Data visualizations Home work 3
Antanas Kaminskas
2022-04-06
Librarys
library(VGAM)## Įkeliamas reikalingas paketas: stats4## Įkeliamas reikalingas paketas: splineslibrary(pracma)## Warning: paketas 'pracma' buvo sukurtas pagal R versiją 4.1.3##
## Pridedamas paketas: 'pracma'## Šie objektai yra užmaskuoti nuo 'package:VGAM':
##
## erf, erfc, expint, logit, loglog, Rank, zetalibrary(ggplot2)
library(plot3D)
library(rgl)
library(viscomplexr)Some theoretical elements of \(\zeta(s)\) functions
Some numbers have the special property that they cannot be expressed as the product of two smaller numbers, e.g., 2, 3, 5, 7, etc. Such numbers are called prime numbers, and they play an important role, both in pure mathematics and its applications. The distribution of such prime numbers among all natural numbers does not follow any regular pattern. However, the German mathematician G.F.B. Riemann (1826 - 1866) observed that the frequency of prime numbers is very closely related to the behavior of an elaborate function
\[ \zeta (s) = \sum\limits_{n = 1}^{\infty} \frac{1}{n^s} \]
\[ \zeta (s) = \frac{1}{1^s} + \frac{1}{2^s} + \frac{1}{3^s} + \dots \]
and it can be expand by the Leonhard Euler infinety product
\[ \zeta (s) = \prod\limits_{p \in \mathbb{P}} \frac{1}{1-p^{-s}} \]
\[ \zeta (s) = \frac{1}{1-2^{-s}} \cdot \frac{1}{1-3^{-s}} \cdot \frac{1}{1-5^{-s}} \cdots \]
where \(s =: \sigma + it\) and \(\sigma\) is real part and \(t\) is imaginary part of complex variable.
called the Riemann Zeta function. The Riemann hypothesis asserts that all interesting solutions of the equation
\[ \zeta (s) = 0 \]
lie on a certain vertical straight line. This has been checked for the first 10,000,000,000,000 solutions. A proof that it is true for every interesting solution would shed light on many of the mysteries surrounding the distribution of prime numbers.
Data
k <- seq(-13, 49.8, by = 0.1)
j <- zeta(0.5 + k*1i)
plot(j,type = "l", lwd = 2, xlab = "Imaginary part (on critical line)" ,
ylab = "Function value", main = "Riemann's Zeta Function along the critical line")
abline(v = 1, lwd = 3, col = "red")
abline(v = 0, lwd = 3, col = "red")
abline(v = 0.5, lwd = 3, lty = 2, col = "darkgreen")
abline(h = 0, lwd = 3, col = "red")
men <- Im(zeta(0.5 + k*1i))
men## [1] 0.655483098 0.685303859 0.710229285 0.730308527 0.745621399
## [6] 0.756275948 0.762405944 0.764168325 0.761740610 0.755318329
## [11] 0.745112472 0.731346988 0.714256350 0.694083207 0.671076133
## [16] 0.645487490 0.617571406 0.587581892 0.555771092 0.522387671
## [21] 0.487675354 0.451871604 0.415206450 0.377901459 0.340168844
## [26] 0.302210716 0.264218463 0.226372259 0.188840692 0.151780510
## [31] 0.115336465 0.079641265 0.044815612 0.010968326 -0.021803460
## [36] -0.053414030 -0.083788766 -0.112863767 -0.140585427 -0.166909982
## [41] -0.191803013 -0.215238939 -0.237200482 -0.257678123 -0.276669552
## [46] -0.294179110 -0.310217242 -0.324799945 -0.337948236 -0.349687621
## [51] -0.360047588 -0.369061117 -0.376764200 -0.383195393 -0.388395383
## [56] -0.392406583 -0.395272745 -0.397038602 -0.397749532 -0.397451244
## [61] -0.396189498 -0.394009830 -0.390957322 -0.387076373 -0.382410509
## [66] -0.377002203 -0.370892715 -0.364121953 -0.356728356 -0.348748776
## [71] -0.340218397 -0.331170648 -0.321637141 -0.311647612 -0.301229875
## [76] -0.290409783 -0.279211197 -0.267655959 -0.255763869 -0.243552666
## [81] -0.231038008 -0.218233457 -0.205150454 -0.191798300 -0.178184127
## [86] -0.164312862 -0.150187187 -0.135807487 -0.121171783 -0.106275654
## [91] -0.091112140 -0.075671626 -0.059941701 -0.043906988 -0.027548944
## [96] -0.010845625 0.006228602 0.023703384 0.041612779 0.059995712
## [101] 0.078896513 0.098365554 0.118459980 0.139244571 0.160792744
## [106] 0.183187713 0.206523819 0.230908055 0.256461777 0.283322588
## [111] 0.311646338 0.341609111 0.373408946 0.407266813 0.443425976
## [116] 0.482148161 0.523703726 0.568350801 0.616294432 0.667609644
## [121] 0.722099744 0.779039078 0.836712596 0.891609918 0.937075501
## [126] 0.961254285 0.944632965 0.859131904 0.674179990 0.376879447
## [131] 0.000000000 -0.376879447 -0.674179990 -0.859131904 -0.944632965
## [136] -0.961254285 -0.937075501 -0.891609918 -0.836712596 -0.779039078
## [141] -0.722099744 -0.667609644 -0.616294432 -0.568350801 -0.523703726
## [146] -0.482148161 -0.443425976 -0.407266813 -0.373408946 -0.341609111
## [151] -0.311646338 -0.283322588 -0.256461777 -0.230908055 -0.206523819
## [156] -0.183187713 -0.160792744 -0.139244571 -0.118459980 -0.098365554
## [161] -0.078896513 -0.059995712 -0.041612779 -0.023703384 -0.006228602
## [166] 0.010845625 0.027548944 0.043906988 0.059941701 0.075671626
## [171] 0.091112140 0.106275654 0.121171783 0.135807487 0.150187187
## [176] 0.164312862 0.178184127 0.191798300 0.205150454 0.218233457
## [181] 0.231038008 0.243552666 0.255763869 0.267655959 0.279211197
## [186] 0.290409783 0.301229875 0.311647612 0.321637141 0.331170648
## [191] 0.340218397 0.348748776 0.356728356 0.364121953 0.370892715
## [196] 0.377002203 0.382410509 0.387076373 0.390957322 0.394009830
## [201] 0.396189498 0.397451244 0.397749532 0.397038602 0.395272745
## [206] 0.392406583 0.388395383 0.383195393 0.376764200 0.369061117
## [211] 0.360047588 0.349687621 0.337948236 0.324799945 0.310217242
## [216] 0.294179110 0.276669552 0.257678123 0.237200482 0.215238939
## [221] 0.191803013 0.166909982 0.140585427 0.112863767 0.083788766
## [226] 0.053414030 0.021803460 -0.010968326 -0.044815612 -0.079641265
## [231] -0.115336465 -0.151780510 -0.188840692 -0.226372259 -0.264218463
## [236] -0.302210716 -0.340168844 -0.377901459 -0.415206450 -0.451871604
## [241] -0.487675354 -0.522387671 -0.555771092 -0.587581892 -0.617571406
## [246] -0.645487490 -0.671076133 -0.694083207 -0.714256350 -0.731346988
## [251] -0.745112472 -0.755318329 -0.761740610 -0.764168325 -0.762405944
## [256] -0.756275948 -0.745621399 -0.730308527 -0.710229285 -0.685303859
## [261] -0.655483098 -0.620750841 -0.581126093 -0.536665042 -0.487462854
## [266] -0.433655238 -0.375419728 -0.312976661 -0.246589805 -0.176566623
## [271] -0.103258123 -0.027058282 0.051596991 0.132231368 0.214330112
## [276] 0.297342590 0.380685314 0.463745493 0.545885099 0.626445444
## [281] 0.704752242 0.780121137 0.851863661 0.919293592 0.981733646
