请天山草网友帮忙计算和画图

qingjiao · 发表于 2011-6-30 11:04

下面引用由天山草在 2011/06/30 10:23am 发表的内容： 先算了 x=2,3,4,5,6,……,1000 这些数。先发上来 x=2,3,4,5,6,……,100 这些数，数据多了网络不给俺发。请楼主先生看一下这些计算结果是否正确？（抽验一下前面的几个数）若不正确，那就是俺对楼主的函数没有正 ...

以前算过一些数，现在一时间找不到，不过看来你的计算应是正确的。正如所料，ψ(x)与x很接近，而且大部分情况ψ(x)-x<0，少数情况>0。如算到40亿，建议还是横坐标取等比数列，纵坐标绝对值取对数。

天山草 · 发表于 2011-6-30 11:07

能发上去啦，再把 r(x) 的前 999 个数据发一下：
x                ψ(x)                r(x)= ψ(x)-x
--------------------------------------------------------------
2          .693147180559945          -1.30685281944005
3          1.79175946922805          -1.20824053077195
4          2.484906649788          -1.515093350212
5          4.0943445622221          -.9056554377779
6          4.0943445622221          -1.9056554377779
7          6.04025471127741          -.959745288722586
8          6.04025471127741          -1.95974528872259
9          7.83201418050547          -1.16798581949453
10          7.83201418050547          -2.16798581949453
11          10.2299094533038          -.77009054669616
12          10.2299094533038          -1.77009054669616
13          12.7948588107654          -.205141189234624
14          12.7948588107654          -1.20514118923462
15          12.7948588107654          -2.20514118923462
16          13.4880059913253          -2.51199400867468
17          16.3212193353815          -.678780664618461
18          16.3212193353815          -1.67878066461846
19          19.265658314548          .265658314547977
20          19.265658314548          -.734341685452023
21          19.265658314548          -1.73434168545202
22          19.265658314548          -2.73434168545202
23          22.4011525304771          -.598847469522873
24          22.4011525304771          -1.59884746952287
25          24.0105904429112          -.98940955708877
26          24.0105904429112          -1.98940955708877
27          24.0105904429112          -2.98940955708877
28          25.1092027315793          -2.89079726842066
29          28.4764985615658          -.523501438434188
30          28.4764985615658          -1.52350143843419
31          31.910485766051          .910485766050957
32          32.6036329466109          .603632946610908
33          32.6036329466109          -.396367053389092
34          32.6036329466109          -1.39636705338909
35          32.6036329466109          -2.39636705338909
36          32.6036329466109          -3.39636705338909
37          36.2145508592551          -.785449140744866
38          36.2145508592551          -1.78544914074487
39          36.2145508592551          -2.78544914074487
40          36.2145508592551          -3.78544914074487
41          39.9281229259594          -1.07187707404056
42          39.9281229259594          -2.07187707404056
43          43.689323041653          .689323041653005
44          43.689323041653          -.310676958346995
45          43.689323041653          -1.310676958347
46          43.689323041653          -2.310676958347
47          47.5394706433631          .53947064336306
48          47.5394706433631          -.46052935663694
49          49.4853807924184          .485380792418368
50          49.4853807924184          -.514619207581632
51          49.4853807924184          -1.51461920758163
52          49.4853807924184          -2.51461920758163
53          53.4556727059705          .455672705970493
54          53.4556727059705          -.544327294029507
55          53.4556727059705          -1.54432729402951
56          53.4556727059705          -2.54432729402951
57          53.4556727059705          -3.54432729402951
58          53.4556727059705          -4.54432729402951
59          57.5332101498762          -1.46678985012379
60          57.5332101498762          -2.46678985012379
61          61.6440840140495          .644084014049525
62          61.6440840140495          -.355915985950475
63          61.6440840140495          -1.35591598595047
64          61.6440840140495          -2.35591598595047
65          62.3372311946095          -2.66276880539053
66          62.3372311946095          -3.66276880539053
67          66.5419238140004          -.458076185999559
68          66.5419238140004          -1.45807618599956
69          66.5419238140004          -2.45807618599956
70          66.5419238140004          -3.45807618599956
71          70.8046036910418          -.195396308958237
72          70.8046036910418          -1.19539630895824
73          75.0950631321901          2.09506313219015
74          75.0950631321901          1.09506313219015
75          75.0950631321901          9.50631321901483E-02
76          75.0950631321901          -.904936867809852
77          75.0950631321901          -1.90493686780985
78          75.0950631321901          -2.90493686780985
79          79.4645109846572          .464510984657167
80          79.4645109846572          -.535489015342833
81          80.5631232733253          -.436876726674726
82          80.5631232733253          -1.43687672667473
83          84.9819638811219          1.98196388112187
84          84.9819638811219          .981963881121871
85          84.9819638811219          -1.80361188781291E-02
86          84.9819638811219          -1.01803611887813
87          84.9819638811219          -2.01803611887813
88          84.9819638811219          -3.01803611887813
89          89.470600250854          .470600250854005
90          89.470600250854          -.529399749145995
91          89.470600250854          -1.52939974914599
92          89.470600250854          -2.52939974914599
93          89.470600250854          -3.52939974914599
94          89.470600250854          -4.52939974914599
95          89.470600250854          -5.52939974914599
96          89.470600250854          -6.52939974914599
97          94.0453112293574          -2.9546887706426
98          94.0453112293574          -3.9546887706426
99          94.0453112293574          -4.9546887706426
100          94.0453112293574          -5.9546887706426
101          98.6604317461987          -2.33956825380135
102          98.6604317461987          -3.33956825380135
103          103.295160734428          .295160734428293
104          103.295160734428          -.704839265571707
105          103.295160734428          -1.70483926557171
106          103.295160734428          -2.70483926557171
107          107.96798956889          .967989568890204
108          107.96798956889          -3.20104311097964E-02
109          112.659337451119          3.65933745111934
110          112.659337451119          2.65933745111934
111          112.659337451119          1.65933745111934
112          112.659337451119          .659337451119342
113          117.386725269832          4.38672526983169
114          117.386725269832          3.38672526983169
115          117.386725269832          2.38672526983169
116          117.386725269832          1.38672526983169
117          117.386725269832          .386725269831686
118          117.386725269832          -.613274730168314
119          117.386725269832          -1.61327473016831
120          117.386725269832          -2.61327473016831
121          119.78462054263          -1.21537945736995
122          119.78462054263          -2.21537945736995
123          119.78462054263          -3.21537945736995
124          119.78462054263          -4.21537945736995
125          121.394058455064          -3.60594154493586
126          121.394058455064          -4.60594154493586
127          126.238245541523          -.76175445847727
128          126.238245541523          -1.76175445847727
129          126.931392722083          -2.06860727791729
130          126.931392722083          -3.06860727791729
131          131.806590045284          .806590045283855
132          131.806590045284          -.193409954716145
133          131.806590045284          -1.19340995471615
134          131.806590045284          -2.19340995471615
135          131.806590045284          -3.19340995471615
136          131.806590045284          -4.19340995471615
137          136.726570971112          -.273429028888017
138          136.726570971112          -1.27342902888802
