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

%I A383224 #53 May 07 2025 05:31:00
%S A383224 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,
%T A383224 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,
%U A383224 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
%N A383224 Decimal expansion Sum_{p primes} log(p)^2*p^2/(p^2-1)^2.
%H A383224 Bill Allombert, <a href="https://pari.math.u-bordeaux.fr/Scripts/sumeulerratlog.gp">SumEulerLog procedure</a>, Pari gp procedures.
%F A383224 Equals 6*(Pi^2*zeta''(2)-6*zeta'(2)^2)/Pi^4.
%F A383224 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.
%F A383224 Equals zeta''(2)/zeta(2)-zeta'(2)^2/zeta(2)^2 see A201994, A073002 and A013661.
%e A383224 0.8844818339635238851965361...
%p A383224 Zeta(2,2)/Zeta(2) -Zeta(1,2)^2/Zeta(2)^2 ; evalf(%) ; # _R. J. Mathar_, May 07 2025
%t A383224 RealDigits[(6 (-6 Zeta'[2]^2 + Pi^2 Zeta''[2]))/Pi^4, 10, 105][[1]]
%o A383224 (PARI)
%o A383224 /* Procedure by Bill Allombert */
%o A383224 default(realprecision, 105);
%o A383224 SumEulerLog(f,s=1,a=2,d=1)=
%o A383224 {
%o A383224   my(p=variable(f));
%o A383224   if(type(d)!="t_INT",error("incorrect type in SumEulerLog"));
%o A383224   if (d<0,
%o A383224     d=-d;
%o A383224     for(i=1,d, f=deriv(f)*p);
%o A383224     (-1)^d*intnum(t=1,[oo,log(2)*s],(t-1)^(d-1)*sumeulerrat(f,t*s,a))/gamma(d)
%o A383224     ,d==0,
%o A383224     sumeulerrat(f,s,a)
%o A383224     ,d>0,
%o A383224     my(S=0,v);
%o A383224     my(prec=getlocalbitprec());
%o A383224     f=subst(f,'p,1/p)+O(p^prec);
%o A383224     for(i=1,d, f=intformal(f/p));
%o A383224     v = valuation(f,p);
%o A383224     f = truncate(f);
%o A383224     for(i=v,prec/(v-1),
%o A383224      S += polcoef(f,i)*derivnum(t=1,sumeulerrat(1/p,t*i*s,a),d));
%o A383224     (-1)^d*S);
%o A383224 }
%o A383224 SumEulerLog(p^2/(p^2-1)^2,,,2)
%Y A383224 Cf. A345364.
%K A383224 nonn,cons
%O A383224 0,1
%A A383224 _Artur Jasinski_, Apr 27 2025