## [286] 1.038522480 1.089021909 1.132624295 1.168760015 1.196904939
## [291] 1.216587816 1.227397489 1.228989831 1.221094304 1.203520049
## [296] 1.176161391 1.139002661 1.092122244 1.035695744 0.969998192
## [301] 0.895405192 0.812392958 0.721537154 0.623510497 0.519079087
## [306] 0.409097436 0.294502192 0.176304563 0.055581491 -0.066534402
## [311] -0.188866006 -0.310203965 -0.429319872 -0.544980265 -0.655961297
## [316] -0.761063893 -0.859129219 -0.949054265 -1.029807322 -1.100443133
## [321] -1.160117493 -1.208101049 -1.243792087 -1.266728046 -1.276595552
## [326] -1.273238755 -1.256665753 -1.227052941 -1.184747112 -1.130265183
## [331] -1.064291443 -0.987672259 -0.901408206 -0.806643629 -0.704653699
## [336] -0.596829038 -0.484658074 -0.369707294 -0.253599629 -0.137991231
## [341] -0.024546965 0.085085053 0.189299488 0.286559980 0.375425086
## [346] 0.454573506 0.522828162 0.579178667 0.622801742 0.653079147
## [351] 0.669612731 0.672236216 0.661023387 0.636292410 0.598606050
## [356] 0.548767621 0.487812587 0.416995792 0.337774379 0.251786557
## [361] 0.160826425 0.066815185 -0.028230883 -0.122235194 -0.213097852
## [366] -0.298734253 -0.377114632 -0.446303846 -0.504500652 -0.550075716
## [371] -0.581607579 -0.597915817 -0.598090670 -0.581518412 -0.547901852
## [376] -0.497275372 -0.430014013 -0.346836243 -0.248800101 -0.137292595
## [381] -0.014012302 0.119054695 0.259665290 0.405355859 0.553484255
## [386] 0.701276622 0.845878242 0.984407513 1.114012046 1.231925807
## [391] 1.335526156 1.422389604 1.490345082 1.537523542 1.562402744
## [396] 1.563846131 1.541134806 1.493991728 1.422597383 1.327596354
## [401] 1.210094384 1.071645736 0.914230851 0.740224509 0.552354961
## [406] 0.353654648 0.147403396 -0.062934864 -0.273780622 -0.481511152
## [411] -0.682535200 -0.873368375 -1.050707549 -1.211502567 -1.353023561
## [416] -1.472922229 -1.569285528 -1.640680371 -1.686188094 -1.705427656
## [421] -1.698566799 -1.666320639 -1.609937473 -1.531171864 -1.432245417
## [426] -1.315795923 -1.184815907 -1.042581856 -0.892575697 -0.738400333
## [431] -0.583691215 -0.432026102 -0.286835253 -0.151314346 -0.028342389
## [436] 0.079593148 0.170461924 0.242745145 0.295477159 0.328272903
## [441] 0.341340165 0.335476057 0.312047460 0.272955635 0.220585636
## [446] 0.157741574 0.087569196 0.013467615 -0.061007603 -0.132247524
## [451] -0.196696393 -0.250958748 -0.291904133 -0.316766429 -0.323234893
## [456] -0.309534137 -0.274490538 -0.217582867 -0.138975317 -0.039531572
## [461] 0.079190980 0.214967183 0.364949996 0.525735486 0.693445574
## [466] 0.863826586 1.032361200 1.194390895 1.345245646 1.480377318
## [471] 1.595492988 1.686684361 1.750549426 1.784302651 1.785870233
## [476] 1.753967294 1.688154321 1.588870744 1.457444118 1.296074070
## [481] 1.107790927 0.896389626 0.666340318 0.422677758 0.170872282
## [486] -0.083314234 -0.333983489 -0.575262613 -0.801468087 -1.007266931
## [491] -1.187830305 -1.338974817 -1.457287101 -1.540227623 -1.586210267
## [496] -1.594654873 -1.566010723 -1.501749784 -1.404329455 -1.277125485
## [501] -1.124336689 -0.950863978 -0.762167071 -0.564103032 -0.362751395
## [506] -0.164231170 