Delta-method

Summary

mprime[u_]:=Simplify[Residue[Residue[Zeta[s+1]^3Zeta[w+1]^3u^(s+w)A[s,w],{s,0}],{w,0}]];
m3[X_]:=Integrate[mprime[u],u]//.u->X;
m3[X]

The constants

A[0,0]:=0.2177787166195363783230075141194468131307977550013559376482764035236264911122526205579254438235637656918339357748032`100.;
Derivative[0,1][SS][0,0]:=2.529066173580929929259587129301894592300092239944399761188992562701357800662864774853625094702443`75.;
Derivative[0,2][SS][0,0]:=-2.793739632789949812117690423089539370154084193816941952109962433096011953452217952624829356255765`75.;
Derivative[1,0][SS][0,0]:=Derivative[0,1][SS][0,0];
Derivative[2,0][SS][0,0]:=Derivative[0,2][SS][0,0];
Derivative[2,1][SS][0,0]:=13.924949838246429023458888451222757226018087649989983039084909096447051662291585052499046880855312`75.;
Derivative[1,1][SS][0,0]:=-6.489224086802580787969531603193559497143899957312797032696395178579677645073585920960349813071894`75.;
Derivative[1,2][SS][0,0]:=Derivative[2,1][SS][0,0];
Derivative[2,2][SS][0,0]:=-51.561612317854622568503183873771816289674440542630587844482317245914668069010854192175426106119782`75.;
Derivative[0,1][A][s_,w_]:=A[s,w]*Derivative[0,1][SS][s,w];
Derivative[0,2][A][s_,w_]:=A[s,w] ((Derivative[0,1][SS][s,w])^2)+A[s,w] Derivative[0,2][SS][s,w];
Derivative[1,0][A][s_,w_]:=A[s,w]*Derivative[1,0][SS][s,w];
Derivative[2,0][A][s_,w_]:=A[s,w] ((Derivative[1,0][SS][s,w])^2)+A[s,w] Derivative[2,0][SS][s,w];
Derivative[2,1][A][s_,w_]:=A[s,w] Derivative[0,1][SS][s,w] ((Derivative[1,0][SS][s,w])^2)+2 A[s,w] Derivative[1,0][SS][s,w] Derivative[1,1][SS][s,w]+A[s,w] Derivative[0,1][SS][s,w] Derivative[2,0][SS][s,w]+A[s,w] Derivative[2,1][SS][s,w];
Derivative[1,1][A][s_,w_]:=A[s,w] Derivative[0,1][SS][s,w] Derivative[1,0][SS][s,w]+A[s,w] Derivative[1,1][SS][s,w];
Derivative[1,2][A][s_,w_]:=A[s,w] ((Derivative[0,1][SS][s,w])^2) Derivative[1,0][SS][s,w]+A[s,w] Derivative[0,2][SS][s,w] Derivative[1,0][SS][s,w]+2 A[s,w] Derivative[0,1][SS][s,w] Derivative[1,1][SS][s,w]+A[s,w] Derivative[1,2][SS][s,w];
Derivative[2,2][A][s_,w_]:=A[s,w] ((Derivative[0,1][SS][s,w])^2) ((Derivative[1,0][SS][s,w])^2)+A[s,w] Derivative[0,2][SS][s,w] ((Derivative[1,0][SS][s,w])^2)+4 A[s,w] Derivative[0,1][SS][s,w] Derivative[1,0][SS][s,w] Derivative[1,1][SS][s,w]+2 A[s,w] ((Derivative[1,1][SS][s,w])^2)+2 A[s,w] Derivative[1,0][SS][s,w] Derivative[1,2][SS][s,w]+A[s,w] ((Derivative[0,1][SS][s,w])^2) Derivative[2,0][SS][s,w]+A[s,w] Derivative[0,2][SS][s,w] Derivative[2,0][SS][s,w]+2 A[s,w] Derivative[0,1][SS][s,w] Derivative[2,1][SS][s,w]+A[s,w] Derivative[2,2][SS][s,w];
Derivative[0,1][A][0,0]:=Derivative[0,1][A][s,w]//.{s->0,w->0};
Derivative[0,2][A][0,0]:=Derivative[0,2][A][s,w]//.{s->0,w->0};
Derivative[1,0][A][0,0]:=Derivative[1,0][A][s,w]//.{s->0,w->0};
Derivative[2,0][A][0,0]:=Derivative[2,0][A][s,w]//.{s->0,w->0};
Derivative[2,1][A][0,0]:=Derivative[2,1][A][s,w]//.{s->0,w->0};
Derivative[1,1][A][0,0]:=Derivative[1,1][A][s,w]//.{s->0,w->0};
Derivative[1,2][A][0,0]:=Derivative[1,2][A][s,w]//.{s->0,w->0};
Derivative[2,2][A][0,0]:=Derivative[2,2][A][s,w]//.{s->0,w->0};

