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, 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

Table of n, a(n) for n=1..90.

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

Cf. A067698, A073004, A000203, A001620, A357330.

Sequence in context: A204911 A087516 A194428 * A299695 A135089 A127310

Adjacent sequences: A357328 A357329 A357330 * A357332 A357333 A357334

Keyword

nonn,cons

Author

Jianing Song, Sep 24 2022

Status

approved