Prime Zeta Function

Author

Eric W. Weisstein
April 5, 2011

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

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

©2011 Wolfram Research, Inc. except for portions noted otherwise

Code

PrimeZeta2[n_,nmax_]:=Sum[MoebiusMu[k]/k Log[Zeta[k n]],{k,nmax}]
PrimeZeta2[n_]:=PrimeZeta2[n,Infinity]

MyPrimeZeta[z_,nmax_:100,prec_:20]:=Total[N[1,prec]/Prime[Range[nmax]]^z]

Notation

PrimeZetaP[s]//TraditionalForm

Sums

(∑_n^∞)(1/((Prime[n])^s))==(PrimeZetaP[s])

V7

Sum[1/Prime[n]^s,{n,∞}]

Sum[((Prime[n])^(-s)),{n,∞}]

Sum[(((1/Prime[n]))^s),{n,∞}]

SumConvergence[(1/((Prime[n])^s)),n]

Numerical

N[PrimeZetaP[1],20]

PrimeZetaP[-1]//N

(∑_n^∞)((Piecewise[(({{1, n∈ℙ}}))])/(n^s))==(PrimeZetaP[s])

V7

Sum[Piecewise[{{1,Element[n,Primes]}}]/n^s,{n,∞}]//InputForm

With[{s=2},Sum[Piecewise[{{1,Element[n,Primes]}}]/n^s,{n,10^5}]]//N

Sum[Boole[Element[n,Primes]]/n^s,{n,∞}]//InputForm

With[{s=2},Sum[Boole[Element[n,Primes]]/n^s,{n,10^5}]]//N

PrimeZetaP[2]//N

V8

Sum[Piecewise[{{1,Element[n,Primes]}}]/n^s,{n,∞}]

(∑_n^∞)(1/((Prime[n])^2))==(PrimeZetaP[2])

V6

Sum[(1/((Prime[n])^2)),{n,∞}]

NSum[(1/((Prime[n])^2)),{n,∞}]

PrimeZeta[2,10^5,30]

Sum[((MoebiusMu[n]Log[Zeta[2n]])/n),{n,∞}]

N[Sum[((MoebiusMu[n]Log[Zeta[2n]])/n),{n,500}],100]

NSum[((MoebiusMu[n]Log[Zeta[2n]])/n),{n,∞}]

V7

Sum[(1/((Prime[n])^2)),{n,∞}]

SumConvergence[(1/((Prime[n])^2)),n]

Numerical

N[PrimeZetaP[2],20]

NSum can't handle this

TimeConstrained[With[{n=2},NSum[MoebiusMu[k]/k Log[Zeta[k n]],{k,∞}]],60]//Timing

V8

Sum[(1/((Prime[n])^2)),{n,∞}]

(∑_k^∞)(((MoebiusMu[k]) ln(ζ(k n)))/k)==(PrimeZetaP[n])

V7

Sum[MoebiusMu[k]/k Log[Zeta[k n]],{k,∞}]

With[{n=1},Sum[MoebiusMu[k]/k Log[Zeta[k n]],{k,∞}]]

V8

Sum[MoebiusMu[k]/k Log[Zeta[k n]],{k,∞}]

(∑_k^∞)((PrimeZetaP[k n])/k)==ln(ζ(n))

V7

TimeConstrained[Sum[((PrimeZetaP[k n])/k),{k,∞}],60]//Timing

V8

Sum[((PrimeZetaP[k n])/k),{k,∞}]

-(∑_n^∞)log(1-((((Prime[n])))^(-s)))==log((Zeta[s]))

-Sum[Log[1-((Prime[n])^(-s))],{n,∞}]

(∑_k^∞)(∑_n^∞)(((((Prime[n])))^(-k s))/k)==log((Zeta[s]))

V7

TimeConstrained[Sum[(((Prime[n])^(-k s))/k),{k,∞},{n,∞}],120]//Timing

With[{s=2},Sum[(((Prime[n])^(-k s))/k),{k,20},{n,20}]]//N

Log[Zeta[2.]]