0.025484656 0.200751313 0.356425355 0.488023082
## [511] 0.591859085 0.665160244 0.706151285 0.714108819 0.689381810
## [516] 0.633377450 0.548512598 0.438132015 0.306395810 0.158139539
## [521] -0.001288637 -0.166208465 -0.330799879 -0.489300634 -0.636202610
## [526] -0.766440070 -0.875563320 -0.959891632 -1.016639935 -1.044014589
## [531] -1.041274596 -1.008755734 -0.947856395 -0.860985214 -0.751471960
## [536] -0.623444478 -0.481675719 -0.331406044 -0.178146975 -0.027473281
## [541] 0.115189102 0.244771599 0.356760231 0.447374084 0.513718114
## [546] 0.553904205 0.567135432 0.553749794 0.515221047 0.454115881
## [551] 0.374008235 0.279353179 0.175324359 0.067620410 -0.037752959
## [556] -0.134717472 -0.217364858 -0.280198479 -0.318364861 -0.327866135
## [561] -0.305744939 -0.250234182 -0.160865207 -0.038529339 0.114510554
## [566] 0.294660088 0.497084610 0.715849429 0.944104853 1.174310107
## [571] 1.398488614 1.608505774 1.796359365 1.954471995 2.075974763
## [576] 2.154971385 2.186772591 2.168091508 2.097192050 1.973983969
## [581] 1.800060107 1.578673513 1.314654328 1.014268622 0.685023618
## [586] 0.335425775 -0.025299920 -0.387519425 -0.741516590 -1.077825505
## [591] -1.387555877 -1.662699113 -1.896403454 -2.083207472 -2.219222834
## [596] -2.302259038 -2.331885072 -2.309425329 -2.237889693 -2.121840291
## [601] -1.967199921 -1.781009545 -1.571144298 -1.345999194 -1.114157018
## [606] -0.884051683 -0.663640594 -0.460099302 -0.279550874 -0.126841071
## [611] -0.005368590 0.083022585 0.138084262 0.161136793 0.154974675
## [616] 0.123708953 0.072553139 0.007561801 -0.064666945 -0.137312145
## [621] -0.203659938 -0.257428195 -0.293071483 -0.306052655 -0.293068747
## [626] -0.252220713 -0.183118907 -0.086918968 0.033714190rel <- Re(zeta(0.5 + k*1i))
rel## [1] 4.430048e-01 5.001672e-01 5.583469e-01 6.171411e-01 6.761634e-01
## [6] 7.350455e-01 7.934391e-01 8.510175e-01 9.074764e-01 9.625351e-01
## [11] 1.015937e+00 1.067449e+00 1.116864e+00 1.163998e+00 1.208693e+00
## [16] 1.250813e+00 1.290247e+00 1.326905e+00 1.360720e+00 1.391644e+00
## [21] 1.419650e+00 1.444730e+00 1.466891e+00 1.486160e+00 1.502574e+00
## [26] 1.516188e+00 1.527066e+00 1.535286e+00 1.540933e+00 1.544102e+00
## [31] 1.544895e+00 1.543420e+00 1.539791e+00 1.534124e+00 1.526540e+00
## [36] 1.517160e+00 1.506109e+00 1.493509e+00 1.479484e+00 1.464155e+00
## [41] 1.447642e+00 1.430065e+00 1.411536e+00 1.392168e+00 1.372069e+00
## [46] 1.351342e+00 1.330087e+00 1.308399e+00 1.286368e+00 1.264080e+00
## [51] 1.241615e+00 1.219049e+00 1.196452e+00 1.173890e+00 1.151424e+00
## [56] 1.129109e+00 1.106997e+00 1.085133e+00 1.063561e+00 1.042317e+00
## [61] 1.021434e+00 1.000943e+00 9.808688e-01 9.612331e-01 9.420545e-01
## [66] 9.233481e-01 9.051260e-01 8.873973e-01 8.701685e-01 8.534435e-01
## [71] 8.372238e-01 8.215089e-01 8.062962e-01 7.915812e-01 7.773577e-01
## [76] 7.636180e-01 7.503529e-01 7.375520e-01 7.252037e-01 7.132950e-01
## [81] 7.018124e-01 6.907410e-01 6.800654e-01 6.697692e-01 6.598352e-01
## [86] 6.502456e-01 6.409817e-01 6.320242e-01 6.233529e-01 6.149467e-01
## [91] 6.067838e-01 5.988412e-01 5.910948e-01 5.835194e-01 