A391785 - OEIS

A391785

Decimal expansion of zeta(2) * Product_{p prime} (1 - 2/p^2 + 1/p^(2/5) - 1/p^(3/2) + 1/p^4).

%I #9 Dec 20 2025 10:16:06

%S 7,6,3,3,5,6,4,3,0,2,1,6,2,2,9,5,9,0,0,9,9,9,4,4,3,9,1,6,7,3,2,0,9,1,

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

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

%N Decimal expansion of zeta(2) * Product_{p prime} (1 - 2/p^2 + 1/p^(2/5) - 1/p^(3/2) + 1/p^4).

%H Hendrik Jager, <a href="https://doi.org/10.1007/978-3-0348-5438-2_32">The avarage order of Gaussian sums (sic)</a>, Studies in Pure Mathematics: To the Memory of Paul Turán, Basel: Birkhäuser Basel, 1983, pp. 381-384.

%e 0.763356430216229590099944391673209107742458366621501...

%t res[m_, dig_] := Block[{$MaxExtraPrecision = 2*dig, ps = {2, 3, 5}, c = LinearRecurrence[{0, 0, 0, 2, -1, 0, 1, -1}, {0, 0, 0, -8, 5, 0, -7, -8}, m]}, RealDigits[Zeta[2] * Product[(1 - 2/p^2 + 1/p^(5/2) - 1/p^(7/2) + 1/p^4), {p, ps}] * Exp[NSum[Indexed[c, n]*(PrimeZetaP[n/2] - Sum[1/p^(n/2), {p, ps}])/n, {n, 4, m}, NSumTerms -> m, WorkingPrecision -> m]], 10, dig][[1]]]; With[{dig = 120}, res[FixedPoint[# + 100 &, 10, SameTest -> (Equal[res[#1, dig], res[#2, dig]] &)], dig]]

%o (PARI) zeta(2) * prodeulerrat(1 - 2/p^4 + 1/p^5 - 1/p^7 + 1/p^8, 1/2)

%Y Cf. A013661.

%K nonn,cons

%O 0,1

%A _Amiram Eldar_, Dec 20 2025