%I #7 Jun 18 2021 21:59:04

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

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

%U 8,0,8,4,4,0,3,7,1,5,8,3,2,8,8,1,9,5,4

%N Decimal expansion of exp(gamma + M)*(G - 7*zeta(3)/(4*Pi))/4, where gamma is Euler's constant (A001620), M is Mertens's constant (A077761) and G is Catalan's constant (A006752).

%C This constant is notable for being the asymptotic limit in a formula derived by Sinha and Wolf (2010) which "brings together the elements from nine different topics of number theory" (see the Formula section).

%H Nilotpal Kanti Sinha and Marek Wolf, <a href="https://arxiv.org/abs/1009.4810">On a unified theory of numbers</a>, arXiv:1009.4810 [math.NT], 2010-2011. See section 8, p. 11, eq. 37.

%F Equals lim_{n->oo} (1/log(n)^2) * Sum_{k=1..n} (1/gamma_k) * (1/k + 1/prime(k)) * (arctan(gamma_k/gamma_n))^2 * exp(H(k) + Sum_{i=1..k} 1/prime(i))), where H(k) = A001008(k)/A002805(k) is the k-th harmonic number, and gamma_k is the imaginary part of the k-th nontrivial zero of the Riemann zeta function.

%e 0.14248676756297667766013119038516485825699065019561...

%t M = EulerGamma - NSum[PrimeZetaP[k]/k, {k, 2, Infinity}, WorkingPrecision -> 300, NSumTerms -> 300]; RealDigits[Exp[EulerGamma + M]*(Catalan - 7*Zeta[3]/(4*Pi))/4, 10, 100][[1]]

%Y Cf. A001620, A002117, A006752, A077761.

%Y Cf. A001008, A002805.

%K nonn,cons

%O 0,2

%A _Amiram Eldar_, Jun 18 2021