139          141.661044904243          2.66104490424269
140          141.661044904243          1.66104490424269
141          141.661044904243          .661044904242686
142          141.661044904243          -.338955095757314
143          141.661044904243          -1.33895509575731
144          141.661044904243          -2.33895509575731
145          141.661044904243          -3.33895509575731
146          141.661044904243          -4.33895509575731
147          141.661044904243          -5.33895509575731
148          141.661044904243          -6.33895509575731
149          146.664991210188          -2.33500878981187
150          146.664991210188          -3.33500878981187
151          151.682271047003          .68227104700307
152          151.682271047003          -.31772895299693
153          151.682271047003          -1.31772895299693
154          151.682271047003          -2.31772895299693
155          151.682271047003          -3.31772895299693
156          151.682271047003          -4.31772895299693
157          156.738516852351          -.261483147648619
158          156.738516852351          -1.26148314764862
159          156.738516852351          -2.26148314764862
160          156.738516852351          -3.26148314764862
161          156.738516852351          -4.26148314764862
162          156.738516852351          -5.26148314764862
163          161.832267053158          -1.16773294684185
164          161.832267053158          -2.16773294684185
165          161.832267053158          -3.16773294684185
166          161.832267053158          -4.16773294684185
167          166.950260865575          -4.97391344250957E-02
168          166.950260865575          -1.0497391344251
169          169.515210223036          .515210223036433
170          169.515210223036          -.484789776963567
171          169.515210223036          -1.48478977696357
172          169.515210223036          -2.48478977696357
173          174.668501817534          1.66850181753421
174          174.668501817534          .668501817534207
175          174.668501817534          -.331498182465793
176          174.668501817534          -1.33149818246579
177          174.668501817534          -2.33149818246579
178          174.668501817534          -3.33149818246579
179          179.855887623375          .855887623374969
180          179.855887623375          -.144112376625031
181          185.054384654641          4.05438465464078
182          185.054384654641          3.05438465464078
183          185.054384654641          2.05438465464078
184          185.054384654641          1.05438465464078
185          185.054384654641          5.43846546407849E-02
186          185.054384654641          -.945615345359215
187          185.054384654641          -1.94561534535922
188          185.054384654641          -2.94561534535922
189          185.054384654641          -3.94561534535922
190          185.054384654641          -4.94561534535922
191          190.306658082687          -.69334191731258
192          190.306658082687          -1.69334191731258
193          195.569348271592          2.5693482715923
194          195.569348271592          1.5693482715923
195          195.569348271592          .569348271592304
196          195.569348271592          -.430651728407696
197          200.85255200033          3.85255200033029
198          200.85255200033          2.85255200033029
199          206.145856825055          7.14585682505478
200          206.145856825055          6.14585682505478
201          206.145856825055          5.14585682505478
202          206.145856825055          4.14585682505478
203          206.145856825055          3.14585682505478
204          206.145856825055          2.14585682505478
205          206.145856825055          1.14585682505478
206          206.145856825055          .145856825054778
207          206.145856825055          -.854143174945222
208          206.145856825055          -1.85414317494522
209          206.145856825055          -2.85414317494522
210          206.145856825055          -3.85414317494522
211          211.497714958531          .49771495853085
212          211.497714958531          -.50228504146915
213          211.497714958531          -1.50228504146915
214          211.497714958531          -2.50228504146915
215          211.497714958531          -3.50228504146915
216          211.497714958531          -4.50228504146915
217          211.497714958531          -5.50228504146915
218          211.497714958531          -6.50228504146915
219          211.497714958531          -7.50228504146915
220          211.497714958531          -8.50228504146915
221          211.497714958531          -9.50228504146915
222          211.497714958531          -10.5022850414692
223          216.904886729991          -6.09511327000902
224          216.904886729991          -7.09511327000902
225          216.904886729991          -8.09511327000902
226          216.904886729991          -9.09511327000902
227          222.329836747472          -4.67016325252763
228          222.329836747472          -5.67016325252763
229          227.763558751027          -1.23644124897339
230          227.763558751027          -2.23644124897339
231          227.763558751027          -3.23644124897339
232          227.763558751027          -4.23644124897339
233          233.214597204592          .214597204592309
234          233.214597204592          -.785402795407691
235          233.214597204592          -1.78540279540769
236          233.214597204592          -2.78540279540769
237          233.214597204592          -3.78540279540769
238          233.214597204592          -4.78540279540769
239          238.691060756524          -.308939243476175
240          238.691060756524          -1.30893924347617
241          244.175857690014          3.17585769001448
242          244.175857690014          2.17585769001448
243          244.175857690014          1.17585769001448
244          245.274469978683          1.27446997868259
245          245.274469978683          .274469978682589
246          245.274469978683          -.725530021317411
247          245.274469978683          -1.72553002131741
248          245.274469978683          -2.72553002131741
249          245.274469978683          -3.72553002131741
250          245.274469978683          -4.72553002131741
251          250.799922917814          -.200077082185629
252          250.799922917814          -1.20007708218563
253          250.799922917814          -2.20007708218563
254          250.799922917814          -3.20007708218563
255          250.799922917814          -4.20007708218563
256          251.493070098374          -4.50692990162568
257          257.04214618327          4.21461832695513E-02
258          257.04214618327          -.957853816730449
259          257.04214618327          -1.95785381673045
260          257.04214618327          -2.95785381673045
261          257.04214618327          -3.95785381673045
262          257.04214618327          -4.95785381673045
263          262.614300215447          -.385699784552685
264          262.614300215447          -1.38569978455268
265          262.614300215447          -2.38569978455268
266          262.614300215447          -3.38569978455268
267          262.614300215447          -4.38569978455268
268          262.614300215447          -5.38569978455268
269          268.209011595049          -.790988404950838
270          268.209011595049          -1.79098840495084
271          273.811130415929          2.81113041592886
272          273.811130415929          1.81113041592886
273          273.811130415929          .811130415928858
274          273.811130415929          -.188869584071142
275          273.811130415929          -1.18886958407114
276          273.811130415929          -2.18886958407114
277          279.435147922116          2.43514792211619
278          279.435147922116          1.43514792211619
279          279.435147922116          .435147922116187
280          279.435147922116          -.564852077883813
281          285.07350259145          4.07350259144994
