白新岭同学没有话说？

qingjiao · 发表于 2011-7-1 22:36

[这个贴子最后由qingjiao在 2011/07/21 08:16pm 第 2 次编辑]

纵坐标取(Li(x)-π(x))/(√x/lnx)，横坐标按上次的方法取等比数列，但由x0=10算起，因为x0太小的话Li(x)的计算可能有较大误差。同样分成300~400个点，算到你能算的位置为止。
Li(x)的公式按陆教授发的帖子，如下图，但注意这里是不定积分，定积分的下界取x=2，上界就是x0，x1，x2，....。π(x)的精确值你应该有的。
估计x足够大时，纵坐标将在1.5附近振荡，波动区间约为(1,2)。这个结果或许可以使本坛某些人的脑袋清醒一点。

qingjiao · 发表于 2011-7-1 22:48

[这个贴子最后由qingjiao在 2011/07/01 10:55pm 第 1 次编辑]

另外，令(lnx)^(k-1)/((k-1)*(k-1)!)>(lnx)^k/(k*k!)，
可得k>lnx-1。说明在第lnx-1项之前各项的值递增，之后递减。
所以算到x=10亿的话，建议k至少取2*(ln(10^9)-1)，保证其后的各项和很小，不影响结果。

天山草 · 发表于 2011-7-2 12:17

好的。我再算算这个题目玩玩。Li(x) 以前算过，要查一查以前是如何算的。
再过一个星期我要带上小孙子回山西避暑去了，到九月份才能回北京。山西老家的电脑没有交网费，那台电脑本身也太陈旧了，所以就暂时不能跟网友们相聚了。争取在回家之前把这个问题算完。若算不完，回来之后再补课吧。

天山草 · 发表于 2011-7-2 12:21

问一句，楼主有没有 42亿至100亿的素数表？或者，这些数据可以通过什么途径得到？

qingjiao · 发表于 2011-7-2 13:51

下面引用由天山草在 2011/07/02 00:21pm 发表的内容：
问一句，楼主有没有 42亿至100亿的素数表？或者，这些数据可以通过什么途径得到？

我没有，只有1000万以内的素数表，还是跟白新岭要的。
一亿内的估计也很大，当时就没有要。
但如果你有txt格式的可以发到我邮箱，因为txt格式的要小得多。
我收到后再转换成excel格式。

