A356693 Decimal expansion of the constant B(2) = Sum_{n>=1} Sum_{m>=n+1} 1/(z(n)*z(m))^2 where z(n) is the imaginary part of the n-th nontrivial zero of the Riemann zeta function.

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

Offset

0,4

Links

Table of n, a(n) for n=0..117.

Formula

Equals (A332645^2 - A335815)/2.

Example

0.000248334053789144...

Mathematica

Join[{0, 0, 0}, RealDigits[N[-4*Catalan + Catalan^2/2 - Pi^2/2 + (Catalan*Pi^2)/8 + Pi^4/128 + (1/64)*Zeta[4, 1/4] + (2*Zeta'[1/2]^2)/Zeta[1/2]^2 - (Catalan Zeta'[1/2]^2)/(2 Zeta[1/2]^2) - (Pi^2 Zeta'[1/2]^2)/(16*Zeta[1/2]^2) - Zeta'[1/2]^4/(8*Zeta[1/2]^4) - (2 Zeta''[1/2])/Zeta[1/2] + (Catalan Zeta''[1/2])/(2 Zeta[1/2]) + (Pi^2 Zeta''[1/2])/(16*Zeta[1/2]) + Zeta'[1/2]^2*Zeta''[1/2]/(4 Zeta[1/2]^3) - Zeta'[1/2] Zeta'''[1/2]/(6 Zeta[1/2]^2) + Zeta''''[1/2]/(24  Zeta[1/2]), 115]][[1]]]

Crossrefs

Cf. A355283 (B(3)).

Cf. A013629, A074760, A104539, A104540, A104541, A104542, A245275, A245276, A306339, A306340, A306341, A332645, A333360, A335814, A335815, A355283.

Sequence in context: A344915 A285334 A086317 * A110217 A338196 A243062

Adjacent sequences: A356690 A356691 A356692 * A356694 A356695 A356696

Keyword

nonn,cons

Author

Artur Jasinski, Aug 23 2022

Status

approved