V8

Sum[(((Prime[n])^(-k s))/k),{k,∞},{n,∞}]//Timing

(∑_k^∞)(((∑_n^∞)((((Prime[n])))^(-k s)))/k)==log((Zeta[s]))

V7

Sum[(1/k)Sum[((Prime[n])^(-k s)),{n,∞}],{k,∞}]//Timing

With[{s=2},Sum[(1/k)Sum[((Prime[n])^(-k s)),{n,20}],{k,20}]]//N

Log[Zeta[2.]]

V8

$Version

Sum[(1/k)Sum[((Prime[n])^(-k s)),{n,∞}],{k,∞}]

(∑_k^∞)((PrimeZetaP[k n])/k)==log((Zeta[s]))

V7

Sum[((PrimeZetaP[k n])/k),{k,∞}]

V8

Sum[((PrimeZetaP[k n])/k),{k,∞}]

Derivative and Integral

Derivative

PrimeZetaP'[s]

Sum[1/Prime[n]^s,{n,∞}]

D[1/Prime[n]^s,s]

Sum[-Log[Prime[n]] ((Prime[n])^(-s)),{n,∞}]//Timing

Integral

Integrate[PrimeZetaP[s],s]

Series

Series[PrimeZetaP[s],{s,0,5}]

Series[PrimeZetaP[s],{s,1,5}]

Sum[((MoebiusMu[n])/n)Log[Zeta[n]],{n,2,∞}]

NSum[((MoebiusMu[n])/n)Log[Zeta[n]],{n,2,∞}]

NSum[((MoebiusMu[n])/n)Log[Zeta[n]],{n,2,∞},WorkingPrecision->100]

Values

Table[N[PrimeZetaP[n],20],{n,2,10}]//Column

Numerical Evaluation

PrimeZetaP[x+I y]/.{x->.5,y->.5}

l=Table[PrimeZetaP[10.^-n]//Timing,{n,2,5}]

ListLogPlot[First/@l,Joined->True]

Plot[Re[PrimeZetaP[s]],{s,.01,.1}]

Plot

<<Utilities`Plot`
<<Utilities`Typesetting`

Real

p1=Plot[{Re[PrimeZetaP[s]],Im[PrimeZetaP[s]]},{s,.01,1.2},PlotPoints->100,AxesLabel->{s,P[s]},AxesOrigin->{0,0}]

p2=PlotDiscontinuous[PrimeZetaP[s],{s,1,10},PlotStyle->{Red,Thickness[.005]},VerticalAsymptotes->{1},PlotRange->{{0,10},{-.02,5}},
AxesLabel->{s,P[s]},Method->{"AxesInFront"->False}]

GraphicsGrid[{{p1,p2}},ImageSize->400]

Real, Im Parts

p1=Plot[Through[{Re,Im}[PrimeZetaP[1/2+I t]]],{t,0,20},Evaluated->True,AxesLabel->{t,PrimeZetaP[1/2+I t]}]

p2=Plot[{Re[PrimeZetaP[1/2+I t]],Im[PrimeZetaP[1+I t]]},{t,0,20},Evaluated->True,AxesLabel->{t,PrimeZetaP[1+I t]}]

GraphicsGrid[{{p1,p2}},ImageSize->400]

Complex Plot

ComplexPlot3D[PrimeZetaP[z],{z,-I,I+2}]

ComplexPlot3D[PrimeZetaP[z],{z,-I,I+2},PlotPoints->200,PlotRange->{-3,5},"PolygonReduction"->.01]

Complex Contour Plot

V5

ComplexContourPlot[PrimeZeta[z],{z,-3-I,-1+I}]//Timing

V7

ComplexContourPlot[PrimeZetaP[z],{z,-I,I+2}]

ComplexContourPlot[PrimeZetaP[z],{z,-I,I+2},Contours->10,"MeshRegions"->False,"PackGraphicsComplex"->True]

Zeros

From http://groups.yahoo.com/group/primenumbers/files/Pari-GP%20code/pzeros.txt.

