Code for Electronic Supplement

FindRoots

FindRoots[f_,{x_,a_,b_},opts___]:=Module[{p,seeds},
p=Plot[f,{x,a-((b-a)/100),b+((b-a)/100)},Mesh->{{0}},MeshFunctions->(f/.x->#1&),Evaluate[Sequence@@FilterRules[{opts},First/@Options[Graphics]]]];seeds=Cases[Normal[p],Point[{z_,_}]:>z,∞]⋃(SameTest->(Abs[#1-#2]<(10^(-6))&));Select[If[seeds=={},{},Table[Re[x]/.FindRoot[f==0,{x,seed},Evaluate[Sequence@@FilterRules[{opts},First/@Options[FindRoot]]]],{seed,seeds}]⋃(SameTest->(Abs[#1-#2]<(1/(10^6))&))],a≤#1≤b&]]

Bornemann Formula for R(x), by A. Pavlyk (included in version 7 of Matheamtica)

Clear[CompensatedRiSeries];
CompensatedRiSeries[logx_]:=Module[{pest,serr=0,its=1,err=0,tsc,maxtsc=-Infinity,mintsc=Infinity,evalprec,oldsum=0,sum=1,term=1,pow=1,prec=Internal`PrecAccur[logx],pwr=1},evalprec=prec;
If[Abs[logx]>1,evalprec+=Abs[logx]];
While[!PossibleZeroQ[N[sum,prec]-N[oldsum,prec]],
term=N[Zeta[its+1],evalprec];
NumericalMath`FixedPrecisionEvaluate[pwr*=logx/its;
oldsum=sum;
term=Divide[pwr,its*term];
If[RealExponent[sum]>(tsc=RealExponent[term]),
{sum,err}=Developer`CompensatedAdd[sum,term,err],
{sum,err}=Developer`CompensatedAdd[term,sum,err]
];
its++;
serr=Max[serr,If[term==0,Abs[err],Abs[err/term]]];
{maxtsc,mintsc}={Max[maxtsc,tsc],Min[mintsc,tsc]};,evalprec];
];
pest=Min[prec,evalprec-(Ceiling[Log[2,Max[2,its]]]+If[SameQ[serr,0],0,RealExponent[serr]])];
N[sum,pest]
]

PlainRiSeries[logx_]:=Module[{n,pow,prev,res},
{res,prev,pow,n}={1,0,1,1};
While[(res-prev)≠0,
prev=res;
pow*=(logx/n);
res+=pow/(n Zeta[n+1]);
n++;
];
res
];

AsymptoticRiNearZero[tau_Real/;Positive[tau]]:=Module[{trep=Divide[-1,tau^2],sum=0,prev=0,pow,prec,k=3},
prec=Internal`PrecAccur[tau];
pow=Divide[1,Pi tau^3];
sum=Divide[pow,N[3 Zeta[3],prec]];
While[sum-prev≠0,prev=sum;
pow*=trep;
k=k+2;
sum+=Divide[pow,N[k Zeta[k],prec]];
];
prev=sum;
k=1;
z=N[ZetaZero[1],prec];
sum+=Re[Divide[tau^(-z),z Cos[Pi z/2] Zeta'[z]]];
While[sum-prev≠0,k++;
prev=sum;
z=N[ZetaZero[k],prec];
sum+=Re[Divide[tau^(-z),z Cos[Pi z/2] Zeta'[z]]];];
sum]

SetAttributes[RiemannR,{NumericFunction,Listable}]

RiemannR[1]=1;

RiemannR[x_?InexactNumberQ]:=Module[{res,prev,logx,n=1,pow=1},logx=Log[x];
(If[Negative[logx]||Abs[logx]>5,If[Negative[Re[logx]]&&(Re[logx]<-Pi*300||(MachineNumberQ[logx]&&Re[logx]<-7)),AsymptoticRiNearZero[-logx/(2*Pi)],CompensatedRiSeries[logx]],PlainRiSeries[logx]])/;NumberQ[logx]]

The TData set cached

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The Riemann Zeta Function

Code for COA

Column[{im1, im2}, Spacings->2]

.  The Riemann Zeta Function

Zeta[Range[2, 16, 2]]

a=Zeta[5](π^(-5));
Rationalize[a,(10^(-6))]

N[Denominator[Rationalize[((Zeta[5])/(π^5)),(10^(-20000))]],6]

GraphicsRow[{Plot[Zeta[x],{x,-10,10},Frame->True,PlotStyle->Thick,Axes->True,Exclusions->1,FrameTicks->{Range[-10,10,5],{-1,0,1,2},{},{}}],Plot[Zeta[x],{x,-10,0},Frame->True,PlotStyle->Thick,Axes->True,PlotRange->{{-10,0},{-0.07,0.05}},FrameTicks->{Range[-8,0,2],{-0.04,0,0.04},{},{}}]}]

Plot3D[Abs[Zeta[x+I y]],{x,-4,5},{y,-10,40},MeshFunctions->(#3&),MeshStyle->Brown,Mesh->10,PlotPoints->60,MaxRecursion->3,ViewPoint->{8,1,3},Boxed->False,BoxRatios->{5,10,2},AxesLabel->{"x","y",None},AxesEdge->{{1,-1},Automatic,Automatic},Ticks->{Automatic,Range[0,30,10],Range[0,4]},ClippingStyle->None,PlotRange->{0,5}]

