Riemann-von Mangoldt Formula

Author

Eric W. Weisstein
June 26, 2004

This notebook downloaded from http://mathworld.wolfram.com/notebooks/PrimeNumbers/Riemann-vonMangoldtFormula.nb.

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

©2005 Wolfram Research, Inc. except for portions noted otherwise

Zeros

Length[zeros=ReadList["/Volumes/Data/Math/ZetaZeros.txt"]]//Timing

Max[zeros]

<<Graphics`Colors`

data=FoldList[Plus,0,Length/@Split[Floor[zeros/1000]]];

ListPlot[data,PlotJoined->True,PlotStyle->Red,Epilog->{Blue,Line[{{1,data[[1]]},{Length[data],data[[-1]]}}]}]

Solve[y==x Log[x]-x,x]

FullSimplify[%,y>1]

Plot[(E^(1+ProductLog[(y/E)])),{y,0,2}]

data2=Exp[1+ProductLog[#/E]]&/@data;

ListPlot[data2,PlotJoined->True,PlotStyle->Red,
Epilog->{Blue,Line[{{1,data[[1]]},{Length[data],data2[[-1]]}}]},
AxesLabel->TraditionalForm/@{T,n[T]<1000n}]

Plot 2

Length[zeros=Take[ReadList["/Volumes/Data/Math/ZetaZeros.txt"]],82]//Timing

zeros={14.134725142`,21.022039639`,25.01085758`,30.424876126`,32.935061588`,37.586178159`,40.918719012`,43.327073281`,48.005150881`,49.773832478`,52.970321478`,56.446247697`,59.347044003`,60.831778525`,65.112544048`,67.079810529`,69.546401711`,72.067157674`,75.704690699`,77.144840069`,79.33737502`,82.910380854`,84.735492981`,87.425274613`,88.809111208`,92.491899271`,94.651344041`,95.870634228`,98.831194218`,101.317851006`,103.72553804`,105.446623052`,107.168611184`,111.029535543`,111.874659177`,114.320220915`,116.226680321`,118.790782866`,121.370125002`,122.946829294`,124.256818554`,127.51668388`,129.5787042`,131.087688531`,133.497737203`,134.756509753`,138.116042055`,139.736208952`,141.123707404`,143.111845808`,146.000982487`,147.422765343`,150.053520421`,150.925257612`,153.024693811`,156.112909294`,157.597591818`,158.849988171`,161.188964138`,163.030709687`,165.537069188`,167.184439978`,169.094515416`,169.911976479`,173.41153652`,174.754191523`,176.441434298`,178.377407776`,179.91648402`,182.207078484`,184.874467848`,185.598783678`,187.228922584`,189.416158656`,192.026656361`,193.079726604`,195.26539668`,196.876481841`,198.015309676`,201.264751944`,202.493594514`,204.189671803`,205.394697202`,207.906258888`,209.576509717`,211.690862595`,213.34791936`,214.547044783`,216.169538508`,219.067596349`,220.714918839`,221.430705555`,224.007000255`,224.98332467`,227.42144428`,229.337413306`,231.2501887`,231.987235253`,233.693404179`,236.524229666`,237.769820481`,239.555477573`,241.049157796`,242.823271934`,244.070898497`,247.136990075`,248.10199006`,249.573689645`,251.014947795`,253.069986748`,255.306256455`,256.380713694`,258.610439492`,259.87440699`,260.805084505`,263.573893905`,265.557851839`,266.614973782`,267.921915083`,269.970449024`,271.494055642`,273.459609188`,275.587492649`,276.452049503`,278.25074353`,279.229250928`,282.465114765`,283.211185733`,284.835963981`,286.667445363`,287.911920501`,289.579854929`,291.846291329`,293.558434139`,294.965369619`,295.573254879`,297.979277062`,299.840326054`,301.649325462`,302.69674959`,304.864371341`,305.728912602`,307.219496128`,310.109463147`,311.16514153`,312.427801181`,313.985285731`,315.475616089`,317.734805942`,318.853104256`,321.160134309`,322.144558672`,323.466969558`,324.862866052`,327.443901262`,329.03307