5.760878e-01
## [96] 5.687714e-01 5.615393e-01 5.543582e-01 5.471914e-01 5.399991e-01
## [101] 5.327367e-01 5.253544e-01 5.177959e-01 5.099973e-01 5.018846e-01
## [106] 4.933721e-01 4.843595e-01 4.747277e-01 4.643350e-01 4.530103e-01
## [111] 4.405457e-01 4.266859e-01 4.111146e-01 3.934362e-01 3.731508e-01
## [116] 3.496209e-01 3.220256e-01 2.892978e-01 2.500384e-01 2.024006e-01
## [121] 1.439364e-01 7.140564e-02 -1.943208e-02 -1.339558e-01 -2.785079e-01
## [126] -4.593029e-01 -6.791953e-01 -9.305658e-01 -1.184325e+00 -1.383390e+00
## [131] -1.460355e+00 -1.383390e+00 -1.184325e+00 -9.305658e-01 -6.791953e-01
## [136] -4.593029e-01 -2.785079e-01 -1.339558e-01 -1.943208e-02 7.140564e-02
## [141] 1.439364e-01 2.024006e-01 2.500384e-01 2.892978e-01 3.220256e-01
## [146] 3.496209e-01 3.731508e-01 3.934362e-01 4.111146e-01 4.266859e-01
## [151] 4.405457e-01 4.530103e-01 4.643350e-01 4.747277e-01 4.843595e-01
## [156] 4.933721e-01 5.018846e-01 5.099973e-01 5.177959e-01 5.253544e-01
## [161] 5.327367e-01 5.399991e-01 5.471914e-01 5.543582e-01 5.615393e-01
## [166] 5.687714e-01 5.760878e-01 5.835194e-01 5.910948e-01 5.988412e-01
## [171] 6.067838e-01 6.149467e-01 6.233529e-01 6.320242e-01 6.409817e-01
## [176] 6.502456e-01 6.598352e-01 6.697692e-01 6.800654e-01 6.907410e-01
## [181] 7.018124e-01 7.132950e-01 7.252037e-01 7.375520e-01 7.503529e-01
## [186] 7.636180e-01 7.773577e-01 7.915812e-01 8.062962e-01 8.215089e-01
## [191] 8.372238e-01 8.534435e-01 8.701685e-01 8.873973e-01 9.051260e-01
## [196] 9.233481e-01 9.420545e-01 9.612331e-01 9.808688e-01 1.000943e+00
## [201] 1.021434e+00 1.042317e+00 1.063561e+00 1.085133e+00 1.106997e+00
## [206] 1.129109e+00 1.151424e+00 1.173890e+00 1.196452e+00 1.219049e+00
## [211] 1.241615e+00 1.264080e+00 1.286368e+00 1.308399e+00 1.330087e+00
## [216] 1.351342e+00 1.372069e+00 1.392168e+00 1.411536e+00 1.430065e+00
## [221] 1.447642e+00 1.464155e+00 1.479484e+00 1.493509e+00 1.506109e+00
## [226] 1.517160e+00 1.526540e+00 1.534124e+00 1.539791e+00 1.543420e+00
## [231] 1.544895e+00 1.544102e+00 1.540933e+00 1.535286e+00 1.527066e+00
## [236] 1.516188e+00 1.502574e+00 1.486160e+00 1.466891e+00 1.444730e+00
## [241] 1.419650e+00 1.391644e+00 1.360720e+00 1.326905e+00 1.290247e+00
## [246] 1.250813e+00 1.208693e+00 1.163998e+00 1.116864e+00 1.067449e+00
## [251] 1.015937e+00 9.625351e-01 9.074764e-01 8.510175e-01 7.934391e-01
## [256] 7.350455e-01 6.761634e-01 6.171411e-01 5.583469e-01 5.001672e-01
## [261] 4.430048e-01 3.872767e-01 3.334112e-01 2.818453e-01 2.330217e-01
## [266] 1.873849e-01 1.453783e-01 1.074395e-01 7.399691e-02 4.546529e-02
## [271] 2.224114e-02 4.698400e-03 -6.816218e-03 -1.198782e-02 -1.053796e-02
## [276] -2.229099e-03 1.313091e-02 3.568468e-02 6.552127e-02 1.026726e-01
## [281] 1.471099e-01 1.987413e-01 2.574089e-01 3.228873e-01 3.948821e-01
## [286] 4.730298e-01 5.568971e-01 6.459825e-01 7.397177e-01 8.374699e-01
## [291] 9.385454e-01 1.042194e+00 1.147612e+00 1.253953e+00 1.360327e+00
## [296] 1.465816e+00 1.569476e+00 1.670349e+00 1.767473e+00 1.859889e+00
## [301] 1.946655e+00 2.026854e+00 2.099607e+00 2.164085e+00 2.219516e+00
## [306] 2.265200e+00 