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A338462
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Decimal expansion of the constant Pi(5,4).
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0
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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
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OFFSET
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1,2
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COMMENTS
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For general definition of constants Pi(q,a) see Alessandro Languasco and Alessandro Zaccagnini, 2010, p. 19 eq (4).
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LINKS
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FORMULA
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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)).
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EXAMPLE
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1.834460850926379503244479431...
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MATHEMATICA
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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["-----"]]
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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