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%I #13 Jan 17 2020 05:40:06
%S 4,0,9,5,0,6,9,0,3,4,1,1,8,9,5,7,6,8,2,4,5,1,1,6,3,9,5,1,8,3,7,9,7,6,
%T 3,7,0,4,3,1,9,9,5,2,9,0,9,8,4,7,1,6,6,3,2,3,4,8,9,0,9,7,6,6,8,2,7,2,
%U 5,6,9,2,7,8,0,6,3,7,6,8,8,9,2,1,2,7,2,9,8,5,0,7,0,4,4,6,0,5,2,8,7,7,5
%N Decimal expansion of the Euler-Kronecker constant (as named by P. Moree) for non-hypotenuse numbers.
%D Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 2.3 Landau-Ramanujan constants, p. 99.
%H Steven R. Finch, <a href="http://arxiv.org/abs/2001.00578">Errata and Addenda to Mathematical Constants.</a> p. 11.
%H Pieter Moree, <a href="http://arxiv.org/pdf/1110.0708.pdf">Counting numbers in multiplicative sets: Landau versus Ramanujan.</a> p. 13, arXiv:1110.0708v1 [math.NT] 4 Oct 2011
%H D. Shanks, <a href="http://dx.doi.org/10.1090/S0025-5718-1964-0159174-9">The second-order term in the asymptotic expansion of B(x)</a>, Mathematics of Computation 18 (1964), pp. 75-86.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Landau-RamanujanConstant.html">Landau-Ramanujan Constant</a>
%F 1 - 2*A244662.
%e -0.40950690341189576824511639518379763704319952909847166323489...
%t digits = 103; m0 = 5; dm = 5; beta[x_] := 1/4^x*(Zeta[x, 1/4] - Zeta[x, 3/4]); L = Pi^(3/2)/Gamma[3/4]^2*2^(1/2)/2; Clear[f]; f[m_] := f[m] = 1/2*(1 - Log[Pi*E^EulerGamma/(2*L)]) - 1/4*NSum[Zeta'[2^k]/Zeta[2^k] - beta'[2^k]/beta[2^k] + Log[2]/(2^(2^k) - 1), {k, 1, m}, WorkingPrecision -> digits + 10]; f[m0]; f[m = m0 + dm]; While[RealDigits[f[m], 10, digits] != RealDigits[f[m - dm], 10, digits], m = m + dm]; RealDigits[1 - 2*f[m] - EulerGamma + Log[Pi] - 4*Log[Gamma[3/4]], 10, digits] // First
%Y Cf. A242013, A244662.
%K nonn,cons
%O 0,1
%A _Jean-François Alcover_, Aug 11 2014
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