2.300520e+00 2.324946e+00 2.338053e+00 2.339523e+00
## [311] 2.329155e+00 2.306872e+00 2.272724e+00 2.226897e+00 2.169707e+00
## [316] 2.101607e+00 2.023186e+00 1.935161e+00 1.838376e+00 1.733796e+00
## [321] 1.622496e+00 1.505650e+00 1.384523e+00 1.260453e+00 1.134838e+00
## [326] 1.009117e+00 8.847535e-01 7.632170e-01 6.459615e-01 5.344060e-01
## [331] 4.299139e-01 3.337721e-01 2.471716e-01 1.711874e-01 1.067604e-01
## [336] 5.468083e-02 1.557240e-02 -1.012043e-02 -2.214351e-02 -2.044125e-02
## [341] -5.162065e-03 2.333922e-02 6.450220e-02 1.175650e-01 1.815717e-01
## [346] 2.553834e-01 3.376926e-01 4.270402e-01 5.218368e-01 6.203855e-01
## [351] 7.209083e-01 8.215744e-01 9.205296e-01 1.015928e+00 1.105966e+00
## [356] 1.188910e+00 1.263134e+00 1.327149e+00 1.379632e+00 1.419454e+00
## [361] 1.445707e+00 1.457726e+00 1.455107e+00 1.437718e+00 1.405715e+00
## [366] 1.359541e+00 1.299931e+00 1.227900e+00 1.144734e+00 1.051978e+00
## [371] 9.514046e-01 8.449942e-01 7.348990e-01 6.234079e-01 5.129058e-01
## [376] 4.058304e-01 3.046261e-01 2.116968e-01 1.293580e-01 5.978866e-02
## [381] 4.984593e-03 -3.328589e-02 -5.352350e-02 -5.453529e-02 -3.546700e-02
## [386] 4.170312e-03 6.448011e-02 1.451674e-01 2.455335e-01 3.644782e-01
## [391] 5.005112e-01 6.517704e-01 8.160493e-01 9.908321e-01 1.173335e+00
## [396] 1.360556e+00 1.549328e+00 1.736380e+00 1.918398e+00 2.092090e+00
## [401] 2.254258e+00 2.401855e+00 2.532059e+00 2.642326e+00 2.730454e+00
## [406] 2.794629e+00 2.833471e+00 2.846066e+00 2.831998e+00 2.791358e+00
## [411] 2.724752e+00 2.633298e+00 2.518601e+00 2.382730e+00 2.228179e+00
## [416] 2.057815e+00 1.874818e+00 1.682619e+00 1.484823e+00 1.285133e+00
## [421] 1.087267e+00 8.948742e-01 7.114537e-01 5.402740e-01 3.842969e-01
## [426] 2.461082e-01 1.278569e-01 3.120338e-02 -4.272019e-02 -9.333757e-02
## [431] -1.206423e-01 -1.251977e-01 -1.081223e-01 -7.106164e-02 -1.614511e-02
## [436] 5.406893e-02 1.366606e-01 2.284259e-01 3.259627e-01 4.257630e-01
## [441] 5.243092e-01 6.181721e-01 7.041076e-01 7.791499e-01 8.406986e-01
## [446] 8.865969e-01 9.151986e-01 9.254223e-01 9.167907e-01 8.894532e-01
## [451] 8.441919e-01 7.824090e-01 7.060965e-01 6.177892e-01 5.204998e-01
## [456] 4.176402e-01 3.129284e-01 2.102851e-01 1.137221e-01 2.722486e-02
## [461] -4.536685e-02 -1.004768e-01 -1.349059e-01 -1.459376e-01 -1.314322e-01
## [466] -8.990672e-02 -2.059843e-02 7.649141e-02 2.005741e-01 3.500724e-01
## [471] 5.226420e-01 7.152153e-01 9.240676e-01 1.144904e+00 1.372962e+00
## [476] 1.603136e+00 1.830109e+00 2.048494e+00 2.252984e+00 2.438501e+00
## [481] 2.600342e+00 2.734318e+00 2.836877e+00 2.905220e+00 2.937388e+00
## [486] 2.932335e+00 2.889968e+00 2.811168e+00 2.697776e+00 2.552559e+00
## [491] 2.379138e+00 2.181903e+00 1.965890e+00 1.736651e+00 1.500100e+00
## [496] 1.262342e+00 1.029506e+00 8.075650e-01 6.021644e-01 4.184549e-01
## [501] 2.609412e-01 1.333472e-01 3.850424e-02 -2.173474e-02 -4.655171e-02
## [506] -3.618879e-02 8.058948e-03 8.388344e-02 1.880353e-01 3.164254e-01
## [511] 4.642533e-01 6.261591e-01 7.963936e-01 9.690017e-01 1.138013e+00
## [516] 1.297635e+00 1.442437e+00 1.567532e+00 