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

1,2

Comments

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

Links

Table of n, a(n) for n=1..104.

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

Cf. A336802.

Sequence in context: A145594 A194731 A021849 * A201754 A070474 A070597

Adjacent sequences: A338459 A338460 A338461 * A338463 A338464 A338465

Keyword

nonn,cons

Author

Artur Jasinski, Jan 31 2021

Status

approved