A357331 -id:A357331 - cp's OEIS frontend

A357330 Decimal expansion of sigma(N) / (N * log(log(N))) for N = 5040, where sigma = A000203.

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, 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, 7, 0, 4, 6, 9, 5, 5, 7, 6, 9, 5, 2, 2, 9, 7, 7, 9, 3, 4, 2, 3, 5

Offset: 1

Jianing Song, Sep 24 2022

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.

			sigma(5040) / (5040 * log(log(5040))) = 1.79097336653488113336... In comparison, exp(gamma) = 1.78107241799019798523...

Eric Weisstein's World of Mathematics, Robin's Theorem.
Wikipedia, Riemann hypothesis.

Cf. A067698, A073004, A000203, A001620, A357331.

RealDigits[DivisorSigma[-1, 5040] / Log[Log[5040]], 10, 120][[1]] (* Amiram Eldar, Jun 19 2023 *)

PARI
```
sigma(5040) / (5040 * log(log(5040)))
```

Equals 403 / (105 * log(log(5040))).

cp's OEIS Frontend

A357330 Decimal expansion of sigma(N) / (N * log(log(N))) for N = 5040, where sigma = A000203.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula