Gram Point

Author

Eric W. Weisstein
September 22, 2007

This notebook downloaded from http://mathworld.wolfram.com/notebooks/SpecialFunctions/GramPoint.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/GramPoint.html.

©2007 Wolfram Research, Inc. except for portions noted otherwise

Starting Point

Solve[t/2 Log[t/(2π)]-t/2==n π,t]//FullSimplify

Solve[t/2 Log[t/(2π)]-t/2-π/8==n π,t]//FullSimplify

Solve[t/2 Log[t/(2π)]-t/2-π/8-1/(48t)==n π,t]//FullSimplify

Plot

ListPlot[Table[{n,GramPoint[n]},{n,0,20}],PlotStyle->{PointSize[.02],Red},AxesLabel->TraditionalForm/@{n,(Subscript[g, n])}]

Enumeration

Table

Table[{n,((2 n π)/(ProductLog[(n/E)]))//N,expr1=2 (E^(1+ProductLog[((1+8 n)/(8 E))])) π//N,expr2=GramPoint[n,WorkingPrecision->20,PrecisionGoal->10],expr1-expr2},{n,0,10}]//TableForm

Round

Round[Table[GramPoint[n],{n,0,50}]]

Specific Examples

GramPoint[#,WorkingPrecision->20]&/@{126,134,137,268}

(Subscript[g, n])==n

g[x_?NumericQ]:=GramPoint[x]-x

FindRoot[g[x],{x,8000,12000},WorkingPrecision->100]

Solve[RiemannSiegelTheta[x]==π x,x]

FindRoot[RiemannSiegelTheta[x]-π x,{x,9000},WorkingPrecision->100]

Transcendental Root

V7

Reduce[RiemannSiegelTheta[n]==π n&&9100<n<9200,n,Reals]

(((-1))^n)Z((Subscript[g, n]))>0

Position[Table[(((-1))^n)RiemannSiegelZ[GramPoint[n,WorkingPrecision->10,PrecisionGoal->5]],{n,1000}],_?Negative,1]//Flatten//Timing

<<Utilities`Plot`

to refer to the stronger statement that there are precisely $n+1$ zeros of $Z(t)$ between 0 
and $g_n$ (Edwards 2001, p.~171).

PlotRoots[RiemannSiegelZ[t],{t,0,50},PlotStyle->Red,Epilog->{{Blue,Line[{{#,-10},{#,10}}]&/@GramPoint/@Range[0,10]}}]

GramPoint[126]

PlotRoots[RiemannSiegelZ[t],{t,275,290},PlotStyle->Red,Epilog->{{Blue,Line[{{#,-10},{#,10}}]&/@GramPoint/@Range[120,140]}}]

PlotRoots[RiemannSiegelZ[GramPoint[t]],{t,125,127},PlotStyle->Red,Epilog->{{Blue,Line[{{#,-10},{#,10}}]&/@GramPoint/@Range[120,140]}}]

GramPoint[134]

PlotRoots[RiemannSiegelZ[t],{t,290,300},PlotStyle->Red,Epilog->{{Blue,Line[{{#,-10},{#,10}}]&/@GramPoint/@Range[130,140]}}]