请天山草网友帮忙计算和画图

qingjiao · 发表于 2011-6-29 16:06

[这个贴子最后由qingjiao在 2011/06/29 04:08pm 第 1 次编辑] 数论中有一个重要函数ψ(x)，它可以计算如下： ψ(x)=[lnx/ln2]*ln2+[lnx/ln3]*ln3+[lnx/ln5]*ln5+...+[lnx/lnp]*lnp，p<=x。显然exp(ψ(x))=1,2,3,4,...,x的最小公倍数。 ψ(x)~x与素数定理等价，这一点数学界早已证明。未能解决的是误差项大小。设ψ(x)=x+r(x)，实际验算的r(x)的阶在O(√x)范围，而目前理论上能确定的要比它大得多。现请天山草网友帮忙计算一下r(x)=ψ(x)-x的值，由2到你所掌握的最大素数（最好超过10亿），按等比数列划分成300~400个区间，然后画出曲线图。注：x的取值可以不是整数，起点和终点也不一定是2和p，可以小于2和大于p，主要照顾作图方便，保证p是<=x的最大素数就行。我估计r(x)的图像是一个缓慢震荡的波形，多数情况r(x)<0，少数情况r(x)>0，主振幅在(1-γ)*√x左右。谢谢！

wangyangkee · 发表于 2011-6-29 16:20

qingjiao，，，，素数分布规律不符合独立随机事件的前提，理论和实践都不成立，不能应用概率原理求素数个数。这是民科们最容易犯的错误。
沉迷“概率素数论”将浪费你们的时间和生命！

wangyangkee · 发表于 2011-6-29 16:26

请数学爱好者注意：
素数分布规律不符合独立随机事件的前提，理论和实践都不成立，不能应用概率原理求素数个数。这是民科们最容易犯的错误。
沉迷“概率素数论”将浪费你们的时间和生命！

qingjiao · 发表于 2011-6-29 16:44

当x很大时，√x也不小，这样前面的波动就被掩盖了，看不清楚。建议天山草网友在画图时纵坐标也取对数（取等比数列=取对数，只是底不一定为e）。另外还有个θ(x)=ln2+ln3+ln5+...+lnp，p<=x。θ(x)~x也和素数定理等价。令θ(x)=x+g(x)，g(x)的阶同样目前理论上能确定的依然太大。天山草网友有时间可以再算一下g(x)的值和画出它的曲线。我估计g(x)很可能始终<0，最大振幅约1.5√x。

qingjiao · 发表于 2011-6-29 22:07

提起来。希望喜欢计算的白新岭同学也能算一算，画一画。

天山草 · 发表于 2011-6-30 08:49

下面引用由qingjiao在 2011/06/29 04:06pm 发表的内容：
数论中有一个重要函数ψ(x)，……
现请天山草网友帮忙计算一下r(x)=ψ(x)-x的值，由2到你所掌握的最大素数（最好超过10亿），按等比数列划分成300~400个区间，然后画出曲线图。

好的，我试一下看看，尽力而为吧。

天山草 · 发表于 2011-6-30 10:23

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确理解了：
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 [br][br]-=-=-=-=- 以下内容由 天山草 在时添加 -=-=-=-=-
若是计算无误，俺就继续算下去，看看能不能算到 40 亿为止。[br][br]-=-=-=-=- 以下内容由 天山草 在时添加 -=-=-=-=-
请教一下：
哪位同学知道，如何根据上面这个数据表，使用 mathematica 软件自动画出曲线图来？或者使用其它软件画图也行。

天山草 · 发表于 2011-6-30 10:50

请楼主检查一下，下面这个程序是否正确？是否正确描述了那个函数？将各素数依顺序调入 p(i) 数组中，即 p(1)=2,p(2)=3,p(3)=5,等等； For x = 2 To 1000 s = 0 对于每一个 x, 计算结果都清零，从头计算。 For i = 1 To 1000 If p(i) <= x Then s = s + Log(p(i)) * Int(Log(x) / Log(p(i))) If p(i) > x Then Exit For 素数大于 x 时停止计算，打印结果 Next i Print x, s, s - x 打印结果 Next x