282          285.07350259145          3.07350259144994
283          290.718949489093          7.71894948909318
284          290.718949489093          6.71894948909318
285          290.718949489093          5.71894948909318
286          290.718949489093          4.71894948909318
287          290.718949489093          3.71894948909318
288          290.718949489093          2.71894948909318
289          293.552162833149          4.5521628331494
290          293.552162833149          3.5521628331494
291          293.552162833149          2.5521628331494
292          293.552162833149          1.5521628331494
293          299.232335442166          6.23233544216646
294          299.232335442166          5.23233544216646
295          299.232335442166          4.23233544216646
296          299.232335442166          3.23233544216646
297          299.232335442166          2.23233544216646
298          299.232335442166          1.23233544216646
299          299.232335442166          .232335442166459
300          299.232335442166          -.767664557833541
301          299.232335442166          -1.76766455783354
302          299.232335442166          -2.76766455783354
303          299.232335442166          -3.76766455783354
304          299.232335442166          -4.76766455783354
305          299.232335442166          -5.76766455783354
306          299.232335442166          -6.76766455783354
307          304.959183189754          -2.04081681024633
308          304.959183189754          -3.04081681024633
309          304.959183189754          -4.04081681024633
310          304.959183189754          -5.04081681024633
311          310.698976101933          -.301023898067115
312          310.698976101933          -1.30102389806711
313          316.445179292473          3.44517929247303
314          316.445179292473          2.44517929247303
315          316.445179292473          1.44517929247303
316          316.445179292473          .445179292473028
317          322.20408106635          5.20408106635028
318          322.20408106635          4.20408106635028
319          322.20408106635          3.20408106635028
320          322.20408106635          2.20408106635028
321          322.20408106635          1.20408106635028
322          322.20408106635          .204081066350284
323          322.20408106635          -.795918933649716
324          322.20408106635          -1.79591893364972
325          322.20408106635          -2.79591893364972
326          322.20408106635          -3.79591893364972
327          322.20408106635          -4.79591893364972
328          322.20408106635          -5.79591893364972
329          322.20408106635          -6.79591893364972
330          322.20408106635          -7.79591893364972
331          328.006199441727          -2.99380055827265
332          328.006199441727          -3.99380055827265
333          328.006199441727          -4.99380055827265
334          328.006199441727          -5.99380055827265
335          328.006199441727          -6.99380055827265
336          328.006199441727          -7.99380055827265
337          333.82628237208          -3.17371762792027
338          333.82628237208          -4.17371762792027
339          333.82628237208          -5.17371762792027
340          333.82628237208          -6.17371762792027
341          333.82628237208          -7.17371762792027
342          333.82628237208          -8.17371762792027
343          333.82628237208          -9.17371762792027
344          335.772192521135          -8.22780747886497
345          335.772192521135          -9.22780747886497
346          335.772192521135          -10.227807478865
347          341.621517301082          -5.37848269891811
348          341.621517301082          -6.37848269891811
349          347.476589223284          -1.52341077671565
350          347.476589223284          -2.52341077671565
351          347.476589223284          -3.52341077671565
352          347.476589223284          -4.52341077671565
353          353.343057280218          .343057280217636
354          353.343057280218          -.656942719782364
355          353.343057280218          -1.65694271978236
356          353.343057280218          -2.65694271978236
357          353.343057280218          -3.65694271978236
358          353.343057280218          -4.65694271978236
359          359.226379668706          .226379668705931
360          359.226379668706          -.773620331294069
361          362.170818647872          1.17081864787235
362          362.170818647872          .170818647872352
363          362.170818647872          -.829181352127648
364          362.170818647872          -1.82918135212765
365          362.170818647872          -2.82918135212765
366          362.170818647872          -3.82918135212765
367          368.076180495927          1.07618049592691
368          368.076180495927          7.61804959269057E-02
369          368.076180495927          -.923819504073094
370          368.076180495927          -1.92381950407309
371          368.076180495927          -2.92381950407309
372          368.076180495927          -3.92381950407309
373          373.997758915571          .997758915570728
374          373.997758915571          -2.24108442927218E-03
375          373.997758915571          -1.00224108442927
376          373.997758915571          -2.00224108442927
377          373.997758915571          -3.00224108442927
378          373.997758915571          -4.00224108442927
379          379.935295120653          .935295120653166
380          379.935295120653          -6.47048793468343E-02
381          379.935295120653          -1.06470487934683
382          379.935295120653          -2.06470487934683
383          385.883330109834          2.88333010983382
384          385.883330109834          1.88333010983382
385          385.883330109834          .883330109833821
386          385.883330109834          -.116669890166179
387          385.883330109834          -1.11666989016618
388          385.883330109834          -2.11666989016618
389          391.846909453452          2.84690945345227
390          391.846909453452          1.84690945345227
391          391.846909453452          .84690945345227
392          391.846909453452          -.15309054654773
393          391.846909453452          -1.15309054654773
394          391.846909453452          -2.15309054654773
395          391.846909453452          -3.15309054654773
396          391.846909453452          -4.15309054654773
397          397.830845734139          .830845734139473
398          397.830845734139          -.169154265860527
399          397.830845734139          -1.16915426586053
400          397.830845734139          -2.16915426586053
401          403.824807161446          2.82480716144602
402          403.824807161446          1.82480716144602
403          403.824807161446          .824807161446017
404          403.824807161446          -.175192838553983
405          403.824807161446          -1.17519283855398
406          403.824807161446          -2.17519283855398
407          403.824807161446          -3.17519283855398
408          403.824807161446          -4.17519283855398
409          409.838522317489          .83852231748881
410          409.838522317489          -.16147768251119
411          409.838522317489          -1.16147768251119
412          409.838522317489          -2.16147768251119
413          409.838522317489          -3.16147768251119
414          409.838522317489          -4.16147768251119
415          409.838522317489          -5.16147768251119
416          409.838522317489          -6.16147768251119
417          409.838522317489          -7.16147768251119
418          409.838522317489          -8.16147768251119
419          415.876393237411          -3.12360676258908
420          415.876393237411          -4.12360676258908
421          421.919026071093          .919026071093299
422          421.919026071093          -8.09739289067011E-02
423          421.919026071093          -1.0809739289067
424          421.919026071093          -2.0809739289067
425          421.919026071093          -3.0809739289067