168`,329.953239728`,331.474467583`,333.645378525`,334.211354833`,336.841850428`,338.339992851`,339.858216725`,341.042261111`,342.05487751`,344.66170294`,346.347870566`,347.272677584`,349.316260871`,350.408419349`,351.878649025`,353.488900489`,356.017574977`,357.151302252`,357.952685102`,359.743754953`,361.289361696`,363.331330579`,364.736024114`,366.212710288`,367.993575482`,368.968438096`,370.050919212`,373.061928372`,373.864873911`,375.825912767`,376.324092231`,378.43668025`,379.872975347`,381.484468617`,383.44352945`,384.956116815`,385.861300846`,387.222890222`,388.846128354`,391.456083564`,392.24508334`,393.427743844`,395.582870011`,396.381854223`,397.91873621`,399.985119876`,401.839228601`,402.861917764`,404.2364418`,405.13438746`,407.581460387`,408.947245502`,410.513869193`,411.972267804`,413.26273607`,415.018809755`,415.455214996`,418.38770579`,419.861364818`,420.643827625`,422.076710059`,423.716579627`,425.069882494`,427.208825084`,428.127914077`,430.328745431`,431.301306931`,432.138641735`,433.889218481`,436.161006433`,437.581698168`,438.621738656`,439.918442214`,441.683199201`,442.904546303`,444.319336278`,446.860622696`,447.441704194`,449.148545685`,450.12694578`,451.403308445`,453.986737807`,454.974683769`,456.328426689`,457.903893064`,459.513415281`,460.087944422`,462.065367275`,464.057286911`,465.671539211`,466.570286931`,467.43904621`,469.536004559`,470.773655478`,472.799174662`,473.835232345`,475.600339369`,476.769015237`,478.075263767`,478.942181535`,481.830339376`,482.834782791`,483.851427212`,485.539148129`,486.528718262`,488.38056709`,489.661761578`,491.398821594`,493.314441582`,493.957997805`,495.358828822`,496.429696216`,498.58078243`,500.309084942`,501.604446965`,502.276270327`,504.499773313`,505.415231742`,506.46415271`,508.800700336`,510.264227944`,511.5622897`,512.623144531`,513.668985555`,515.435057167`,517.589668572`,518.234223148`,520.106310412`,521.525193449`,522.456696178`,523.960530892`,525.077385687`,527.903641601`,528.406213852`,529.806226319`,530.866917884`,532.688183028`,533.779630754`,535.664314076`,537.069759083`,538.428526176`,540.213166376`,540.631390247`,541.847437121`,544.323890101`,545.636833249`,547.010912058`,547.931613364`,549.497567563`,550.970010039`,552.049572201`,553.764972119`,555.792020562`,556.899476407`,557.564659172`,559.316237029`,560.240807497`,562.559207616`,564.160879111`,564.506055938`,566.698787683`,567.731757901`,568.92395518`,570.051114782`,572.419984132`,573.614610527`,575.093886014`,575.807247141`,577.039003472`,579.098834672`,580.136959362`,581.946576266`,583.236088219`,584.561705903`,585.984563205`,586.742771891`,588.139663266`,590.660397517`,591.725858065`,592.5713583`,593.974714682`,595.7281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<<Graphics`Colors`
<<Graphics`Graphics`

Solve[t/(2π) Log[t/(2π)]-t/(2π)==0,t]

curve=Plot[t/(2π) Log[t/(2π)]-t/(2π),{t,2π E,800},PlotStyle->Red][[1,2]];

Length[FoldList[Plus,0,Length/@Split[Sort[Floor[zeros/10]]]]]

MakeBoxes[label,TraditionalForm]:=RowBox[{"N","(","T",")"}]

Histogram[FoldList[Plus,0,Length/@Split[Sort[Floor[zeros/10]]]],
HistogramCategories->Range[0,820,10],FrequencyData->True,Epilog->{{Blue,Thickness[.01],curve}},
AxesLabel->TraditionalForm/@{T,label}]