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%I #25 May 25 2023 07:27:12
%S 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,
%T 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,
%U 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
%N Decimal expansion of Nicolas's constant in his condition for the Riemann Hypothesis (RH).
%C 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).
%H Jeffrey C. Lagarias, <a href="https://doi.org/10.1090/S0273-0979-2013-01423-X">Euler's constant: Euler's work and modern developments</a>, Bull. A.M.S., 50 (2013), 527-628; see p. 574.
%H Jean-Louis Nicolas, <a href="http://dx.doi.org/10.4064%2Faa155-3-7">Small values of the Euler function and the Riemann hypothesis</a>, Acta Arith., Vol. 155, No. 3 (2012), pp. 311-321; <a href="http://arxiv.org/abs/1202.0729">arXiv preprint</a>, arXiv:1202.0729 [math.NT], 2012.
%F Equals e^gamma*(4 + gamma - log(4*Pi)), where gamma is the Euler-Mascheroni constant.
%F Equals e^gamma*(2 + beta), where beta = Sum 1/(rho*(1-rho)), where rho runs over all nonreal zeros of the zeta function.
%e 3.64441509640737014106511619283514816005226024664324245685246375826374...
%t RealDigits[Exp[EulerGamma]*(4 + EulerGamma - Log[4*Pi]), 10, 120][[1]] (* _Amiram Eldar_, May 25 2023 *)
%o (PARI) exp(Euler)*(4 + Euler - log(4*Pi)) \\ _Charles R Greathouse IV_, Mar 10 2016
%Y Cf. A000010, A001620, A195423, A216868, A218245.
%K nonn,cons
%O 1,1
%A _Jonathan Sondow_, Dec 19 2013
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