426          421.919026071093          -4.0809739289067
427          421.919026071093          -5.0809739289067
428          421.919026071093          -6.0809739289067
429          421.919026071093          -7.0809739289067
430          421.919026071093          -8.0809739289067
431          427.985134161197          -3.01486583880296
432          427.985134161197          -4.01486583880296
433          434.0558718892             1.05587188919952
434          434.0558718892             5.58718891995227E-02
435          434.0558718892          -.944128110800477
436          434.0558718892          -1.94412811080048
437          434.0558718892          -2.94412811080048
438          434.0558718892          -3.94412811080048
439          440.140371302275          1.14037130227467
440          440.140371302275          .140371302274673
441          440.140371302275          -.859628697725327
442          440.140371302275          -1.85962869772533
443          446.23394107232          3.23394107231979
444          446.23394107232          2.23394107231979
445          446.23394107232          1.23394107231979
446          446.23394107232          .233941072319794
447          446.23394107232          -.766058927680206
448          446.23394107232          -1.76605892768021
449          452.340963960062          3.34096396006203
450          452.340963960062          2.34096396006203
451          452.340963960062          1.34096396006203
452          452.340963960062          .340963960062027
453          452.340963960062          -.659036039937973
454          452.340963960062          -1.65903603993797
455          452.340963960062          -2.65903603993797
456          452.340963960062          -3.65903603993797
457          458.465647350956          1.46564735095626
458          458.465647350956          .46564735095626
459          458.465647350956          -.53435264904374
460          458.465647350956          -1.53435264904374
461          464.599045393953          3.59904539395291
462          464.599045393953          2.59904539395291
463          470.736772448039          7.73677244803912
464          470.736772448039          6.73677244803912
465          470.736772448039          5.73677244803912
466          470.736772448039          4.73677244803912
467          476.883101705708          9.88310170570804
468          476.883101705708          8.88310170570804
469          476.883101705708          7.88310170570804
470          476.883101705708          6.88310170570804
471          476.883101705708          5.88310170570804
472          476.883101705708          4.88310170570804
473          476.883101705708          3.88310170570804
474          476.883101705708          2.88310170570804
475          476.883101705708          1.88310170570804
476          476.883101705708          .883101705708043
477          476.883101705708          -.116898294291957
478          476.883101705708          -1.11689829429196
479          483.054802303119          4.05480230311895
480          483.054802303119          3.05480230311895
481          483.054802303119          2.05480230311895
482          483.054802303119          1.05480230311895
483          483.054802303119          5.48023031189473E-02
484          483.054802303119          -.945197696881053
485          483.054802303119          -1.94519769688105
486          483.054802303119          -2.94519769688105
487          489.243066426202          2.24306642620155
488          489.243066426202          1.24306642620155
489          489.243066426202          .243066426201551
490          489.243066426202          -.756933573798449
491          495.439510553996          4.43951055399606
492          495.439510553996          3.43951055399606
493          495.439510553996          2.43951055399606
494          495.439510553996          1.43951055399606
495          495.439510553996          .439510553996058
496          495.439510553996          -.560489446003942
497          495.439510553996          -1.56048944600394
498          495.439510553996          -2.56048944600394
499          501.652116649748          2.6521166497476
500          501.652116649748          1.6521166497476
501          501.652116649748          .652116649747597
502          501.652116649748          -.347883350252403
503          507.872706819847          4.87270681984734
504          507.872706819847          3.87270681984734
505          507.872706819847          2.87270681984734
506          507.872706819847          1.87270681984734
507          507.872706819847          .872706819847338
508          507.872706819847          -.127293180152662
509          514.105154836398          5.1051548363979
510          514.105154836398          4.1051548363979
511          514.105154836398          3.1051548363979
512          514.798302016958          2.79830201695779
513          514.798302016958          1.79830201695779
514          514.798302016958          .798302016957791
515          514.798302016958          -.201697983042209
516          514.798302016958          -1.20169798304221
517          514.798302016958          -2.20169798304221
518          514.798302016958          -3.20169798304221
519          514.798302016958          -4.20169798304221
520          514.798302016958          -5.20169798304221
521          521.054052058711          5.40520587111359E-02
522          521.054052058711          -.945947941288864
523          527.313633522776          4.31363352277606
524          527.313633522776          3.31363352277606
525          527.313633522776          2.31363352277606
526          527.313633522776          1.31363352277606
527          527.313633522776          .313633522776058
528          527.313633522776          -.686366477223942
529          530.449127738705          1.44912773870522
530          530.449127738705          .449127738705215
531          530.449127738705          -.550872261294785
532          530.449127738705          -1.55087226129478
533          530.449127738705          -2.55087226129478
534          530.449127738705          -3.55087226129478
535          530.449127738705          -4.55087226129478
536          530.449127738705          -5.55087226129478
537          530.449127738705          -6.55087226129478
538          530.449127738705          -7.55087226129478
539          530.449127738705          -8.55087226129478
540          530.449127738705          -9.55087226129478
541          536.742547017552          -4.25745298244829
542          536.742547017552          -5.25745298244829
543          536.742547017552          -6.25745298244829
544          536.742547017552          -7.25745298244829
545          536.742547017552          -8.25745298244829
546          536.742547017552          -9.25745298244829
547          543.046995819974          -3.95300418002626
548          543.046995819974          -4.95300418002626
549          543.046995819974          -5.95300418002626
550          543.046995819974          -6.95300418002626
551          543.046995819974          -7.95300418002626
552          543.046995819974          -8.95300418002626
553          543.046995819974          -9.95300418002626
554          543.046995819974          -10.9530041800263
555          543.046995819974          -11.9530041800263
556          543.046995819974          -12.9530041800263
557          549.369561059901          -7.630438940099
558          549.369561059901          -8.630438940099
559          549.369561059901          -9.630438940099
560          549.369561059901          -10.630438940099
561          549.369561059901          -11.630438940099
562          549.369561059901          -12.630438940099
563          555.702840688041          -7.29715931195926
564          555.702840688041          -8.29715931195926
565          555.702840688041          -9.29715931195926
566          555.702840688041          -10.2971593119593
567          555.702840688041          -11.2971593119593
568          555.702840688041          -12.2971593119593
569          562.046721122167          -6.95327887783299
