A383224 Decimal expansion Sum_{p primes} log(p)^2*p^2/(p^2-1)^2.

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

Offset: 0

Author

Artur Jasinski, Apr 27 2025

Keywords

• nonn
• cons

Examples

			0.8844818339635238851965361...
		

Links

• Bill Allombert, SumEulerLog procedure, Pari gp procedures.

Crossrefs

Cf. A345364.

Programs

Maple

Zeta(2,2)/Zeta(2) -Zeta(1,2)^2/Zeta(2)^2 ; evalf(%) ; # R. J. Mathar, May 07 2025

Mathematica

RealDigits[(6 (-6 Zeta'[2]^2 + Pi^2 Zeta''[2]))/Pi^4, 10, 105][[1]]

PARI

/* Procedure by Bill Allombert */
default(realprecision, 105);
SumEulerLog(f,s=1,a=2,d=1)=
{
  my(p=variable(f));
  if(type(d)!="t_INT",error("incorrect type in SumEulerLog"));
  if (d<0,
    d=-d;
    for(i=1,d, f=deriv(f)*p);
    (-1)^d*intnum(t=1,[oo,log(2)*s],(t-1)^(d-1)*sumeulerrat(f,t*s,a))/gamma(d)
    ,d==0,
    sumeulerrat(f,s,a)
    ,d>0,
    my(S=0,v);
    my(prec=getlocalbitprec());
    f=subst(f,'p,1/p)+O(p^prec);
    for(i=1,d, f=intformal(f/p));
    v = valuation(f,p);
    f = truncate(f);
    for(i=v,prec/(v-1),
     S += polcoef(f,i)*derivnum(t=1,sumeulerrat(1/p,t*i*s,a),d));
    (-1)^d*S);
}
SumEulerLog(p^2/(p^2-1)^2,,,2)

Formula

Equals 6*(Pi^2*zeta''(2)-6*zeta'(2)^2)/Pi^4.

Equals 6*(Pi^2*zeta''(2)-6*(zeta[2]*(gamma + log(2*Pi) - 12*log(A)))^2)/Pi^4 where A is Glaisher-Kinkelin constant A074962.

Equals zeta''(2)/zeta(2)-zeta'(2)^2/zeta(2)^2 see A201994, A073002 and A013661.