(5040) / (exp(Euler) * 5040 * log(log(5040)))
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
Jianing Song, Sep 24 2022
STATUS
approved