570          562.046721122167          -7.95327887783299
571          568.394110331823          -2.60588966817693
572          568.394110331823          -3.60588966817693
573          568.394110331823          -4.60588966817693
574          568.394110331823          -5.60588966817693
575          568.394110331823          -6.60588966817693
576          568.394110331823          -7.60588966817693
577          574.751952598331          -2.24804740166883
578          574.751952598331          -3.24804740166883
579          574.751952598331          -4.24804740166883
580          574.751952598331          -5.24804740166883
581          574.751952598331          -6.24804740166883
582          574.751952598331          -7.24804740166883
583          574.751952598331          -8.24804740166883
584          574.751952598331          -9.24804740166883
585          574.751952598331          -10.2480474016688
586          574.751952598331          -11.2480474016688
587          581.126977418159          -5.8730225818407
588          581.126977418159          -6.8730225818407
589          581.126977418159          -7.8730225818407
590          581.126977418159          -8.8730225818407
591          581.126977418159          -9.8730225818407
592          581.126977418159          -10.8730225818407
593          587.512171817157          -5.48782818284303
594          587.512171817157          -6.48782818284303
595          587.512171817157          -7.48782818284303
596          587.512171817157          -8.48782818284303
597          587.512171817157          -9.48782818284303
598          587.512171817157          -10.487828182843
599          593.907433415272          -5.09256658472759
600          593.907433415272          -6.09256658472759
601          600.306028349808          -.693971650192339
602          600.306028349808          -1.69397165019234
603          600.306028349808          -2.69397165019234
604          600.306028349808          -3.69397165019234
605          600.306028349808          -4.69397165019234
606          600.306028349808          -5.69397165019234
607          606.714557140867          -.285442859132786
608          606.714557140867          -1.28544285913279
609          606.714557140867          -2.28544285913279
610          606.714557140867          -3.28544285913279
611          606.714557140867          -4.28544285913279
612          606.714557140867          -5.28544285913279
613          613.132922076803          .132922076803425
614          613.132922076803          -.867077923196575
615          613.132922076803          -1.86707792319658
616          613.132922076803          -2.86707792319658
617          619.557791100709          2.55779110070887
618          619.557791100709          1.55779110070887
619          625.985896373393          6.98589637339342
620          625.985896373393          5.98589637339342
621          625.985896373393          4.98589637339342
622          625.985896373393          3.98589637339342
623          625.985896373393          2.98589637339342
624          625.985896373393          1.98589637339342
625          627.595334285827          2.59533428582745
626          627.595334285827          1.59533428582745
627          627.595334285827          .595334285827448
628          627.595334285827          -.404665714172552
629          627.595334285827          -1.40466571417255
630          627.595334285827          -2.40466571417255
631          634.042640148369          3.04264014836861
632          634.042640148369          2.04264014836861
633          634.042640148369          1.0426401483686
634          634.042640148369          .042640148368605
635          634.042640148369          -.957359851631395
636          634.042640148369          -1.9573598516314
637          634.042640148369          -2.9573598516314
638          634.042640148369          -3.9573598516314
639          634.042640148369          -4.9573598516314
640          634.042640148369          -5.9573598516314
641          640.505669605289          -.494330394710687
642          640.505669605289          -1.49433039471069
643          646.971814329527          3.97181432952698
644          646.971814329527          2.97181432952698
645          646.971814329527          1.97181432952698
646          646.971814329527          .971814329526978
647          653.444160624028          6.44416062402786
648          653.444160624028          5.44416062402786
649          653.444160624028          4.44416062402786
650          653.444160624028          3.44416062402786
651          653.444160624028          2.44416062402786
652          653.444160624028          1.44416062402786
653          659.925737753304          6.92573775330425
654          659.925737753304          5.92573775330425
655          659.925737753304          4.92573775330425
656          659.925737753304          3.92573775330425
657          659.925737753304          2.92573775330425
658          659.925737753304          1.92573775330425
659          666.416461287807          7.41646128780678
660          666.416461287807          6.41646128780678
661          672.910215127658          11.9102151276585
662          672.910215127658          10.9102151276585
663          672.910215127658          9.91021512765849
664          672.910215127658          8.91021512765849
665          672.910215127658          7.91021512765849
666          672.910215127658          6.91021512765849
667          672.910215127658          5.91021512765849
668          672.910215127658          4.91021512765849
669          672.910215127658          3.91021512765849
670          672.910215127658          2.91021512765849
671          672.910215127658          1.91021512765849
672          672.910215127658          .910215127658489
673          679.421960457303          6.42196045730327
674          679.421960457303          5.42196045730327
675          679.421960457303          4.42196045730327
676          679.421960457303          3.42196045730327
677          685.939631730216          8.93963173021552
678          685.939631730216          7.93963173021552
679          685.939631730216          6.93963173021552
680          685.939631730216          5.93963173021552
681          685.939631730216          4.93963173021552
682          685.939631730216          3.93963173021552
683          692.466126589786          9.46612658978631
684          692.466126589786          8.46612658978631
685          692.466126589786          7.46612658978631
686          692.466126589786          6.46612658978631
687          692.466126589786          5.46612658978631
688          692.466126589786          4.46612658978631
689          692.466126589786          3.46612658978631
690          692.466126589786          2.46612658978631
691          699.004266413554          8.00426641355398
692          699.004266413554          7.00426641355398
693          699.004266413554          6.00426641355398
694          699.004266413554          5.00426641355398
695          699.004266413554          4.00426641355398
696          699.004266413554          3.00426641355398
697          699.004266413554          2.00426641355398
698          699.004266413554          1.00426641355398
699          699.004266413554          4.26641355397805E-03
700          699.004266413554          -.995733586446022
701          705.556774300589          4.55677430058859
702          705.556774300589          3.55677430058859
703          705.556774300589          2.55677430058859
704          705.556774300589          1.55677430058859
705          705.556774300589          .556774300588586
706          705.556774300589          -.443225699411414
707          705.556774300589          -1.44322569941141
708          705.556774300589          -2.44322569941141
709          712.120629827121          3.12062982712075
710          712.120629827121          2.12062982712075
711          712.120629827121          1.12062982712075
712          712.120629827121          .120629827120752
713          712.120629827121          -.879370172879248