Calculating the constants

Definitions

AA[s_,w_]:=Log[1-(1-((p (((1-(p^(-1-s))))^3))/(-1+p))) (1-((p (((1-(p^(-1-w))))^3))/(-1+p)))];

A[0,0]

A[p_]:=1-4/p^2+4/p^3-1/p^4;
N[Product[A[p]//.p->Prime[n],{n,1,10^4}],10]
Block[{$MaxExtraPrecision=250},Do[CC=Join[{0},Series[Log[A[p]//.p->1/x],{x,0,t}][[3]]];
Print[N[Exp[Sum[CC[[k]]*(PrimeZetaP[k]-1/2^k),{k,2,Length[CC]}]]*A[2],100]],{t,100,1000,100}]]

Derivative[0,1][AA][0,0]

Simplify[Derivative[0,1][AA][0,0]]

N[Sum[Derivative[0,1][AA][0,0]//.p->Prime[n],{n,1,10^4}],10]
Block[{$MaxExtraPrecision=1000},Do[CC=Join[{0},Series[Derivative[0,1][AA][0,0]/Log[p]//.p->1/x,{x,0,t}][[3]]];
Print[N[-Sum[CC[[k]]*(PrimeZetaP'[k]+Log[2]/2^k),{k,1,Length[CC]}]+Derivative[0,1][AA][0,0]//.p->2,100]],{t,1000,1100,100}]]

N[Sum[Derivative[0,1][AA][0,0]//.p->Prime[n],{n,1,10^4}],20]
Block[{$MaxExtraPrecision=1000},Do[CC=Join[{0},Series[Derivative[0,1][AA][0,0]/Log[p]//.p->1/x,{x,0,t}][[3]]];
Print[N[-Sum[CC[[k]]*(PrimeZetaP'[k]+Log[2]/2^k),{k,1,Length[CC]}]+Derivative[0,1][AA][0,0]//.p->2,75]],{t,1000,1500,100}]]

Derivative[0,1][SS][0,0]:=2.529066173580929929259587129301894592300092239944399761188992562701357800662864774853625094702443`75.;

Derivative[0,2][AA][0,0]

Simplify[Derivative[0,2][AA][0,0]]

N[Sum[Derivative[0,2][AA][0,0]//.p->Prime[n],{n,1,10^4}],20]
Block[{$MaxExtraPrecision=1000},Do[CC=Join[{0},Series[Derivative[0,2][AA][0,0]/Log[p]^2//.p->1/x,{x,0,t}][[3]]];
Print[N[Sum[CC[[k]]*(PrimeZetaP''[k]-Log[2]^2/2^k),{k,1,Length[CC]}]+Derivative[0,2][AA][0,0]//.p->2,100]],{t,950,1000,25}]]

N[Sum[Derivative[0,2][AA][0,0]//.p->Prime[n],{n,1,10^4}],20]
Block[{$MaxExtraPrecision=1000},Do[CC=Join[{0},Series[Derivative[0,2][AA][0,0]/Log[p]^2//.p->1/x,{x,0,t}][[3]]];
Print[N[Sum[CC[[k]]*(PrimeZetaP''[k]-Log[2]^2/2^k),{k,1,Length[CC]}]+Derivative[0,2][AA][0,0]//.p->2,75]],{t,500,1000,100}]]

Derivative[0,2][SS][0,0]:=-2.793739632789949812117690423089539370154084193816941952109962433096011953452217952624829356255765`75.;

Derivative[1,0][AA][0,0]

Simplify[Derivative[1,0][AA][0,0]]

Derivative[1,0][SS][0,0]:=Derivative[0,1][SS][0,0]

Simplify[Derivative[2,0][AA][0,0]]

Derivative[2,0][SS][0,0]:=Derivative[0,2][SS][0,0]

Simplify[Derivative[2,1][AA][0,0]]

N[Sum[Derivative[2,1][AA][0,0]//.p->Prime[n],{n,1,10^4}],10]
Block[{$MaxExtraPrecision=1000},Do[CC=Join[{0},Series[Simplify[Derivative[2,1][AA][0,0]/Log[p]^3]//.p->1/x,{x,0,t}][[3]]];
Print[N[-Sum[CC[[k]]*(PrimeZetaP'''[k]+Log[2]^3/2^k),{k,1,Length[CC]}]+Derivative[2,1][AA][0,0]//.p->2,100]],{t,950,1000,50}]]

N[Sum[Derivative[2,1][AA][0,0]//.p->Prime[n],{n,1,10^4}],20]
Block[{$MaxExtraPrecision=1000},Do[CC=Join[{0},Series[Simplify[Derivative[2,1][AA][0,0]/Log[p]^3]//.p->1/x,{x,0,t}][[3]]];
Print[N[-Sum[CC[[k]]*(PrimeZetaP'''[k]+Log[2]^3/2^k),{k,1,Length[CC]}]+Derivative[2,1][AA][0,0]//.p->2,75]],{t,500,1000,100}]]

