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

%I A338462 #36 Dec 22 2024 02:30:55
%S A338462 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,
%T A338462 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,
%U A338462 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
%N A338462 Decimal expansion of the constant Pi(5,4).
%C A338462 For general definition of constants Pi(q,a) see Alessandro Languasco and Alessandro Zaccagnini, 2010, p. 19 eq (4).
%H A338462 Alessandro Languasco and Alessandro Zaccagnini, <a href="https://projecteuclid.org/euclid.facm/1269437065">On the constant in the Mertens product for arithmetic progressions. I. Identities</a>, Funct. Approx. Comment. Math. Volume 42, Number 1 (2010), 17-27.
%H A338462 For more links see A340711.
%F A338462 Let
%F A338462 A = Pi(5,1) = 2.9425847722692714928420688949... see A336802.
%F A338462 B = Pi(5,2) = 0.2707208383746805812341970398...
%F A338462 C = Pi(5,3) = 0.68429108588000504123233810749...
%F A338462 D = Pi(5,4) = 1.834460850926379503244479431... this constant.
%F A338462 Then
%F A338462 A*B*C*D = 1 (rule for all Pi(q,n) when product taken by all available q such that gcd(n,q)=1).
%F A338462 A*D = 5/(4*arccsch(2)^2) = 5/(4*log((1+sqrt(5))/2)^2).
%F A338462 B*C = 4*arccsch(2)^2/5 = (4/5)*log((1+sqrt(5))/2)^2.
%F A338462 A/D = 5^4/(4*Pi^4).
%F A338462 A = 25*sqrt(5)/(4*Pi^2*log((1+sqrt(5))/2)).
%F A338462 D = sqrt(5)*Pi^2/(25*log((1+sqrt(5))/2)).
%F A338462 (* formulas of Pascal Sebah personal communicated to _Artur Jasinski_ Feb 01 2021 *)
%F A338462 B = (2/5)*sqrt(5)*log((1 + sqrt(5))/2)/exp(arctan(1/2)).
%F A338462 C = 2*sqrt(5)*exp(arctan(1/2))*log((1 + sqrt(5))/2)/5.
%F A338462 C/B = exp(2*arctan(1/2)) = exp(2*arccot(2)).
%e A338462 1.834460850926379503244479431...
%t A338462 RealDigits[N[Sqrt[5] Pi^2/(25 Log[(1 + Sqrt[5])/2]), 104]][[1]]
%t A338462 (* 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 *)
%t A338462 Lvalue[q_, j_] := (-1/q)*Sum[DirichletCharacter[q, j, b]*PolyGamma[b/q*1.0`150], {b, 1, q - 1}];
%t A338462 Lprod[q_] := For[a = 1, a < q, a++,If[GCD[a, q] == 1,
%t A338462    Print["PI(", q, ",", a, ") = ",Re[Exp[-Sum[Log[Lvalue[q, j]]*(Conjugate[
%t A338462            DirichletCharacter[q, j, a]]), {j, 2, EulerPhi[q]}]]]],]];For[r = 3, r <= 24, r++, Print["q = ", r]; Lprod[r]; Print["-----"]]
%Y A338462 Cf. A336802.
%K A338462 nonn,cons
%O A338462 1,2
%A A338462 _Artur Jasinski_, Jan 31 2021