天山草 · 发表于 2011-7-2 19:12

我有到 42 亿的，每 2 亿一个文件，都是 TXT 的，每个约 100 兆。0-2 亿的是 113 M，越往后越小，40-42 亿的是 98 M。不知能不能给你发过去。你邮箱号多少？我试试。
你这个题目，我打算分三步走，第一步是把 x 分成 367 个数，并算出每个 x 的 π(x)值；第二步是算出 Li(x) 值；第三步就简单了，按题目要求算出最终结果来。
我已完成了第一步，见以下数据，π(x)数据用 mathetica 抽查过，没有误差。第一个数是 10，末一个数是 41.3 亿。
序号                x                   π(x)
--------------------------------------------------------------
1          10.0002190582697          4
2          10.5572312598154          4
3          11.1452690409871          5
4          11.7660605265701          5
5          12.4214300979             5
6          13.1133037543531          6
7          13.8437147734705          6
8          14.6148096863528          6
9          15.4288545858827          6
10          16.2882417863163          6
11          17.1954968538142          7
12          18.1532860285716          7
13          19.1644240603631          8
14          20.2318824805253          8
15          21.3587983346905          8
16          22.5484834019328          8
17          23.8044339274205          9
18          25.1303408971778          9
19          26.5301008851506          9
20          28.0078275044535          9
21          29.5678634964516          10
22          31.2147934932039          11
23          32.9534574907754          11
24          34.7889650730116          11
25          36.7267104275783          11
26          38.7723881983944          12
27          40.932010221045          12
28          43.2119231903572          14
29          45.6188273120601          14
30          48.1597959933419          15
31          50.842296630171          15
32          53.6742125524715          16
33          56.6638661916442          16
34          59.8200435385188          17
35          63.1520199636143          18
36          66.6695874755876          18
37          70.3830834979778          19
38          74.3034212488152          21
39          78.4421218123742          21
40          82.8113479973235          22
41          87.4239400807744          23
42          92.2934535432735          24
43          97.4341989056339          25
44          102.861283784678          26
45          108.590657291484          28
46          114.63915690262          30
47          121.024557942096          30
48          127.765625819471          31
49          134.882171177615          32
50          142.395108112208          34
51          150.326515634058          35
52          158.699702554875          37
53          167.539275987182          39
54          176.871213659668          40
55          186.722940260512          42
56          197.123408033022          45
57          208.103181860461          46
58          219.694529090089          47
59          231.931514360407          50
60          244.850099710282          53
61          258.488250264145          55
62          272.886045803857          58
63          288.085798555132          61
64          304.132177534653          62
65          321.072339823333          66
66          338.956069151493          68
67          357.835922203231          71
68          377.767383069951          74
69          398.809026306948          78
70          421.022689072245          82
71          444.473652853569          86
72          469.230835317512          91
73          495.366992844698          94
74          522.958934346148          98
75          552.087746989228          101
76          582.839034496528          106
77          615.303168717985          112
78          649.575555215577          118
79          685.756913641084          124
80          723.953573730893          128
81          764.277787787704          135
82          806.848060567479          139
83          851.789497541087          146
84          899.234172554126          154
85          949.321515965391          161
86          1002.19872440466          168
87          1058.021193354             177
88          1116.95297382382          186
89          1179.16725446581          193
90          1244.84687053955          203
91          1314.18484122861          214
92          1387.38493688504          221
93          1464.66227786954          232
94          1546.24396674687          243
95          1632.36975569467          258
96          1723.29275108686          269
97          1819.2801573224          280
98          1920.61406208526          293
99          2027.59226534341          307
100          2140.52915452304          322
101          2259.75662842997          335
102          2385.62507263352          354
103          2518.50438917921          368
104          2658.78508365649          384
105          2806.87941281616          409
106          2963.22259611002          427
107          3128.27409471335          445
108          3302.51896178888          464
109          3486.46926796052          487
110          3680.66560618592          514
111          3885.67868045048          538
112          4102.11098295157          565
113          4330.59856470197          591
114          4571.81290475587          619
115          4826.46288355078          649
116          5095.29686616455          680
117          5379.10490160992          708
118          5678.72104462959          747
119          5995.02580681546          783
120          6328.94874425508          823
121          6681.47118931009          861
122          7053.62913455467          906
123          7446.51627734936          942
124          7861.28723399772          992
125          8299.16093293139          1042
126          8761.42419689567          1093