Plot3D[(1/(Abs[Zeta[x+I y]])),{x,-4,5},{y,-10,40},MeshFunctions->(#3&),MeshStyle->Brown,Mesh->10,PlotPoints->60,MaxRecursion->3,ViewPoint->{8,1,3},Boxed->False,BoxRatios->{5,10,2},AxesLabel->{"x","y",None},AxesEdge->{{1,-1},Automatic,Automatic},Ticks->{Automatic,Range[0,30,10],Range[0,4]},ClippingStyle->None,PlotRange->{0,5}]

ζ[s_]:=({Re[#1],Im[#1]}&)[Zeta[N[s]]]
ParametricPlot[ζ[(1/2)+I t],{t,0,26},PlotStyle->Thick]
ParametricPlot[ζ[(1/2)+I t],{t,7004.1,7005.32},PlotStyle->Thick]

Plot[{Abs[Zeta[(1/2)+I t]],RiemannSiegelZ[t]},{t,0,100},AspectRatio->1/4, PlotStyle->{{Thickness[0.01], Lighter[Blue]},{ Thickness[0.004], Red}}]

GraphicsRow[{
Plot[RiemannSiegelZ[t],{t,7004.1,7005.3},Ticks->{{7004.1,7004.7,7005.3},{-1.4,-1,-0.6}},PlotStyle->Thick],
Show[Plot[RiemannSiegelZ[t],{t,7005,7005.15},PlotRange->{-0.06,0.015},AxesOrigin->{7005,0},PlotStyle->Thick,Ticks->{{7005.03,7005.08,7005.13},{-0.1,-0.04,-0.02,0.01}}]]}]

FindRoots[RiemannSiegelZ[t],{t,0,145},WorkingPrecision->30,AccuracyGoal->20]

ζZeros=N[Im[ZetaZero[Range[50]]],30]

Plot[RiemannSiegelZ[t], {t, 0, 145}, Epilog-> {PointSize[0.018],Red, Point[{#,0}&/@ ζZeros]}, Ticks->None, Axes->None, Frame->True, PlotStyle->Thick]

.  The Influence of the Zeros of ζ on the Distribution of Primes

π0[x_]:=(1/2) (PrimePi[x+0.001]+PrimePi[x-0.001]);
Plot[{RiemannR[x],π0[x]},{x,0,100},PlotPoints->100,PlotStyle->{{Thick,Lighter[Red]},{Black,Thickness[0.004]}},Exclusions->(x==#&)/@Prime[Range[25]],Epilog->{PointSize[0.01],Point[Table[{Prime[i],i-(1/2)},{i,25}]]}]

Round[LogIntegral[(10^100)]]

ρ=N[ZetaZero[Range[100]]]

μValues=Table[((MoebiusMu[n])/n),{n,154}];
μIndices=Flatten[Position[μValues,_?(#≠0 &)]];
μ=-2 μValues[[μIndices]];

T[x_,k_]:=μ . Re[ExpIntegralEi[((ρ[[k]] Log[N[x]])/(μIndices))]]
Attributes[T]=Listable;

domain=Range[12.,100,(88/200)];

TData=Table[T[domain,i],{i,100}];

Manipulate[ListLinePlot[Transpose[{domain,TData[[i]]}],Frame->True,Axes->False,PlotStyle->Thick,PlotRange->{-0.5,0.5}],{i,1,100,1}]

RiemannData=RiemannR[domain]+(1/π)ArcTan[(π/(Log[domain]))]-(1/(Log[domain]));

TDataEnhanced=Table[Transpose[{domain,ss}],{ss,FoldList[Plus, RiemannData,TData]}];

π0Plot=Plot[π0[x],{x,0,100},PlotPoints->100,PlotStyle->{Red,Thickness[0.006]},Exclusions->(x==#&)/@Prime[Range[25]], Epilog->{PointSize[0.01],Red, Point[Table[{Prime[i],i-1/2}, {i,25}]]}];

Manipulate[Show[π0Plot,
ListLinePlot[TDataEnhanced[[i]],PlotStyle->{Thickness[0.005], Blue}],
Graphics[{PointSize[0.01], Table[Point[{Prime[i], i-1/2}], {i,25}]}],Ticks->{Range[20,100,10],Automatic}, PlotRange->{{10,100},{5,27}},AxesOrigin->{10,5}],{i,1,50,1}]

.  A Backwards Look at Riemann's R(x)

R=RiemannR;
Plot[R[x], {x, 0,1000}]

Table[(((Log[(10.^(-100))])^m)/(m! m Zeta[m+1])),{m,16}]

Plot[R[x],{x,0,(1/(10^8))}]

Plot[Log[Abs[R[(10^x)]]],{x,-45000,2},Frame->True,PlotStyle->{Black,Thickness[0.006]},Axes->False,PlotPoints->50,MaxRecursion->1,PlotRange->{-35,2}]

Plot[R[(10^x)],{x,-20000,-7400},GridLines->{{},{0}},Frame->True,Axes->False]

(10^t)/.FindRoot[R[(10^t)],{t,-15000}]

data=Table[{x,R[(10^x)]},{x,-500000.,-10000,(1/600) (-10+500000)}];

ListLinePlot[data,Frame->True,GridLines->{{-36168},{}},FrameTicks->{{-14000,-100000,-300000,-500000},{-3.,0,6.}(10^(-13)),{-36168},{}},PlotRange->All]

N[FindMinimum[R[(10^t)],{t,-35000},WorkingPrecision->40]]