714          712.120629827121          -1.87937017287925
715          712.120629827121          -2.87937017287925
716          712.120629827121          -3.87937017287925
717          712.120629827121          -4.87937017287925
718          712.120629827121          -5.87937017287925
719          718.698491184842          -.301508815158172
720          718.698491184842          -1.30150881515817
721          718.698491184842          -2.30150881515817
722          718.698491184842          -3.30150881515817
723          718.698491184842          -4.30150881515817
724          718.698491184842          -5.30150881515817
725          718.698491184842          -6.30150881515817
726          718.698491184842          -7.30150881515817
727          725.287417662375          -1.71258233762467
728          725.287417662375          -2.71258233762467
729          725.287417662375          -3.71258233762467
730          726.386029951044          -3.6139700489565
731          726.386029951044          -4.6139700489565
732          726.386029951044          -5.6139700489565
733          732.98317565293          -1.68243470698144E-02
734          732.98317565293          -1.01682434706981
735          732.98317565293          -2.01682434706981
736          732.98317565293          -3.01682434706981
737          732.98317565293          -4.01682434706981
738          732.98317565293          -5.01682434706981
739          739.588473573878          .588473573878332
740          739.588473573878          -.411526426121668
741          739.588473573878          -1.41152642612167
742          739.588473573878          -2.41152642612167
743          746.199169618596          3.1991696185961
744          746.199169618596          2.1991696185961
745          746.199169618596          1.1991696185961
746          746.199169618596          .199169618596102
747          746.199169618596          -.800830381403898
748          746.199169618596          -1.8008303814039
749          746.199169618596          -2.8008303814039
750          746.199169618596          -3.8008303814039
751          752.82057527036          1.82057527036022
752          752.82057527036          .820575270360223
753          752.82057527036          -.179424729639777
754          752.82057527036          -1.17942472963978
755          752.82057527036          -2.17942472963978
756          752.82057527036          -3.17942472963978
757          759.449938523798          2.44993852379764
758          759.449938523798          1.44993852379764
759          759.449938523798          .449938523797641
760          759.449938523798          -.550061476202359
761          766.084571881659          5.08457188165937
762          766.084571881659          4.08457188165937
763          766.084571881659          3.08457188165937
764          766.084571881659          2.08457188165937
765          766.084571881659          1.08457188165937
766          766.084571881659          8.45718816593717E-02
767          766.084571881659          -.915428118340628
768          766.084571881659          -1.91542811834063
769          772.729662851165          3.72966285116502
770          772.729662851165          2.72966285116502
771          772.729662851165          1.72966285116502
772          772.729662851165          .729662851165017
773          779.379941899752          6.37994189975245
774          779.379941899752          5.37994189975245
775          779.379941899752          4.37994189975245
776          779.379941899752          3.37994189975245
777          779.379941899752          2.37994189975245
778          779.379941899752          1.37994189975245
779          779.379941899752          .379941899752453
780          779.379941899752          -.620058100247547
781          779.379941899752          -1.62005810024755
782          779.379941899752          -2.62005810024755
783          779.379941899752          -3.62005810024755
784          779.379941899752          -4.62005810024755
785          779.379941899752          -5.62005810024755
786          779.379941899752          -6.62005810024755
787          786.04817014817          -.951829851830098
788          786.04817014817          -1.9518298518301
789          786.04817014817          -2.9518298518301
790          786.04817014817          -3.9518298518301
791          786.04817014817          -4.9518298518301
792          786.04817014817          -5.9518298518301
793          786.04817014817          -6.9518298518301
794          786.04817014817          -7.9518298518301
795          786.04817014817          -8.9518298518301
796          786.04817014817          -9.9518298518301
797          792.72902482696          -4.27097517303991
798          792.72902482696          -5.27097517303991
799          792.72902482696          -6.27097517303991
800          792.72902482696          -7.27097517303991
801          792.72902482696          -8.27097517303991
802          792.72902482696          -9.27097517303991
803          792.72902482696          -10.2709751730399
804          792.72902482696          -11.2709751730399
805          792.72902482696          -12.2709751730399
806          792.72902482696          -13.2709751730399
807          792.72902482696          -14.2709751730399
808          792.72902482696          -15.2709751730399
809          799.424823744019          -9.57517625598143
810          799.424823744019          -10.5751762559814
811          806.123091798134          -4.87690820186606
812          806.123091798134          -5.87690820186606
813          806.123091798134          -6.87690820186606
814          806.123091798134          -7.87690820186606
815          806.123091798134          -8.87690820186606
816          806.123091798134          -9.87690820186606
817          806.123091798134          -10.8769082018661
818          806.123091798134          -11.8769082018661
819          806.123091798134          -12.8769082018661
820          806.123091798134          -13.8769082018661
821          812.833614907586          -8.16638509241363
822          812.833614907586          -9.16638509241363
823          819.546571108263          -3.45342889173662
824          819.546571108263          -4.45342889173662
825          819.546571108263          -5.45342889173662
826          819.546571108263          -6.45342889173662
827          826.264375803287          -.735624196712934
828          826.264375803287          -1.73562419671293
829          832.984595958422          3.98459595842235
830          832.984595958422          2.98459595842235
831          832.984595958422          1.98459595842235
832          832.984595958422          .984595958422346
833          832.984595958422          -1.54040415776535E-02
834          832.984595958422          -1.01540404157765
835          832.984595958422          -2.01540404157765
836          832.984595958422          -3.01540404157765
837          832.984595958422          -4.01540404157765
838          832.984595958422          -5.01540404157765
839          839.71680666489          .716806664889532
840          839.71680666489          -.283193335110468
841          843.084102494876          2.08410249487599
842          843.084102494876          1.08410249487599
843          843.084102494876          8.41024948759923E-02
844          843.084102494876          -.915897505124008
845          843.084102494876          -1.91589750512401
846          843.084102494876          -2.91589750512401
847          843.084102494876          -3.91589750512401
848          843.084102494876          -4.91589750512401
849          843.084102494876          -5.91589750512401
850          843.084102494876          -6.91589750512401
851          843.084102494876          -7.91589750512401
852          843.084102494876          -8.91589750512401
853          849.832862042368          -3.16713795763235
854          849.832862042368          -4.16713795763235
855          849.832862042368          -5.16713795763235
856          849.832862042368          -6.16713795763235
