A199401 Decimal expansion of constant Product_{p>=3} (1 - (-1)^((p-1)/2)/(p-1)). Hardy-Littlewood constant of x^2 + 1.

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, 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, 4, 2, 2, 2, 6, 7, 1, 7, 3, 7, 0, 2, 3, 1, 7, 2, 7, 7, 1

Offset: 1

Author

N. J. A. Sloane, Nov 05 2011

Keywords

• nonn
• cons

Comments

Arises in studying A002496.

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

Examples

			1.372813462818246009112192696727...
		

References

• Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 2.1, p. 85.
• Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See p. 264.

Links

• T. Amdeberhan, L. A. Median, and V. H. Moll, Arithmetical properties of a sequence arising from an arctangent sum, J. Numb. Theory 128 (2008) 1807-1846, eq. (1.10).
• Karim Belabas and Henri Cohen, Computation of the Hardy-Littlewood constant for quadratic polynomials, PARI/GP script, 2020.
• Henri Cohen, High-precision computation of Hardy-Littlewood constants, (1998). [pdf copy, with permission]
• G. H. Hardy and J. E. Littlewood, Some problems of 'Partitio numerorum'; III: On the expression of a number as a sum of primes, Acta Math., Vol. 44, No. 1 (1923), pp. 1-70. See Section 5.41.
• Richard J. Mathar, Table of Dirichlet L-Series and Prime Zeta Modulo Functions for Small Moduli, arXiv:1008.2547 [math.NT], 2010-2015.
• Marek Wolf, Search for primes of the form m^2+1, arXiv:0803.1456 [math.NT], 2008-2010.

Crossrefs

Cf. A002496.

Cf. A001037, A003881, A083844, A175647, A221712, A331942.

Equals 2*constant given by A331941.

Programs

PARI

\\ See Belabas, Cohen link. Run as HardyLittlewood2(x^2+1) after setting the required precision.

Extensions

Extended title, a(30) and beyond from Hugo Pfoertner, Feb 16 2020