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

Author

Jonathan Sondow, Dec 19 2013

Keywords

• nonn
• cons

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).

Examples

			3.64441509640737014106511619283514816005226024664324245685246375826374...
		

Links

• 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.

Crossrefs

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

Programs

Mathematica

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

PARI

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

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.