857          856.586299960965          -.413700039034552
858          856.586299960965          -1.41370003903455
859          863.34206888295          4.34206888294966
860          863.34206888295          3.34206888294966
861          863.34206888295          2.34206888294966
862          863.34206888295          1.34206888294966
863          870.102483574033          7.10248357403304
864          870.102483574033          6.10248357403304
865          870.102483574033          5.10248357403304
866          870.102483574033          4.10248357403304
867          870.102483574033          3.10248357403304
868          870.102483574033          2.10248357403304
869          870.102483574033          1.10248357403304
870          870.102483574033          .102483574033045
871          870.102483574033          -.897516425966955
872          870.102483574033          -1.89751642596696
873          870.102483574033          -2.89751642596696
874          870.102483574033          -3.89751642596696
875          870.102483574033          -4.89751642596696
876          870.102483574033          -5.89751642596696
877          876.878990566405          -.121009433594736
878          876.878990566405          -1.12100943359474
879          876.878990566405          -2.12100943359474
880          876.878990566405          -3.12100943359474
881          883.660048192341          2.66004819234149
882          883.660048192341          1.66004819234149
883          890.443373392945          7.44337339294543
884          890.443373392945          6.44337339294543
885          890.443373392945          5.44337339294543
886          890.443373392945          4.44337339294543
887          897.231218375255          10.231218375255
888          897.231218375255          9.23121837525503
889          897.231218375255          8.23121837525503
890          897.231218375255          7.23121837525503
891          897.231218375255          6.23121837525503
892          897.231218375255          5.23121837525503
893          897.231218375255          4.23121837525503
894          897.231218375255          3.23121837525503
895          897.231218375255          2.23121837525503
896          897.231218375255          1.23121837525503
897          897.231218375255          .231218375255025
898          897.231218375255          -.768781624744975
899          897.231218375255          -1.76878162474497
900          897.231218375255          -2.76878162474497
901          897.231218375255          -3.76878162474497
902          897.231218375255          -4.76878162474497
903          897.231218375255          -5.76878162474497
904          897.231218375255          -6.76878162474497
905          897.231218375255          -7.76878162474497
906          897.231218375255          -8.76878162474497
907          904.04136082537          -2.95863917462987
908          904.04136082537          -3.95863917462987
909          904.04136082537          -4.95863917462987
910          904.04136082537          -5.95863917462987
911          910.85590372263          -.144096277369954
912          910.85590372263          -1.14409627736995
913          910.85590372263          -2.14409627736995
914          910.85590372263          -3.14409627736995
915          910.85590372263          -4.14409627736995
916          910.85590372263          -5.14409627736995
917          910.85590372263          -6.14409627736995
918          910.85590372263          -7.14409627736995
919          917.679189844986          -1.32081015501421
920          917.679189844986          -2.32081015501421
921          917.679189844986          -3.32081015501421
922          917.679189844986          -4.32081015501421
923          917.679189844986          -5.32081015501421
924          917.679189844986          -6.32081015501421
925          917.679189844986          -7.32081015501421
926          917.679189844986          -8.32081015501421
927          917.679189844986          -9.32081015501421
928          917.679189844986          -10.3208101550142
929          924.5132985838          -4.48670141620039
930          924.5132985838          -5.48670141620039
931          924.5132985838          -6.48670141620039
932          924.5132985838          -7.48670141620039
933          924.5132985838          -8.48670141620039
934          924.5132985838          -9.48670141620039
935          924.5132985838          -10.4867014162004
936          924.5132985838          -11.4867014162004
937          931.355981866038          -5.64401813396194
938          931.355981866038          -6.64401813396194
939          931.355981866038          -7.64401813396194
940          931.355981866038          -8.64401813396194
941          938.202925005623          -2.79707499437654
942          938.202925005623          -3.79707499437654
943          938.202925005623          -4.79707499437654
944          938.202925005623          -5.79707499437654
945          938.202925005623          -6.79707499437654
946          938.202925005623          -7.79707499437654
947          945.056224098809          -1.94377590119052
948          945.056224098809          -2.94377590119052
949          945.056224098809          -3.94377590119052
950          945.056224098809          -4.94377590119052
951          945.056224098809          -5.94377590119052
952          945.056224098809          -6.94377590119052
953          951.915839002464          -1.08416099753629
954          951.915839002464          -2.08416099753629
955          951.915839002464          -3.08416099753629
956          951.915839002464          -4.08416099753629
957          951.915839002464          -5.08416099753629
958          951.915839002464          -6.08416099753629
959          951.915839002464          -7.08416099753629
960          951.915839002464          -8.08416099753629
961          955.349826206949          -5.65017379305107
962          955.349826206949          -6.65017379305107
963          955.349826206949          -7.65017379305107
964          955.349826206949          -8.65017379305107
965          955.349826206949          -9.65017379305107
966          955.349826206949          -10.6501737930511
967          962.224024702402          -4.77597529759782
968          962.224024702402          -5.77597529759782
969          962.224024702402          -6.77597529759782
970          962.224024702402          -7.77597529759782
971          969.102351170694          -1.89764882930649
972          969.102351170694          -2.89764882930649
973          969.102351170694          -3.89764882930649
974          969.102351170694          -4.89764882930649
975          969.102351170694          -5.89764882930649
976          969.102351170694          -6.89764882930649
977          975.986837822736          -1.01316217726367
978          975.986837822736          -2.01316217726367
979          975.986837822736          -3.01316217726367
980          975.986837822736          -4.01316217726367
981          975.986837822736          -5.01316217726367
982          975.986837822736          -6.01316217726367
983          982.877446942883          -.122553057116534
984          982.877446942883          -1.12255305711653
985          982.877446942883          -2.12255305711653
986          982.877446942883          -3.12255305711653
987          982.877446942883          -4.12255305711653
988          982.877446942883          -5.12255305711653
989          982.877446942883          -6.12255305711653
990          982.877446942883          -7.12255305711653
991          989.776161477213          -1.22383852278654
992          989.776161477213          -2.22383852278654
993          989.776161477213          -3.22383852278654
994          989.776161477213          -4.22383852278654
995          989.776161477213          -5.22383852278654
996          989.776161477213          -6.22383852278654
997          996.680912247175          -.319087752824657
998          996.680912247175          -1.31908775282466
999          996.680912247175          -2.31908775282466
1000       996.680912247175          -3.31908775282466