127          9249.43552466276          1146
128          9764.62908338648          1203
129          10308.5189233311          1264
130          10882.7034273607          1322
131          11488.8700082646          1384
132          12128.800067725          1452
133          12804.3742314973          1526
134          13517.5778761917          1601
135          14270.5069638955          1675
136          15065.3742017845          1759
137          15904.5155448239          1853
138          16790.3970606706          1939
139          17725.62217695             2034
140          18712.9393322061          2135
141          19755.25005301             2236
142          20855.6174809626          2345
143          22017.2753746522          2466
144          23243.6376130204          2592
145          24538.3082280656          2722
146          25905.0919963689          2849
147          27348.0056205666          2991
148          28871.2895336322          3144
149          30479.4203606555          3290
150          32177.124074744          3451
151          33969.3898857072          3637
152          35861.4849023412          3809
153          37858.9696114016          4003
154          39967.7142187566          4199
155          42193.9159007414          4411
156          44544.1170164127          4631
157          47025.2243342269          4852
158          49644.5293296433          5099
159          52409.7296133044          5355
160          55328.9515527655          5617
161          58410.7741542545          5913
162          61664.2542746465          6204
163          65098.9532377443          6502
164          68724.9649330867          6830
165          72552.9454798596          7176
166          76594.1445430878          7533
167          80860.4383941378          7914
168          85364.3648126913          8310
169          90119.1599327582          8727
170          95138.7971410129          9171
171          100438.028141767          9627
172          106032.426309264          10109
173          111938.43245469          10614
174          118173.403142416          11148
175          124755.661697449          11713
176          131704.552053996          12300
177          139040.495603404          12928
178          146785.051208514          13572
179          154960.978560828          14270
180          163592.305066666          14979
181          172704.396458879          15730
182          182324.031341639          16522
183          192479.479887368          17367
184          203200.586917095          18236
185          214518.859608377          19172
186          226467.560088563          20140
187          239081.803185496          21148
188          252398.659622929          22234
189          266457.264963926          23349
190          281298.934622416          24542
191          296967.285280885          25768
192          313508.36307103          27072
193          330970.778894087          28482
194          349405.851278487          29929
195          368867.757194699          31447
196          389413.691270444          33016
197          411104.033874208          34698
198          434002.528561001          36481
199          458176.469401849          38326
200          483696.898747532          40291
201          510638.81600777          42342
202          539081.398059402          44505
203          569108.231931311          46749
204          600807.560449885          49156
205          634272.541566944          51659
206          669601.522132223          54275
207          706898.326914988          57069
208          746272.563724153          59968
209          787839.945523588          63049
210          831722.630489252          66294
211          878049.581007503          69675
212          926956.942669621          73256
213          978588.444376319          76973
214          1033095.82072808          80917
215          1090639.25794263          85019
216          1151387.86461004          89409
217          1215520.16866882          94030
218          1283224.64206367          98844
219          1354700.25462662          103882
220          1430157.05880932          109211
221          1509816.806985             114871
222          1593913.60313406          120707
223          1682694.59082863          126941
224          1776420.67953779          133423
225          1875367.31138804          140320
226          1979825.27063236          147520
227          2090101.53820658          155140
228          2206520.19388469          163110
229          2329423.36868406          171494
230          2459172.25031976          180311
231          2596148.14466258          189617
232          2740753.59632028          199375
233          2893413.57163532          209673
234          3054576.70757541          220442
235          3224716.63018736          231885
236          3404333.3464888          243847
237          3593954.71388822          256310
238          3794137.9914518          269598
239          4005471.47757566          283502
240          4228576.23887663          298197
241          4464107.93538206          313591
242          4712758.74738284          329791
243          4975259.40961206          346886
244          5252381.35872745          364839
245          5544939.00040857          383659
246          5853792.10273133          403452
247          6179848.32285347          424372
248          6524065.8744364          446297
249          6887456.34364251          469491
250          7271087.6619834          493825
