A221711 Decimal expansion of sum 1/(p^2 * log p) over the primes p=2,3,5,7,11,...

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

Offset

0,1

References

Henri Cohen, Number Theory, Volume II: Analytic and Modern Tools, GTM Vol. 240, Springer, 2007; see pp. 208-209.

Links

Table of n, a(n) for n=0..105.

Karim Belabas and Henri Cohen, Computation of sum_{p prime} 1/(p^s log(p)), PARI/GP script, 2020.

Henri Cohen, High-precision calculation of Hardy-Littlewood constants, (1998).

Henri Cohen, High-precision computation of Hardy-Littlewood constants. [pdf copy, with permission]

R. J. Mathar, Twenty digits of some integrals of the prime zeta function, arXiv:0811.4739 [math.NT], 2009-2018, Section 2.4.

Example

0.50778218785919931877437510379470557...

Mathematica

digits = 106; precision = digits + 15;
tmax = 400; (* integrand considered negligible beyond tmax *)
kmax = 400; (* f(k) considered negligible beyond kmax *)
InLogZeta[k_] := NIntegrate[Log[Zeta[t]], {t, k, tmax},
   WorkingPrecision -> precision, MaxRecursion -> 20,
   AccuracyGoal -> precision];
f[k_] := With[{mu = MoebiusMu[k]}, If[mu == 0, 0, (mu/k^2)*InLogZeta[2k]]];
s = 0; Do[s = s + f[k]; Print[k, " ", s], {k, 1, kmax}];
RealDigits[s][[1]][[1 ;; digits]] (* Jean-François Alcover, Feb 06 2021, updated Jun 23 2022 *)

Prog

(PARI) See Belabas, Cohen link. Run as SumEulerlog(2) after setting the required precision.
(PARI) default(realprecision, 200); s=0; for(k=1, 300, s = s + moebius(k)/k^2 * intnum(x=2*k, [[1], 1], log(zeta(x))); print(s)); \\ Vaclav Kotesovec, Jun 12 2022

Crossrefs

Cf. A137245.

Sequence in context: A021201 A112254 A180660 * A200400 A190147 A346190

Adjacent sequences: A221708 A221709 A221710 * A221712 A221713 A221714

Keyword

nonn,cons

Author

N. J. A. Sloane, Jan 26 2013

Extensions

More terms from Hugo Pfoertner, Feb 01 2020

More digits from Vaclav Kotesovec, Jun 12 2022

Status

approved