qingjiao · 发表于 2011-6-30 11:11

例如：ψ(10)=3ln2+2ln3+ln5+ln7=7.83201418050547
和你的结果一致。

qingjiao · 发表于 2011-6-30 11:30

[这个贴子最后由qingjiao在 2011/06/30 11:33am 第 1 次编辑]

另外，如果x取整数，我建议都+0.5，因为取在素数方幂的边界上可能有误差，如果算出是0.9999...，再取整时很可能丢掉整一个素数，尤其在x较大时容易出现。故宜取[ln25/ln5]*ln5=[ln25.5/ln5]*ln5，等等。

天山草 · 发表于 2011-6-30 15:43

结果应以 [ψ(x)-x]/x 体现才好，其值应无限地趋近于零，负值多而正值少。就像一个人在水里游泳，他离水面越来越近，偶而也会露一下头。
算了三个小时了，算到了 3.3 亿，下面算得会更慢。是每隔 10万个数计算一次。文件大小已到 200 多k，不知能不能发得上来。可以每隔 100 万个数或是 1000 万个数计算一次。先发个样品看看吧：
x                ψ(x)                   [ψ(x)-x]/x
--------------------------------------------------------------
100000       100051.564025658          5.15640256579791E-04
200000       200026.938797705          1.34693988526415E-04
300000       299659.942852716          -1.13352382428168E-03
400000       400108.161383975          2.70403459938534E-04
500000       500113.296873401          2.2659374680242E-04
……………………………………………………………………………………
332800000    332797485.394407          -7.55590623029723E-06
332900000    332898662.630366          -4.01733143349674E-06
333000000    332998212.657541          -5.36739477237782E-06
333100000    333097950.635763          -6.15239939024698E-06
333200000    333197925.635507          -6.22558371109121E-06
333300000    333298422.202592          -4.73386560926343E-06
333400000    333397674.134261          -6.97620197638968E-06
333500000    333496211.281828          -1.13604742797955E-05
333600000    333595770.449465          -1.26785087967805E-05
333700000    333696390.92314          -1.0815333712769E-05
333800000    333796708.72521          -9.86002034220316E-06
333900000    333896625.713991          -1.0105678373627E-05
334000000    333997329.293732          -7.99612655200644E-06
334100000    334096719.412531          -9.81917829632723E-06
334200000    334195718.506122          -1.28111725840506E-05
334300000    334296210.79527          -1.13347434341426E-05
334400000    334395840.995385          -1.24372147585145E-05
334500000    334496081.176953          -1.17154650139345E-05
334600000    334596381.777799          -1.0813575017723E-05
334700000    334696016.525631          -1.19016264382141E-05
334800000    334796889.406897          -9.29089935164867E-06
334900000    334897920.865717          -6.20822419405482E-06
335000000    334997825.162483          -6.49205228962115E-06
335100000    335098192.275248          -5.39458296701681E-06
335200000    335199355.93933          -1.92142204801621E-06
335300000    335298852.557952          -3.42213554468363E-06
335400000    335399901.511713          -2.93644265930924E-07
335500000    335499332.430606          -1.98977464766154E-06
335600000    335598588.188417          -4.20682831657373E-06
335700000    335698375.508818          -4.83911582462696E-06
335800000    335796633.050397          -1.00266515858973E-05
335900000    335897620.961367          -7.08258003253744E-06
336000000    335996185.796576          -1.13517959043384E-05
336100000    336096705.588543          -9.80187877697875E-06
336200000    336197815.889781          -6.49646109268114E-06
336300000    336298240.561668          -5.23175240076974E-06
336400000    336396595.400927          -1.01206869000345E-05
336500000    336497494.326252          -7.44628157002788E-06
336600000    336596804.410763          -9.49372916563734E-06
336700000    336696312.32939          -1.09523926643012E-05
336800000    336796234.085833          -1.11814553651631E-05
336900000    336896304.617734          -1.09687808441741E-05
337000000    336995012.002845          -1.4801178500808E-05

qingjiao · 发表于 2011-6-30 16:26

结果应以 [ψ(x)-x]/x 体现才好，
=================================
这样看不出波动，可能[ψ(x)-x]/√x更好一些。
其实不用几十亿个都算的，取等比数列就能看出大概。当然如果你喜欢每个算也可以。

qingjiao · 发表于 2011-6-30 16:36

这样看不出波动，可能[ψ(x)-x]/√x更好一些。
===========================================
数学上的重要猜想就是│ψ(x)-x│=O(√x)，此猜想等价于│π(x)-Li(x)│=O((√xlnx)。这样结果将在某个常数范围内波动或振荡。

qingjiao · 发表于 2011-6-30 16:50

每10万或100万地取数，效果不一定好于按等比数列取。因为其分界点附近素数稀疏或密集的概率不一定能反映整个40亿内的情况。
由ψ(x)的表达式可以看出，当某个区域素数密集时，ψ(x)增长快，与x的差距小，甚至>x；反之，当素数稀疏时，ψ(x)增长慢，与x的差距拉大。θ(x)同理。
由此我们也可以了解为什么素数定理证明后的100多年，π(x)的进展依然有限。因为无论Li(x)，x/lnx或x/(lnx-1)等等，都是单调连续光滑的函数，是π(x)的稳定部分。而π(x)的不稳定部分，是一个振幅和周期不断变化的函数。这种函数是很难求出来的。

天山草 · 发表于 2011-6-30 19:00

楼主，请您给出一个等比数列来，我按您的这个数列再算一算。
从计算结果看，[ψ(x)-x]/x 大概不会收敛到零。[ψ(x)-x]/√x 就更不会了。

wangyangkee · 发表于 2011-6-30 19:26

借用宝楼：关于结论验算，求助天山草或诸师长------

http://www.mathchina.com/cgi-bin/topic.cgi?forum=5&topic=7469

		自动登录	找回密码
密码			注册

请天山草网友帮忙计算和画图