OFFSET
1,2
COMMENTS
For general definition of constants Pi(q,a) see Alessandro Languasco and Alessandro Zaccagnini, 2010, p. 19 eq (4).
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.
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)).
EXAMPLE
1.834460850926379503244479431...
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["-----"]]
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
Artur Jasinski, Jan 31 2021
STATUS
approved