A233825 Decimal expansion of Nicolas's constant in his condition for the Riemann Hypothesis (RH).

3, 6, 4, 4, 4, 1, 5, 0, 9, 6, 4, 0, 7, 3, 7, 0, 1, 4, 1, 0, 6, 5, 1, 1, 6, 1, 9, 2, 8, 3, 5, 1, 4, 8, 1, 6, 0, 0, 5, 2, 2, 6, 0, 2, 4, 6, 6, 4, 3, 2, 4, 2, 4, 5, 6, 8, 5, 2, 4, 6, 3, 7, 5, 8, 2, 6, 3, 7, 4, 1, 7, 3, 4, 8, 0, 9, 2, 9, 5, 8, 1, 8, 6, 8, 3, 2, 3, 0, 5, 7, 0, 5, 1, 7, 5, 1, 2, 6, 1, 6, 1, 5, 5, 6, 4, 1, 4, 3, 3, 5, 5, 3, 1, 7, 7, 5, 2, 9, 2, 7

Offset

1,1

Comments

Nicolas proved that RH is true if and only if limsup_{n-->infinity} (n/phi(n) - e^gamma*log(log(n)))*sqrt(log(n)) = e^gamma*(4 + gamma - log(4*Pi)), where phi(n) = A000010(n).

Links

Table of n, a(n) for n=1..120.

Jeffrey C. Lagarias, Euler's constant: Euler's work and modern developments, Bull. A.M.S., 50 (2013), 527-628; see p. 574.

Jean-Louis Nicolas, Small values of the Euler function and the Riemann hypothesis, Acta Arith., Vol. 155, No. 3 (2012), pp. 311-321; arXiv preprint, arXiv:1202.0729 [math.NT], 2012.

Formula

Equals e^gamma*(4 + gamma - log(4*Pi)), where gamma is the Euler-Mascheroni constant.

Equals e^gamma*(2 + beta), where beta = Sum 1/(rho*(1-rho)), where rho runs over all nonreal zeros of the zeta function.

Example

3.64441509640737014106511619283514816005226024664324245685246375826374...

Mathematica

RealDigits[Exp[EulerGamma]*(4 + EulerGamma - Log[4*Pi]), 10, 120][[1]] (* Amiram Eldar, May 25 2023 *)

Prog

(PARI) exp(Euler)*(4 + Euler - log(4*Pi)) \\ Charles R Greathouse IV, Mar 10 2016

Crossrefs

Cf. A000010, A001620, A195423, A216868, A218245.

Sequence in context: A090038 A308291 A006464 * A351124 A159354 A196500

Adjacent sequences: A233822 A233823 A233824 * A233826 A233827 A233828

Keyword

nonn,cons

Author

Jonathan Sondow, Dec 19 2013

Status

approved