A338462 Decimal expansion of the constant Pi(5,4).

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

Offset: 1

Author

Artur Jasinski, Jan 31 2021

Keywords

• nonn
• cons

Comments

For general definition of constants Pi(q,a) see Alessandro Languasco and Alessandro Zaccagnini, 2010, p. 19 eq (4).

Examples

			1.834460850926379503244479431...
		

Links

• 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.
• For more links see A340711.

Crossrefs

Cf. A336802.

Programs

Mathematica

RealDigits[N[Sqrt[5] Pi^2/(25 Log[(1 + Sqrt[5])/2]), 104]][[1]]
(* 150 digits accuracy fast procedure of Alessandro Languasco to numerical counting of values Pi(q,a) personal communicated to Artur Jasinski Jan 31 2021 and published by permission *)
Lvalue[q_, j_] := (-1/q)*Sum[DirichletCharacter[q, j, b]*PolyGamma[b/q*1.0`150], {b, 1, q - 1}];
Lprod[q_] := For[a = 1, a < q, a++,If[GCD[a, q] == 1,
   Print["PI(", q, ",", a, ") = ",Re[Exp[-Sum[Log[Lvalue[q, j]]*(Conjugate[
           DirichletCharacter[q, j, a]]), {j, 2, EulerPhi[q]}]]]],]];For[r = 3, r <= 24, r++, Print["q = ", r]; Lprod[r]; Print["-----"]]

Formula

Let

A = Pi(5,1) = 2.9425847722692714928420688949... see A336802.

B = Pi(5,2) = 0.2707208383746805812341970398...

C = Pi(5,3) = 0.68429108588000504123233810749...

D = Pi(5,4) = 1.834460850926379503244479431... this constant.

Then

A*B*C*D = 1 (rule for all Pi(q,n) when product taken by all available q such that gcd(n,q)=1).

A*D = 5/(4*arccsch(2)^2) = 5/(4*log((1+sqrt(5))/2)^2).

B*C = 4*arccsch(2)^2/5 = (4/5)*log((1+sqrt(5))/2)^2.

A/D = 5^4/(4*Pi^4).

A = 25*sqrt(5)/(4*Pi^2*log((1+sqrt(5))/2)).

D = sqrt(5)*Pi^2/(25*log((1+sqrt(5))/2)).

(* formulas of Pascal Sebah personal communicated to Artur Jasinski Feb 01 2021 *)

B = (2/5)*sqrt(5)*log((1 + sqrt(5))/2)/exp(arctan(1/2)).

C = 2*sqrt(5)*exp(arctan(1/2))*log((1 + sqrt(5))/2)/5.

C/B = exp(2*arctan(1/2)) = exp(2*arccot(2)).