251          7676087.24475588          519384
252          8103645.30428878          546247
253          8555018.34773767          574625
254          9031532.86970665          604399
255          9534589.25054932          635747
256          10065665.8718049          668696
257          10626323.4608644          703410
258          11218209.6776346          739982
259          11843063.9566788          778387
260          12502722.6190659          818861
261          13199124.2689478          861346
262          13934315.4907282          906155
263          14710456.8635618          953193
264          15529829.3108622          1002700
265          16394840.8034772          1054833
266          17308033.4362309          1109772
267          18272090.898629          1167590
268          19289846.3616826          1228264
269          20364290.8040283          1292241
270          21498581.8018127          1359539
271          22696052.8081737          1430281
272          23960222.9495889          1504767
273          25294807.367881          1583129
274          26703728.138272          1665730
275          28191125.7955738          1752542
276          29761371.5023872          1843920
277          31419079.8950702          1940204
278          33169122.6452256          2041419
279          35016642.7765647          2147716
280          36967069.7792193          2259708
281          39026135.5659219          2377893
282          41199891.3169437          2502200
283          43494725.2632975          2632695
284          45917381.4604631          2770398
285          48474979.6078109          2914910
286          51175035.971966          3067284
287          54025485.4756045          3227624
288          57034705.0165957          3396374
289          60211538.0860201          3573876
290          63565320.7574114          3760809
291          67105909.1235992          3957639
292          70843708.2617837          4164719
293          74789702.8119651          4382721
294          78955489.2585915          4612018
295          83353310.0102951          4853562
296          87996089.3778685          5107622
297          92897471.5562158          5375183
298          98071860.721897          5656864
299          103534463.364107          5953262
300          109301332.973487          6265488
301          115389417.220111          6593630
302          121816607.759271          6939148
303          128601792.811462          7303255
304          135764912.671061          7686228
305          143327018.306839          8089534
306          151310333.22653          8513877
307          159738318.787248          8960507
308          168635743.143697          9431001
309          178028754.036801          9926516
310          187944955.636651          10447664
311          198413489.665612          10995918
312          209465121.039987          11573590
313          221132328.281914          12181384
314          233449398.967217          12821555
315          246452530.489691          13495743
316          260179936.437967          14205163
317          274671958.897562          14951983
318          289971187.008156          15738181
319          306122582.12451          16565951
320          323173609.948845          17437154
321          341174380.022996          18354530
322          360177792.990277          19320137
323          380239696.059835          20337515
324          401419047.130368          21407706
325          423778088.05553          22534822
326          447382527.560223          23721524
327          472301734.345328          24971042
328          498608940.948362          26286537
329          526381458.959186          27670350
330          555700906.223213          29128822
331          586653446.699846          30663759
332          619330043.681027          32280161
333          653826727.11406          33982055
334          690244875.814314          35773848
335          728691515.397171          37660360
336          769279632.804793          39645685
337          812128508.352021          41737832
338          857364066.267228          43939904
339          905119244.758313          46257327
340          955534386.691351          48699221
341          1008757652.03006          51269955
342          1064945453.24813          53976631
343          1124262914.99405          56827224
344          1186884359.35922          59827076
345          1252993818.17553          62986725
346          1322785573.84791          66313913
347          1396464730.31124          69817652
348          1474247815.78957          73505982
349          1556363419.12905          77391836
350          1643052861.57454          81482256
351          1734570905.96424          85788976
352          1831186505.42645          90324673
353          1933183593.77871          95100020
354          2040861919.95218          100129987
355          2154537928.89352          105425905
356          2274545691.53289          111003798
357          2401237886.55127          116875507
358          2534986836.83217          123060363
359          2676185603.64373          129573148
360          2825249141.76668          136431322
361          2982615518.96309          143653231
362          3148747203.36933          151258073
363          3324132422.597             159267380
364          3509286598.53566          167702290
365          3704753862.07409          176585244
366          3911108652.19162          185939975
367          4128957404.11869          195789743
---------------------------------------------------------