Real

Solve[PrimeZetaP[s]==0,s]

FindRoot[Re[PrimeZetaP[s]]==0,{s,.6},WorkingPrecision->20]

List

Length[zeros=ReadList["/Volumes/Users/Data/Math/PrimeZetaZeros.txt"]]

Values

zeros={0.01781950752876758827661098957`27.250895697428458+0.0001734173454455217104711817777999999999999999999999999999`27.239092534054446 I,0.0114137153293753904509579378`27.057427036565564+0.0004616347035591074040805972982999999999999999999999999999`27.66429844962277 I,0.0288015920541253700192904521700000000000000000000000000001`27.459416494745742+0.0005322777356134537375153800299`27.72613830065999 I,0.10173842970511969511651554120692469854014037748949496092`56.00748503017302+2.81718832353748697981983693913073901306091461552232022464`56.449815879694114 I,0.253751610037541116925066801039228316076296569416665000029`56.40440880650039+4.75811479609140609415952123804264985121502963767405681858`56.67743491578187 I,0.0991871483095256300169915938999999999999999999999999999999`27.996455404212316+5.000162717759452802973050122`27.698984137591054 I,0.03492619084723663312743830533`27.543151222790993+6.1011467269596658792725053080000000000000000000000000000001`27.78541146950029 I,0.0299210321404749248801140574700000000000000000000000000001`27.47597657064902+6.1020164725315015530209250119999999999999999999999999999999`27.78547337569575 I,0.01759109314333626873343878292`27.245292828165123+7.4508702601795074326377704000000000000000000000000000000001`27.87220700122088 I,0.0351515829032572901565998031199999999999999999999999999999`27.545944886416205+7.4613725376399209626369328189999999999999999999999999999999`27.872818724348566 I,0.228746175231902736338424557412148204578475663788372682316`56.35935384114476+7.96607917615808969108659433023687604730737674374420509172`56.901244618618335 I,0.0252339653029628560775528280000000000000000000000000000001`27.40198550156321+8.122212774517992515699654877`27.909674362350362 I,0.03177606618984387317254761581`27.50210013145571+8.216725949953627380561791161`27.914698802318924 I,0.02487871798389443140652078694`27.395827997125476+8.216836410641152599935414418`27.914704640671683 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I,0.0254545351586157788654811678499999999999999999999999999999`27.40576517051925+12.11303974424714320709886576`27.08325314219303 I,0.008635738824959866083745320026`27.936299499271975+12.13715246989083084389633285`27.084116807688915 I,0.0238984999341045129433889363499999999999999999999999999999`27.378370641919425+12.2998338829407007762440576`27.089899246056525 I,0.0741904119392491525068051517`27.870347782490125+12.32964994827894328097154893`27.090950746693952 I,0.03286815505509483073154771598`27.516775327085+15.14140993435345865547017045`27.180166317581453 I,0.03931777420710316565092609922`27.594588924294037+15.16812189333708817419579535`27.1809318100646 I,0.0235851138343245021382072993`27.372637976723322+15.19953282096896037259620393`27.18183023949784 I,0.02332435188793222121604639489`27.367809584877048+15.2186366667947840838590441099999999999999999999999999999999`27.18237574871525 I,0.03673080092288194685504097203`27.565030398331565+15.2389641595388407836404928`27.18295544764103 I,0.0156517372055881622498509993800000000000000000000000000001`27.194562547437116+15.26466446962138038655821599`27.1836872625755 I,0.05206284281212072886723863469`27.716527878408435+16.90892372957604286351335273`27.228115965186053 I,0.04395941667724798732096696939`27.64305192088673+17.21856586938880847853113901`27.23599697633088 I,0.1315285145341544639301831255`27.119019915277388+17.27225842397426600179062729`27.2373491271994 I,0.00782531449429226711938842746`27.893501800581564+17.72945877247286830587283109`27.248695478087313 I,0.0632625629345426356455139659799999999999999999999999999999`27.801146782359698+17.3380804466645808860745481299999999999999999999999999999999`27.239001013695695 I,0.0142157812921552340654269372`27.15277073328762+17.47648644111459621362820902`27.24245412437306 I,0.01126864302911342888856426964`27.05187162141878+17.4948001114110765834554770000000000000000000000000000000001`27.24290898477222 I,0.02399998437374832202315267849`27.38021095894506+17.49903371096225024742170489`27.243014067795816 I,0.0160723790161391657004471316899999999999999999999999999999`27.206080165320394+17.51658734861133402312731659`27.243449499090552 I,0.0178542319`+17.7127606` I,0.01852182948735073882169272627`27.267683881753033+17.71447726101175356196865028`27.248328341201645 I,0.0245094965887636225309987796`27.389334391179975+17.79158834208227789682947915`27.25021472140655 I,0.1652313246620007180398383864`27.218092384620157+18.05505841075746028998208724`27.25659889777135 I,0.05983346306905758259055795318`27.776944139867904+18.21588301989685933827600125`27.26045022863166 I,0.0180106364811657142020792167300000000000000000000000000001`27.25552906070259+18.29702239531373009069450758`27.262380419648395 I,0.0331100569346930779606400729299999999999999999999999999999`27.519959927547653+18.31023734353696715446269041`27.262693973811277 I,0.0233696481`+19.1878702` I,0.0429123527000473509210331483100000000000000000000000000001`27.632582325687135+19.63546433451250730919808556`27.29304117581434 I,0.04052408643952668501532888038`27.60771323305184+19.65004131880781905218632312`27.293363467918134 I,0.0739077106341922605454987969100000000000000000000000000001`27.868689749777705+19.67298132412376807287487876`27.293870179670385 I,0.04259113833591542042874629917`27.6293192476436+19.71767107333764247624397201`27.29485561751552 I,0.03445059080697539024015631356`27.537196674197467+19.7303607552923811139095477100000000000000000000000000000001`27.295135026083436 I,0.0848142675672916611267974990200000000000000000000000000001`27.928468915989484+20.2375631922395704313393655100000000000000000000000000000001`27.306158217857693 I,0.0706492339959872744463977435400000000000000000000000000001`27.849107457435544+20.3374630086457353840182825300000000000000000000000000000001`27.308296776017166 I,0.04468452308214515166388892806`27.65015712707104+20.36496471058515708049780963`27.308883661855777 I,0.1886074524313979232420388183`27.27555884898397+21.85272309635202981551581977`27.339505562680902 I,0.08889287563333905641472644003`27.948866955603368+22.19598401717331620430360655`27.346274403416594 I,0.0745549012783188631045088386999999999999999999999999999999`27.87247619947458+22.38833079285956464541464165`27.35002171508534 I,0.1373748312452847253077699586`27.137907171933556+22.5092531052781292023165930600000000000000000000000000000001`27.35236108462123 I,0.07339568087423594841752199745`27.865670503680693+22.52222167406227123908492193`27.352611228683557 I,0.04487997577290865531276869044`27.652052613807488+22.55637409419630580998569554`27.353269288687105 I,0.03731502089098735059270405267`27.57188368909826+22.58277451588084031752900055`27.353777298213995 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Computation