1.668729e+00 1.742672e+00
## [521] 1.786951e+00 1.800175e+00 1.782019e+00 1.733232e+00 1.655604e+00
## [526] 1.551908e+00 1.425793e+00 1.281663e+00 1.124515e+00 9.597627e-01
## [531] 7.930451e-01 6.300229e-01 4.761759e-01 3.366043e-01 2.158434e-01
## [536] 1.176976e-01 4.509953e-02 1.153240e-06 -1.670082e-02 -5.200370e-03
## [541] 3.322251e-02 9.624719e-02 1.805974e-01 2.821633e-01 3.961562e-01
## [546] 5.172920e-01 6.399946e-01 7.586153e-01 8.676562e-01 9.619935e-01
## [551] 1.037089e+00 1.089183e+00 1.115462e+00 1.114190e+00 1.084806e+00
## [556] 1.027974e+00 9.455865e-01 8.407254e-01 7.175710e-01 5.812690e-01
## [561] 4.377573e-01 2.935570e-01 1.555368e-01 3.065710e-02 -7.429510e-02
## [566] -1.529725e-01 -1.997236e-01 -2.098300e-01 -1.797142e-01 -1.071107e-01
## [571] 8.807491e-03 1.673523e-01 3.662983e-01 6.019165e-01 8.690638e-01
## [576] 1.161326e+00 1.471211e+00 1.790384e+00 2.109941e+00 2.420701e+00
## [581] 2.713523e+00 2.979622e+00 3.210873e+00 3.400105e+00 3.541357e+00
## [586] 3.630097e+00 3.663390e+00 3.640015e+00 3.560512e+00 3.427176e+00
## [591] 3.243978e+00 3.016432e+00 2.751401e+00 2.456855e+00 2.141587e+00
## [596] 1.814897e+00 1.486261e+00 1.164993e+00 8.599115e-01 5.790291e-01
## [601] 3.292727e-01 1.162474e-01 -5.594716e-02 -1.848407e-01 -2.696511e-01
## [606] -3.112765e-01 -3.122198e-01 -2.764501e-01 -2.092051e-01 -1.167447e-01
## [611] -6.063121e-03 1.154251e-01 2.402151e-01 3.610421e-01 4.711984e-01
## [616] 5.648243e-01 6.371591e-01 6.847427e-01 7.055580e-01 6.991080e-01
## [621] 6.664241e-01 6.100051e-01 5.336888e-01 4.424618e-01 3.422156e-01
## [626] 2.394602e-01 1.410056e-01 5.362860e-02 -1.626359e-02modulus <- abs(zeta(0.5 + k*1i))
k## [1] -13.0 -12.9 -12.8 -12.7 -12.6 -12.5 -12.4 -12.3 -12.2 -12.1 -12.0 -11.9
## [13] -11.8 -11.7 -11.6 -11.5 -11.4 -11.3 -11.2 -11.1 -11.0 -10.9 -10.8 -10.7
## [25] -10.6 -10.5 -10.4 -10.3 -10.2 -10.1 -10.0 -9.9 -9.8 -9.7 -9.6 -9.5
## [37] -9.4 -9.3 -9.2 -9.1 -9.0 -8.9 -8.8 -8.7 -8.6 -8.5 -8.4 -8.3
## [49] -8.2 -8.1 -8.0 -7.9 -7.8 -7.7 -7.6 -7.5 -7.4 -7.3 -7.2 -7.1
## [61] -7.0 -6.9 -6.8 -6.7 -6.6 -6.5 -6.4 -6.3 -6.2 -6.1 -6.0 -5.9
## [73] -5.8 -5.7 -5.6 -5.5 -5.4 -5.3 -5.2 -5.1 -5.0 -4.9 -4.8 -4.7
## [85] -4.6 -4.5 -4.4 -4.3 -4.2 -4.1 -4.0 -3.9 -3.8 -3.7 -3.6 -3.5
## [97] -3.4 -3.3 -3.2 -3.1 -3.0 -2.9 -2.8 -2.7 -2.6 -2.5 -2.4 -2.3
## [109] -2.2 -2.1 -2.0 -1.9 -1.8 -1.7 -1.6 -1.5 -1.4 -1.3 -1.2 -1.1
## [121] -1.0 -0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0.0 0.1
## [133] 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3
## [145] 1.4 1.5 1.6 1.7 1.8 1.9 2.0 2.1 2.2 2.3 2.4 2.5
## [157] 2.6 2.7 2.8 2.9 3.0 3.1 3.2 3.3 3.4 3.5 3.6 3.7
## [169] 3.8 3.9 4.0 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9
## [181] 5.0 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9 6.0 6.1
## [193] 6.2 6.3 6.4 6.5 6.6 6.7 6.8 6.9 7.0 7.1 7.2 7.3
## [205] 7.4 7.5 7.6 7.7 7.8 7.9 8.0 8.1 8.2 8.3 8.4 8.5
## [217] 8.6 8.7 8.8 8.9 9.0 9.1 9.2 9.3 9.4 9.5 9.6 9.7
## [229] 9.8 9.9 10.0 10.1 10.2 10.3 10.4 10.5 10.6 10.7 10.8 10.9
## [241] 11.0 11.1 11.2 11.3 11.4 11.5 11.6 11.7 11.8 11.9 12.0 12.1
## [253] 12.2 12.3 12.4 12.5 12.6 12.7 12.8 12.9 