A175647 Decimal expansion of the Product_{primes p == 1 (mod 4)} 1/(1-1/p^2).

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

Offset: 1

Author

R. J. Mathar, Aug 01 2010

Keywords

• cons
• nonn

Comments

The Euler product of the Riemann zeta function at 2 restricted to primes in A002144, which is the inverse of the infinite product (1-1/5^2)*(1-1/13^2)*(1-1/17^2)*(1-1/29^2)*...

There is a complementary Product_{primes p == 3 (mod 4)} 1/(1-1/p^2) = 1.16807558541051428866969673706404040136467... such that (this constant here)*1.16807.../(1-1/2^2) = zeta(2) = A013661.

Examples

			1.0561821217268161417379307653162198905...
		

Links

• S. Ettahri, O. Ramare, L. Surel, Fast multi-precision computation of some Euler products, arxiv:1908.06808 (2019), Section 9.
• R. J. Mathar, Table of Dirichlet L-series and Prime Zeta Modulo Functions for Small Moduli, arXiv:1008.2547 [math.NT], 2010-2015.

Crossrefs

Cf. A004613, A004614, A013661, A175646.

Programs

Mathematica

digits = 105;
LandauRamanujanK = 1/Sqrt[2]*NProduct[((1 - 2^(-2^n))*Zeta[2^n]/  DirichletBeta[2^n])^(1/2^(n+1)), {n, 1, 24}, WorkingPrecision -> digits+5];
RealDigits[1/(4*LandauRamanujanK/Pi)^2, 10, digits][[1]] (* Jean-François Alcover, Jan 12 2021 *)

Formula

Equals 1/A088539. - Vaclav Kotesovec, May 05 2020

From Amiram Eldar, Sep 27 2020: (Start)

Equals Sum_{k>=1} 1/A004613(k)^2.

The complementary product equals Sum_{k>=1} 1/A004614(k)^2. (End)

Extensions

More digits from Vaclav Kotesovec, Jun 27 2020