zeros[[1]]

PrimeZeta[#,10000,50]&/@Take[zeros,1]

zeros[[1]]//InputForm

Plus@@N[1/Prime[Range[10^5]]^zeros[[1]],50]//InputForm//Timing

Chop[PrimeZetaP/@zeros]//Union//Timing

Plot

p1=Graphics[{PointSize[.006],Red,Point[Through[{Re,Im}[#]]&/@zeros]},Axes->True,
AxesLabel->{x,y},PlotRange->{{0,1.5},{0, 1000}},AspectRatio->1/GoldenRatio]

p2=ContourPlot[Evaluate[Through[{Re,Im}[PrimeZetaP[x+I y]]]],{x,.2,2},{y,0,50},ContourShading->None,Contours->{0},PlotPoints->100,MaxRecursion->4,ContourStyle->List/@{Red,Blue},FrameLabel->{x,y},PlotLabel->P[x+i y],Epilog->{PointSize[.03],Point[Through[{Re,Im}[#]]&/@zeros]}]

p3=ContourPlot[Evaluate[Through[{Re,Im}[PrimeZetaP[x+I y]]]],{x,.2,2},{y,0,100},ContourShading->None,Contours->{0},PlotPoints->30,MaxRecursion->4,ContourStyle->List/@{Red,Blue},FrameLabel->{x,y},PlotLabel->P[x+i y],Epilog->{PointSize[.02],Point[{Re[#],Im[#]}&/@zeros]}]

GraphicsGrid[{{p1,p2}},ImageSize->400,Spacings->-30]