A340711 Decimal expansion of Product_{primes p == 3 (mod 5)} (p^2+1)/(p^2-1).

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

Offset

1,2

Links

Vaclav Kotesovec, Table of n, a(n) for n = 1..500

Salma Ettahri, Olivier Ramaré, and Léon Surel, Fast multi-precision computation of some Euler product, arXiv:1908.06808 [math.NT], 2019, p. 20.

Steven Finch, Quartic and Octic Characters Modulo n, arXiv:1008.2547 [math.NT], 2007-2010 p. 11 (formula on kappa(5) and kappa(-5)).

Steven Finch, Greg Martin and Pascal Sebah, Roots of unity and nullity modulo n, Proc. Amer. Math. Soc. Volume 138, Number 8, August 2010, pp. 2729-2743.

Steven Finch and Pascal Sebah, Residue of a Mod 5 Euler Product, arXiv:0912.3677 [math.NT], 2009 (formulas).

Alessandro Languasco and Alessandro Zaccagnini, A note on Mertens' formula for arithmetic progressions, Journal of Number Theory Volume 127, Issue 1, (2007), 37-46.

Alessandro Languasco and Alessandro Zaccagnini, On the constant in the Mertens product for arithmetic progressions. II: Numerical values, Math. Comp. 78 (2009), 315-326.

Alessandro Languasco and Alessandro Zaccagnini, On the constant in the Mertens product for arithmetic progressions. I. Identities, Funct. Approx. Comment. Math. Volume 42, Number 1 (2010), 17-27.

Alessandro Languasco and Alessandro Zaccagnini, Computation of the Mertens constants - more than 100 correct digits, (2007), 1-134 (digital data).

Alessandro Languasco and Alessandro Zaccagnini, Computation of the Mertens constants mod q; 3 <= q <= 100, (2007) (GP-PARI procedure 100 digits accuracy).

Richard J. Mathar,  Table of Dirichlet L-Series and Prime Zeta Modulo Functions for Small Moduli, arXiv:1008.2547 [math.NT], 2010-2015.

G. Niklasch, Some number-theoretical constants arising as products of rational functions of p over primes

Doug S. Phillips and Peter Zvengrowski, Convergence of Dirichlet Series and Euler Products, Contributions Section of Natural Mathematical and Biotechnical Sciences 38(2):153 (2017).

Formula

D = Product_{primes p == 0 (mod 5)} (p^2+1)/(p^2-1) = 13/12.

E = Product_{primes p == 1 (mod 5)} (p^2+1)/(p^2-1) = A340629.

F = Product_{primes p == 2 (mod 5)} (p^2+1)/(p^2-1) = A340710.

G = Product_{primes p == 3 (mod 5)} (p^2+1)/(p^2-1) = this constant.

H = Product_{primes p == 4 (mod 5)} (p^2+1)/(p^2-1) = A340628.

D*E*F*G*H = 5/2.

E*F*G*H = 30/13.

D*E*H = sqrt(5)/2.

D*F*G = 13*sqrt(5)/12.

F*G = sqrt(5).

E*H = 6*sqrt(5)/13.

Equals Sum_{q in A004617} 2^A001221(q)/q^2. - R. J. Mathar, Jan 27 2021

Example

1.273986613206833925158...

Mathematica

(* Using Vaclav Kotesovec's function Z from A301430. *)
$MaxExtraPrecision = 1000; digits = 121;
digitize[c_] := RealDigits[Chop[N[c, digits]], 10, digits - 1][[1]];
digitize[1/(Z[5, 3, 4]/Z[5, 3, 2]^2)]

Crossrefs

Cf. A175646, A175647, A248930, A248938, A301429, A333240, A334826, A335963, A340576, A340577, A340578, A340628, A340629, A340004, A340127, A340710.

Sequence in context: A021891 A256614 A021369 * A242304 A227415 A361207

Adjacent sequences: A340708 A340709 A340710 * A340712 A340713 A340714

Keyword

nonn,cons

Author

Artur Jasinski, Jan 16 2021

Status

approved