Block[{$MaxExtraPrecision=1500},Do[CC=Join[{0},Series[Simplify[Derivative[2,1][AA][0,0]/Log[p]^3]//.p->1/x,{x,0,t}][[3]]];
Print[N[-Sum[CC[[k]]*(PrimeZetaP'''[k]+Log[2]^3/2^k),{k,1,Length[CC]}]+Derivative[2,1][AA][0,0]//.p->2,75]],{t,1000,1100,50}]]

Derivative[2,1][SS][0,0]:=13.924949838246429023458888451222757226018087649989983039084909096447051662291585052499046880855312`75.;

Derivative[1,1][AA][0,0]

Simplify[Derivative[1,1][AA][0,0]]

N[Sum[Simplify[Derivative[1,1][AA][0,0]]//.p->Prime[n],{n,1,10^4}],10]
Block[{$MaxExtraPrecision=1000},Do[CC=Join[{0},Series[Simplify[Derivative[1,1][AA][0,0]/Log[p]^2]//.p->1/x,{x,0,t}][[3]]];
Print[N[Sum[CC[[k]]*(PrimeZetaP''[k]-Log[2]^2/2^k),{k,1,Length[CC]}]+Derivative[1,1][AA][0,0]//.p->2,100]],{t,950,1000,50}]]

N[Sum[Simplify[Derivative[1,1][AA][0,0]]//.p->Prime[n],{n,1,10^4}],20]
Block[{$MaxExtraPrecision=1000},Do[CC=Join[{0},Series[Simplify[Derivative[1,1][AA][0,0]/Log[p]^2]//.p->1/x,{x,0,t}][[3]]];
Print[N[Sum[CC[[k]]*(PrimeZetaP''[k]-Log[2]^2/2^k),{k,1,Length[CC]}]+Derivative[1,1][AA][0,0]//.p->2,75]],{t,500,1000,100}]]

Derivative[1,1][SS][0,0]:=-6.489224086802580787969531603193559497143899957312797032696395178579677645073585920960349813071894`75.;

Derivative[1,2][AA][0,0]

Simplify[Derivative[1,2][AA][0,0]-Derivative[2,1][AA][0,0]]

Derivative[1,2][SS][0,0]:=Derivative[2,1][SS][0,0]

Derivative[0,1][AA][0,0]

Simplify[Derivative[0,1][AA][0,0]-Derivative[1,0][AA][0,0]]

Derivative[0,1][SS][0,0]:=Derivative[1,0][SS][0,0];

Derivative[2,2][AA][0,0]

Simplify[Derivative[2,2][AA][0,0]]

N[Sum[Simplify[Derivative[2,2][AA][0,0]]//.p->Prime[n],{n,1,10^4}],10]
Block[{$MaxExtraPrecision=1000},Do[CC=Join[{0},Series[Simplify[Derivative[2,2][AA][0,0]/Log[p]^4]//.p->1/x,{x,0,t}][[3]]];
Print[N[Sum[CC[[k]]*(PrimeZetaP''''[k]-Log[2]^4/2^k),{k,1,Length[CC]}]+Derivative[2,2][AA][0,0]//.p->2,100]],{t,950,1000,50}]]

N[Sum[Simplify[Derivative[2,2][AA][0,0]]//.p->Prime[n],{n,1,10^4}],20]
Block[{$MaxExtraPrecision=1000},Do[CC=Join[{0},Series[Simplify[Derivative[2,2][AA][0,0]/Log[p]^4]//.p->1/x,{x,0,t}][[3]]];
Print[N[Sum[CC[[k]]*(PrimeZetaP''''[k]-Log[2]^4/2^k),{k,1,Length[CC]}]+Derivative[2,2][AA][0,0]//.p->2,75]],{t,500,1000,100}]]

Block[{$MaxExtraPrecision=1500},Do[CC=Join[{0},Series[Simplify[Derivative[2,2][AA][0,0]/Log[p]^4]//.p->1/x,{x,0,t}][[3]]];
Print[N[Sum[CC[[k]]*(PrimeZetaP''''[k]-Log[2]^4/2^k),{k,1,Length[CC]}]+Derivative[2,2][AA][0,0]//.p->2,75]],{t,1000,1150,50}]]

Derivative[2,2][SS][0,0]:=-51.561612317854622568503183873771816289674440542630587844482317245914668069010854192175426106119782`75.;

A[p_]:=1-(4/(p^2))+(4/(p^3))-(1/(p^4));
N[Product[A[p]//.p->Prime[n],{n,1,10^4}],10]
Block[{$MaxExtraPrecision=350},Do[CC=Join[{0},Series[Log[A[p]//.p->1/x],{x,0,t}][[3]]];
Print[N[Exp[Sum[CC[[k]]*(PrimeZetaP[k]-1/2^k),{k,2,Length[CC]}]]*A[2],100]],{t,1100,1200,50}]]