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%I #30 Dec 25 2021 12:02:34
%S 1,3,7,2,8,1,3,4,6,2,8,1,8,2,4,6,0,0,9,1,1,2,1,9,2,6,9,6,7,2,7,0,1,8,
%T 8,6,8,1,7,8,3,3,3,1,0,1,2,5,5,7,5,9,5,5,7,9,3,6,2,3,4,1,4,7,3,2,7,8,
%U 4,2,2,2,6,7,1,7,3,7,0,2,3,1,7,2,7,7,1
%N Decimal expansion of constant Product_{p>=3} (1 - (-1)^((p-1)/2)/(p-1)). Hardy-Littlewood constant of x^2 + 1.
%C Arises in studying A002496.
%C The constant is Product_{primes p} (1-chi(p)/(p-1)) where chi is the Dirichlet character A101455. Its Euler expansion is (1/(L(m=4,r=2,s=1)* zeta(m=4,n=3,s=2)) *Product_{s>=2} zeta(m=4,n=1,s)^gamma(s), where L and zeta are the functions tabulated in arXiv:1008.2547 and gamma is the sequence A001037. In particular L(m=4,r=2,s=1) = A003881 and zeta(m=4,n=1,s=2)=A175647. - _R. J. Mathar_, Nov 29 2011
%H T. Amdeberhan, L. A. Median, V. H. Moll, <a href="http://dx.doi.org/10.1016/j.jnt.2007.05.008">Arithmetical properties of a sequence arising from an arctangent sum</a>, J. Numb. Theory 128 (2008) 1807-1846, eq. (1.10).
%H Karim Belabas, Henri Cohen, <a href="/A221712/a221712.gp.txt">Computation of the Hardy-Littlewood constant for quadratic polynomials</a>, PARI/GP script, 2020.
%H Henri Cohen, <a href="/A221712/a221712.pdf">High-precision computation of Hardy-Littlewood constants</a>, (1998). [pdf copy, with permission]
%H G. H. Hardy and J. E. Littlewood, <a href="https://doi.org/10.1007/BF02403921">Some problems of 'Partitio numerorum'; III: On the expression of a number as a sum of primes</a>, Acta Math., Vol. 44, No. 1 (1923), pp. 1-70. See Section 5.41.
%H Richard J. Mathar, <a href="https://arxiv.org/abs/1008.2547">Table of Dirichlet L-Series and Prime Zeta Modulo Functions for Small Moduli</a>, arXiv:1008.2547 [math.NT], 2010-2015.
%H Marek Wolf, <a href="http://arXiv.org/abs/0803.1456">Search for primes of the form m^2+1</a>, arXiv:0803.1456 [math.NT], 2008-2010.
%e 1.372813462818246009112192696727...
%o (PARI) See Belabas, Cohen link. Run as HardyLittlewood2(x^2+1) after setting the required precision.
%Y Cf. A002496.
%Y Cf. A001037, A003881, A083844, A175647, A221712, A331942.
%Y Equals 2*constant given by A331941.
%K nonn,cons
%O 1,2
%A _N. J. A. Sloane_, Nov 05 2011
%E Extended title, a(30) and beyond from _Hugo Pfoertner_, Feb 16 2020
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