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

Author

N. J. A. Sloane, Jan 26 2013

Keywords

• nonn
• cons

Examples

			0.50778218785919931877437510379470557...
		

References

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

Links

• 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.

Crossrefs

Cf. A137245.

Programs

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 *)

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

Extensions

More terms from Hugo Pfoertner, Feb 01 2020

More digits from Vaclav Kotesovec, Jun 12 2022