天山草 · 发表于 2011-7-2 19:33

[这个贴子最后由天山草在 2011/07/02 08:21pm 第 1 次编辑]

至于 Li(x) 的计算，以前算过，程序是现成的，计算公式与陆教授的一样：

qingjiao · 发表于 2011-7-2 20:25

[这个贴子最后由qingjiao在 2011/07/02 08:28pm 第 1 次编辑]

我有到 42 亿的，每 2 亿一个文件，都是 TXT 的，每个约 100 兆。0-2 亿的是 113 M，越往后越小，40-42 亿的是 98 M。不知能不能给你发过去。你邮箱号多少？我试试。
=====================================
可能还是太大了，你先试试将2亿以内的压缩成rar（选“最好”或最大压缩项）再发过来，我看能不能收到。
drugengmen@21cn.com

qingjiao · 发表于 2011-7-2 20:27

你的程序有自动控制误差的功能吗？取到第几项？

天山草 · 发表于 2011-7-2 20:34

下面这个是最终结果：
序号    x          π(x)             Li(x)          (Li(x)-π(x))/(√x/lnx)
--------------------------------------------------------------------------------
  1    10.0002       4          5.12052258318897       .81589774304677
  2    10.5572       4          5.35959414467818       .986186678097735
  3    11.1453       5          5.60627467968866       .437848615970672
  4    11.7661       5          5.86088199147018       .618704793892662
  5    12.4214       5          6.12379406963947       .803345141208751
  6    13.1133       6          6.39548191946793       .281071127625926
  7    13.8437       6          6.67631002150975       .477657984430855
  8    14.6148       6          6.96673385726551       .678226703857135
  9    15.4289       6          7.26721815044797       .88275150528923
  10 16.2882       6          7.57816465624587       1.09116040441113
  11 17.1955       7          7.90016460917887       .617508615787195
  12 18.1533       7          8.23367092562627       .839358774267299
  13 19.1644       8          8.57921447562839       .390717838634153
  14 20.2319       8          8.93739746887694       .626724252744634
  15 21.3588       8          9.30875690779956       .8669612680541
  16 22.5485       8          9.69393117794411       1.11144552888736
  17 23.8044       9          10.0935256452389       .710463652876796
  18 25.1303       9          10.5082399666099       .970010976347258
  19 26.5301       9          10.9387692672063       1.23396465224727
  20 28.0078       9          11.3858050087046       1.50232327588888
  21 29.5679       10       11.8501539082709       1.15231966766512
  22 31.2148       11       12.332555020514       .820683980402608
  23 32.9535       11       12.8338858159891       1.11655677244572
  24 34.789          11       13.3549831340883       1.41712999337757
  25 36.7267       11       13.8967578306098       1.72244850007961
  26 38.7724       12       14.460189901965       1.44516155283647
  27 40.932          12       15.0462432451937       1.76738345871466
  28 43.2119       14       15.6559736772199       .948736111421694
  29 45.6188       14       16.2904707350131       1.29554514405137
  30 48.1598       15       16.9508818223834       1.08919854581222
  31 50.8423       15       17.6383834844681       1.45370861150617
  32 53.6742       16       18.3542303839759       1.2798795799985
  33 56.6639       16       19.0997508279356       1.66244419472522
  34 59.82          17       19.876269361704       1.52149990069109
  35 63.152          18       20.6852766969878       1.40080302810611
  36 66.6696       18       21.5282531674955       1.81475983476947
  37 70.3831       19       22.4067643062144       1.72743055868217
  38 74.3034       21       23.3224564032613       1.16073938149207
  39 78.4421       21       24.2770753627014       1.614111019719
  40 82.8113       22       25.272416290754       1.58821145705642
  41 87.4239       23       26.3103886365043       1.58287309087706
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  317 274671958.8976 14951983 14952920.6298937       1.0993130989008
  318 289971187.0082 15738181 15739172.8677512       1.13496708825012
  319 306122582.1245 16565951 16566913.5706478       1.07497482030844
  320 323173609.9488 17437154 17438338.7004452       1.29124028273308
  321 341174380.023 18354530 18355760.7648104       1.30919126432012
  322 360177792.9903 19320137 19321615.0175309       1.5343831543349
  323 380239696.0598 20337515 20338465.9894256       .963502499190248
  324 401419047.1304 21407706 21409014.3674869       1.29367907287781
  325 423778088.0555 22534822 22536104.2408183       1.2373222085874
  326 447382527.5602 23721524 23722730.7330924       1.13641668074312
  327 472301734.3453 24971042 24972048.0420151       .924596929121618
  328 498608940.9484 26286537 26287377.9076923       .754208471043152
  329 526381458.9592 27670350 27672218.5327167       1.63548798633959
  330 555700906.2232 29128822 29130253.978207       1.22316055536431
  331 586653446.6998 30663759 30665364.0611543       1.33793773549399
  332 619330043.681 32280161 32281634.7799408       1.19886723080046
  333 653826727.1141 33982055 33983369.2962302       1.04333247794793
  334 690244875.8143 35773848 35775099.5029925       .96950326543559
  335 728691515.3972 37660360 37661598.2100352       .936044599652476
  336 769279632.8048 39645685 39647891.9799659       1.62810491771621
  337 812128508.352 41737832 41739274.6495081       1.03854078389065
  338 857364066.2672 43939904 43941321.5727108       .995826693079839
  339 905119244.7583 46257327 46259904.6247267       1.76697655678386
  340 955534386.6914 48699221 48701208.0068336       1.32916879967915
  341 1008757652.0301  51269955 51271744.8955497       1.16835591268309
  342 1064945453.2481  53976631 53978374.9810406       1.11084370420036
  343 1124262914.994 56827224 56828322.9423658       .68304118047824
  344 1186884359.3592  59827076 59829197.9096986       1.28693527340969
  345 1252993818.1755  62986725 62989013.9663689       1.35464073457375
  346 1322785573.8479  66313913 66316211.746373       1.32747937112571
  347 1396464730.3112  69817652 69819681.1859523       1.14342499943137
  348 1474247815.7896  73505982 73508785.4910808       1.54145641464038
  349 1556363419.129 77391836 77393386.3858686       .831792965893358
  350 1643052861.5745  81482256 81483870.7105848       .84529958856521
  351 1734570905.9642  85788976 85791178.4414083       1.1250142372199
  352 1831186505.4264  90324673 90326832.2081793       1.0761753535638
  353 1933183593.7787  95100020 95102968.3902894       1.4338560799466
  354 2040861919.9522  100129987 100132369.875243       1.13070983423895
  355 2154537928.8935  105425905 105428500.568974       1.20173562519687
  356 2274545691.5329  111003798 111005541.751725       .787743351871847
  357 2401237886.5513  116875507 116878430.378337       1.28856433468191
  358 2534986836.8322  123060363 123062899.427169       1.09084169518711
  359 2676185603.6437  129573148 129575520.407371       .995506561835587
  360 2825249141.7667  136431322 136433748.140162       .993307171651833
  361 2982615518.9631  143653231 143655967.935968       1.09330780384415
  362 3148747203.3693  151258073 151261545.29584       1.35332460284028
  363 3324132422.597 159267380 159270878.272409       1.33028183335145
  364 3509286598.5357  167702290 167705452.632919       1.1733861196362
  365 3704753862.0741  176585244 176587899.974577       .961425319837485
  366 3911108652.1916  185939975 185942058.950454       .735996730399177
  367 4128957404.1187  195789743 195793039.772686       1.13598275199717
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