天山草 · 发表于 2011-6-30 11:02

下面这个是 g(x) 的计算结果，也是先发上来前 99 个数据：
g(x) 确实都是小于零的。
x                θ(x)                g(x) = θ(x)-x
--------------------------------------------------------------
2          .693147180559945          -1.30685281944005
3          1.79175946922805          -1.20824053077195
4          1.79175946922805          -2.20824053077195
5          3.40119738166216          -1.59880261833784
6          3.40119738166216          -2.59880261833785
7          5.34710753071747          -1.65289246928253
8          5.34710753071747          -2.65289246928253
9          5.34710753071747          -3.65289246928253
10          5.34710753071747          -4.65289246928253
11          7.74500280351584          -3.25499719648416
12          7.74500280351584          -4.25499719648416
13          10.3099521609774          -2.69004783902262
14          10.3099521609774          -3.69004783902262
15          10.3099521609774          -4.69004783902262
16          10.3099521609774          -5.69004783902262
17          13.1431655050336          -3.85683449496641
18          13.1431655050336          -4.85683449496641
19          16.0876044842             -2.91239551579996
20          16.0876044842             -3.91239551579996
21          16.0876044842             -4.91239551579996
22          16.0876044842             -5.91239551579996
23          19.2230987001292          -3.77690129987081
24          19.2230987001292          -4.77690129987081
25          19.2230987001292          -5.77690129987081
26          19.2230987001292          -6.77690129987081
27          19.2230987001292          -7.77690129987081
28          19.2230987001292          -8.77690129987081
29          22.5903945301157          -6.40960546988434
30          22.5903945301157          -7.40960546988434
31          26.0243817346008          -4.97561826539919
32          26.0243817346008          -5.97561826539919
33          26.0243817346008          -6.97561826539919
34          26.0243817346008          -7.97561826539919
35          26.0243817346008          -8.97561826539919
36          26.0243817346008          -9.97561826539919
37          29.635299647245          -7.36470035275497
38          29.635299647245          -8.36470035275497
39          29.635299647245          -9.36470035275497
40          29.635299647245          -10.364700352755
41          33.3488717139493          -7.65112828605066
42          33.3488717139493          -8.65112828605066
43          37.1100718296429          -5.8899281703571
44          37.1100718296429          -6.8899281703571
45          37.1100718296429          -7.8899281703571
46          37.1100718296429          -8.8899281703571
47          40.960219431353          -6.03978056864705
48          40.960219431353          -7.03978056864705
49          40.960219431353          -8.03978056864705
50          40.960219431353          -9.03978056864705
51          40.960219431353          -10.039780568647
52          40.960219431353          -11.039780568647
53          44.9305113449051          -8.06948865509492
54          44.9305113449051          -9.06948865509492
55          44.9305113449051          -10.0694886550949
56          44.9305113449051          -11.0694886550949
57          44.9305113449051          -12.0694886550949
58          44.9305113449051          -13.0694886550949
59          49.0080487888108          -9.9919512111892
60          49.0080487888108          -10.9919512111892
61          53.1189226529841          -7.88107734701589
62          53.1189226529841          -8.88107734701589
63          53.1189226529841          -9.88107734701589
64          53.1189226529841          -10.8810773470159
65          53.1189226529841          -11.8810773470159
66          53.1189226529841          -12.8810773470159
67          57.3236152723751          -9.67638472762492
68          57.3236152723751          -10.6763847276249
69          57.3236152723751          -11.6763847276249
70          57.3236152723751          -12.6763847276249
71          61.5862951494164          -9.41370485058361
72          61.5862951494164          -10.4137048505836
73          65.8767545905648          -7.12324540943521
74          65.8767545905648          -8.12324540943521
75          65.8767545905648          -9.12324540943521
76          65.8767545905648          -10.1232454094352
77          65.8767545905648          -11.1232454094352
78          65.8767545905648          -12.1232454094352
79          70.2462024430318          -8.75379755696819
80          70.2462024430318          -9.75379755696819
81          70.2462024430318          -10.7537975569682
82          70.2462024430318          -11.7537975569682
83          74.6650430508284          -8.3349569491716
84          74.6650430508284          -9.3349569491716
85          74.6650430508284          -10.3349569491716
86          74.6650430508284          -11.3349569491716
87          74.6650430508284          -12.3349569491716
88          74.6650430508284          -13.3349569491716
89          79.1536794205605          -9.84632057943946
90          79.1536794205605          -10.8463205794395
91          79.1536794205605          -11.8463205794395
92          79.1536794205605          -12.8463205794395
93          79.1536794205605          -13.8463205794395
94          79.1536794205605          -14.8463205794395
95          79.1536794205605          -15.8463205794395
96          79.1536794205605          -16.8463205794395
97          83.7283903990639          -13.2716096009361
98          83.7283903990639          -14.2716096009361
99          83.7283903990639          -15.2716096009361
100          83.7283903990639          -16.2716096009361