13.0 13.1 13.2 13.3
## [265] 13.4 13.5 13.6 13.7 13.8 13.9 14.0 14.1 14.2 14.3 14.4 14.5
## [277] 14.6 14.7 14.8 14.9 15.0 15.1 15.2 15.3 15.4 15.5 15.6 15.7
## [289] 15.8 15.9 16.0 16.1 16.2 16.3 16.4 16.5 16.6 16.7 16.8 16.9
## [301] 17.0 17.1 17.2 17.3 17.4 17.5 17.6 17.7 17.8 17.9 18.0 18.1
## [313] 18.2 18.3 18.4 18.5 18.6 18.7 18.8 18.9 19.0 19.1 19.2 19.3
## [325] 19.4 19.5 19.6 19.7 19.8 19.9 20.0 20.1 20.2 20.3 20.4 20.5
## [337] 20.6 20.7 20.8 20.9 21.0 21.1 21.2 21.3 21.4 21.5 21.6 21.7
## [349] 21.8 21.9 22.0 22.1 22.2 22.3 22.4 22.5 22.6 22.7 22.8 22.9
## [361] 23.0 23.1 23.2 23.3 23.4 23.5 23.6 23.7 23.8 23.9 24.0 24.1
## [373] 24.2 24.3 24.4 24.5 24.6 24.7 24.8 24.9 25.0 25.1 25.2 25.3
## [385] 25.4 25.5 25.6 25.7 25.8 25.9 26.0 26.1 26.2 26.3 26.4 26.5
## [397] 26.6 26.7 26.8 26.9 27.0 27.1 27.2 27.3 27.4 27.5 27.6 27.7
## [409] 27.8 27.9 28.0 28.1 28.2 28.3 28.4 28.5 28.6 28.7 28.8 28.9
## [421] 29.0 29.1 29.2 29.3 29.4 29.5 29.6 29.7 29.8 29.9 30.0 30.1
## [433] 30.2 30.3 30.4 30.5 30.6 30.7 30.8 30.9 31.0 31.1 31.2 31.3
## [445] 31.4 31.5 31.6 31.7 31.8 31.9 32.0 32.1 32.2 32.3 32.4 32.5
## [457] 32.6 32.7 32.8 32.9 33.0 33.1 33.2 33.3 33.4 33.5 33.6 33.7
## [469] 33.8 33.9 34.0 34.1 34.2 34.3 34.4 34.5 34.6 34.7 34.8 34.9
## [481] 35.0 35.1 35.2 35.3 35.4 35.5 35.6 35.7 35.8 35.9 36.0 36.1
## [493] 36.2 36.3 36.4 36.5 36.6 36.7 36.8 36.9 37.0 37.1 37.2 37.3
## [505] 37.4 37.5 37.6 37.7 37.8 37.9 38.0 38.1 38.2 38.3 38.4 38.5
## [517] 38.6 38.7 38.8 38.9 39.0 39.1 39.2 39.3 39.4 39.5 39.6 39.7
## [529] 39.8 39.9 40.0 40.1 40.2 40.3 40.4 40.5 40.6 40.7 40.8 40.9
## [541] 41.0 41.1 41.2 41.3 41.4 41.5 41.6 41.7 41.8 41.9 42.0 42.1
## [553] 42.2 42.3 42.4 42.5 42.6 42.7 42.8 42.9 43.0 43.1 43.2 43.3
## [565] 43.4 43.5 43.6 43.7 43.8 43.9 44.0 44.1 44.2 44.3 44.4 44.5
## [577] 44.6 44.7 44.8 44.9 45.0 45.1 45.2 45.3 45.4 45.5 45.6 45.7
## [589] 45.8 45.9 46.0 46.1 46.2 46.3 46.4 46.5 46.6 46.7 46.8 46.9
## [601] 47.0 47.1 47.2 47.3 47.4 47.5 47.6 47.7 47.8 47.9 48.0 48.1
## [613] 48.2 48.3 48.4 48.5 48.6 48.7 48.8 48.9 49.0 49.1 49.2 49.3
## [625] 49.4 49.5 49.6 49.7 49.8scatter3D(x = rel, y = men, z = k, bty = "b2", pch = 20, cex = 2,
ticktype = "detailed", clab = "Positive step by 0.1",
xlab = "Real part", ylab = "Imaginary part", zlab = " t", phi = 10,
theta = 150 )
kk <- seq(-30, 30, len = 629)
xx <- Re(zeta(.5 + kk*1i))
yy <- Im(zeta(.5 + kk*1i))
length(xx)## [1] 629length(yy)## [1] 629Zeta function for real part
plot(kk, xx, type = "l", col = "darkgreen", lty = 1, lwd = 3, xlim = c(-30, 30),
xlab = " Real part", ylab = "Function value", main = expression(paste("Real part of","", zeta, "(s)")))
abline(v = 0, lty = 2 ,lwd = 2)
abline(h = 0, lty = 2 ,lwd = 2)
Zeta function for imaginary part
plot(kk, yy, type = "l", col = "red", lty = 1, lwd = 3, xlim = c(-30, 30),
xlab = " Imaginary part",ylab = " Funtion values", main = expression(paste("Imaginary part of","", zeta, "(s)")))
abline(v = 0, lty = 2 ,lwd = 2)
abline(h = 0, lty = 2 ,lwd = 2)
3D plot which only can be look from R
fig <- plot3d(x = xx, y = yy, z = kk, type = "l", xlab = "Real part", ylab = "Imaginary part", zlab = "t" )
fig