A319231 Decimal expansion of Sum_{p = prime} 1/(p*log(p)^2).

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

Offset

1,2

Comments

Computed by expanding the formalism of arXiv:0811.4739 to double integrals over the Riemann zeta function.

Links

Table of n, a(n) for n=1..105.

R. J. Mathar, Twenty digits of some integrals of the prime zeta function, arXiv:0811.4739 [math.NT], 2008-2018.

Example

1/(2*A253191) + 1/(3*A175478) +1/(5*2.59029...) +1/(7*3.7865)+ ... = 1.52097043...

Mathematica

digits = 105; precision = digits + 10;
tmax = 500; (* integrand considered negligible beyond tmax *)
kmax = 500; (* f(k) considered negligible beyond kmax *)
InLogZeta[k_] := NIntegrate[(t - k) Log[Zeta[t]], {t, k, tmax}, WorkingPrecision -> precision, MaxRecursion -> 20, AccuracyGoal -> precision];
f[k_] := With[{mu = MoebiusMu[k]}, If[mu == 0, 0, (mu/k^3)*InLogZeta[k]]];
s = 0;
Do[s = s + f[k]; Print[k, " ", s], {k, 1, kmax}];
RealDigits[s][[1]][[1 ;; digits]] (* Jean-François Alcover, Jun 21 2022, after Vaclav Kotesovec *)

Prog

(PARI) default(realprecision, 200); s=0; for(k=1, 500, s=s+moebius(k)/k^3 * intnum(x=k, [[1], 1], (x-k)*log(zeta(x))); print(s)); \\ Vaclav Kotesovec, Jun 12 2022

Crossrefs

Cf. A137245, A115563, A221711, A319232, A354917.

Sequence in context: A021196 A257406 A269980 * A058512 A111560 A371026

Adjacent sequences: A319228 A319229 A319230 * A319232 A319233 A319234

Keyword

nonn,cons

Author

R. J. Mathar, Sep 14 2018

Extensions

More digits from Vaclav Kotesovec, Jun 12 2022

Status

approved