天山草 · 发表于 2011-6-30 11:04

发前 999 个数据，看看能不能发上来哈;
x                θ(x)                g(x) = θ(x)-x
--------------------------------------------------------------
2          .693147180559945          -1.30685281944005
3          1.79175946922805          -1.20824053077195
4          1.79175946922805          -2.20824053077195
5          3.40119738166216          -1.59880261833784
6          3.40119738166216          -2.59880261833785
7          5.34710753071747          -1.65289246928253
8          5.34710753071747          -2.65289246928253
9          5.34710753071747          -3.65289246928253
10          5.34710753071747          -4.65289246928253
11          7.74500280351584          -3.25499719648416
12          7.74500280351584          -4.25499719648416
13          10.3099521609774          -2.69004783902262
14          10.3099521609774          -3.69004783902262
15          10.3099521609774          -4.69004783902262
16          10.3099521609774          -5.69004783902262
17          13.1431655050336          -3.85683449496641
18          13.1431655050336          -4.85683449496641
19          16.0876044842             -2.91239551579996
20          16.0876044842             -3.91239551579996
21          16.0876044842             -4.91239551579996
22          16.0876044842             -5.91239551579996
23          19.2230987001292          -3.77690129987081
24          19.2230987001292          -4.77690129987081
25          19.2230987001292          -5.77690129987081
26          19.2230987001292          -6.77690129987081
27          19.2230987001292          -7.77690129987081
28          19.2230987001292          -8.77690129987081
29          22.5903945301157          -6.40960546988434
30          22.5903945301157          -7.40960546988434
31          26.0243817346008          -4.97561826539919
32          26.0243817346008          -5.97561826539919
33          26.0243817346008          -6.97561826539919
34          26.0243817346008          -7.97561826539919
35          26.0243817346008          -8.97561826539919
36          26.0243817346008          -9.97561826539919
37          29.635299647245          -7.36470035275497
38          29.635299647245          -8.36470035275497
39          29.635299647245          -9.36470035275497
40          29.635299647245          -10.364700352755
41          33.3488717139493          -7.65112828605066
42          33.3488717139493          -8.65112828605066
43          37.1100718296429          -5.8899281703571
44          37.1100718296429          -6.8899281703571
45          37.1100718296429          -7.8899281703571
46          37.1100718296429          -8.8899281703571
47          40.960219431353          -6.03978056864705
48          40.960219431353          -7.03978056864705
49          40.960219431353          -8.03978056864705
50          40.960219431353          -9.03978056864705
51          40.960219431353          -10.039780568647
52          40.960219431353          -11.039780568647
53          44.9305113449051          -8.06948865509492
54          44.9305113449051          -9.06948865509492
55          44.9305113449051          -10.0694886550949
56          44.9305113449051          -11.0694886550949
57          44.9305113449051          -12.0694886550949
58          44.9305113449051          -13.0694886550949
59          49.0080487888108          -9.9919512111892
60          49.0080487888108          -10.9919512111892
61          53.1189226529841          -7.88107734701589
62          53.1189226529841          -8.88107734701589
63          53.1189226529841          -9.88107734701589
64          53.1189226529841          -10.8810773470159
65          53.1189226529841          -11.8810773470159
66          53.1189226529841          -12.8810773470159
67          57.3236152723751          -9.67638472762492
68          57.3236152723751          -10.6763847276249
69          57.3236152723751          -11.6763847276249
70          57.3236152723751          -12.6763847276249
71          61.5862951494164          -9.41370485058361
72          61.5862951494164          -10.4137048505836
73          65.8767545905648          -7.12324540943521
74          65.8767545905648          -8.12324540943521
75          65.8767545905648          -9.12324540943521
76          65.8767545905648          -10.1232454094352
77          65.8767545905648          -11.1232454094352
78          65.8767545905648          -12.1232454094352
79          70.2462024430318          -8.75379755696819
80          70.2462024430318          -9.75379755696819
81          70.2462024430318          -10.7537975569682
82          70.2462024430318          -11.7537975569682
83          74.6650430508284          -8.3349569491716
84          74.6650430508284          -9.3349569491716
85          74.6650430508284          -10.3349569491716
86          74.6650430508284          -11.3349569491716
87          74.6650430508284          -12.3349569491716
88          74.6650430508284          -13.3349569491716
89          79.1536794205605          -9.84632057943946
90          79.1536794205605          -10.8463205794395
91          79.1536794205605          -11.8463205794395
92          79.1536794205605          -12.8463205794395
93          79.1536794205605          -13.8463205794395
94          79.1536794205605          -14.8463205794395
95          79.1536794205605          -15.8463205794395
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1000       956.245265120059          -43.7547348799411

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