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A233825 Decimal expansion of Nicolas's constant in his condition for the Riemann Hypothesis (RH). 3


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

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

%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 J. Lagarias, <a href="http://www.ams.org/journals/bull/2013-50-04/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 J.-L. Nicolas, <a href="http://arxiv.org/abs/1202.0729"> Small values of the Euler function and the Riemann hypothesis</a>, Acta Arith., 155 (2012), 311-321.

%e 3.64441509640737014106511619283514816005226024664324245685246375826374...

%o (PARI) exp(Euler)*(4 + Euler - log(4*pi)) \\ _Charles R Greathouse IV_, Mar 10 2016

%Y Cf. A195423, A216868, A218245.

%K nonn,cons

%O 1,1

%A _Jonathan Sondow_, Dec 19 2013

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Last modified February 16 21:37 EST 2020. Contains 331975 sequences. (Running on oeis4.)