求助:对数平方的倒数积分值
孪生素数猜想有个对数平方的倒数积分:M(X)=1.3203223631∫1÷(lnt)^2dt,1,谁能给出它在10的9次到20次幂范围的积分数值?如:
X M(X) ∫1÷(lnt)^2dt
10^5 1249
10^6 8248
10^7 58754
10^8 44368
10^9
10^1027411417
2,谁能给出它较高精度的近似积分,谢谢! 定理:熊一兵作诗祝贺的的那个哥猜证明的证明人鲁思顺是个二百五。 2C=1.32032236316937 (施承忠),以此系数给出积分式的值。 10^n 孪生素数积分值 增速
5 1246
6 8245 0.661717496
7 58751 0.712565191
8 440364 0.749542987
9 3425302 0.777834246
10 27411387 0.800261904
11 224368646 0.818523506
12 1870558067 0.833698514
13 15834583088 0.846516522
14 1.35780134438000E+11 0.85749106
15 1.17720736083400E+12 0.866995283
16 1.03041826545570E+13 0.875307359
17 9.09487458802810E+13 0.882639108
18 8.08675124543875E+14 0.889154784
19 7.23751110917974E+15 0.894983769
20 6.51542028305949E+16 0.9002294
21 5.89629392174714E+17 0.904975223
22 5.36143831191048E+18 0.909289527
23 4.89621925404855E+19 0.913228684
24 4.48904790547214E+20 0.916839641
25 4.13065048056444E+21 0.920161818
26 3.81353447279810E+22 0.923228555
27 3.53159318182675E+23 0.926068246
28 3.27980904276120E+24 0.928705226
29 3.05402851337041E+25 0.931160465
30 2.85078944668020E+26 0.933452138
31 2.66718741204436E+27 0.935596073
32 2.50077123362640E+28 0.937606117
33 2.34946066800325E+29 0.939494439
34 2.21148101534716E+30 0.941271776
35 2.08531079648932E+31 0.942947636
36 1.96963959453380E+32 0.944530474
37 1.86333386498815E+33 0.946027827
38 1.76540903815721E+34 0.947446441
39 1.67500662411834E+35 0.948792369
40 1.59137532057627E+36 0.950071061
41 1.51385534318979E+37 0.951287433
42 1.44186536507321E+38 0.952445933
43 1.37489158044780E+39 0.953550598
44 1.31247850655544E+40 0.954605094
45 1.25422121506713E+41 0.955612765
46 1.19975874458566E+42 0.956576663
47 1.14876849337192E+43 0.957499579
48 1.10096142905974E+44 0.958384074
49 1.05607798208327E+45 0.959232498
50 1.01388451351306E+46 0.960047014
51 9.74170267270995E+46 0.960829615
52 9.36744732265030E+47 0.961582142
53 9.01435352619005E+48 0.962306295
54 8.68085534469364E+49 0.96300365
55 8.36552906226206E+50 0.963675667
56 8.06707796118192E+51 0.964323703
57 7.78431896549307E+52 0.964949019
58 7.51617089520467E+53 0.96555279
59 7.26164411293595E+54 0.96613611
60 7.01983137746777E+55 0.966700002
61 6.78989974604226E+56 0.96724542
62 6.57108339019085E+57 0.967773257
63 6.36267720917378E+58 0.96828435
64 6.16403114140418E+59 0.968779484
65 5.97454508801403E+60 0.969259394
66 5.79366437441747E+61 0.969724772
67 5.62087568567945E+62 0.970176269
68 5.45570341998804E+63 0.970614496
69 5.29770641178915E+64 0.97104003
70 5.14647498236638E+65 0.971453414
71 5.00162828099669E+66 0.971855163
72 4.86281188441800E+67 0.972245759
73 4.72969562632011E+68 0.972625662
74 4.60197163200857E+69 0.972995304
75 4.47935253637109E+70 0.973355095
76 4.36156986586454E+71 0.973705425
77 4.24837256749265E+72 0.974046662
78 4.13952566970849E+73 0.974379154
79 4.03480906189091E+74 0.974703235
80 3.93401638054531E+75 0.975019219
81 3.83695399169471E+76 0.975327406
82 3.74344006008278E+77 0.975628081
83 3.65330369682672E+78 0.975921516
84 3.56638417805309E+79 0.976207968
85 3.48253022784023E+80 0.976487684
86 3.40159935948868E+81 0.9767609
87 3.32345726975963E+82 0.977027839
88 3.24797728126886E+83 0.977288714
89 3.17503982871025E+84 0.977543731
90 3.10453198501526E+85 0.977793084
91 3.03634702393952E+86 0.97803696
92 2.97038401591085E+87 0.978275537
93 2.90654745427920E+88 0.978508987
94 2.84474690938253E+89 0.978737473
95 2.78489670808767E+90 0.978961151
96 2.72691563668438E+91 0.979180172
97 2.67072666520767E+92 0.979394679
98 2.61625669144009E+93 0.979604812
99 2.56343630300468E+94 0.979810701
100 2.51219955610195E+95 0.980012475
101 2.46248376957344E+96 0.980210256
102 2.41422933309024E+97 0.980404161
103 2.36737952837021E+98 0.980594302
104 2.32188036242199E+99 0.980780789
105 2.27768041190015E+100 0.980963726
106 2.23473067773315E+101 0.981143213
107 2.19298444925669E+102 0.981319347
108 2.15239717714849E+103 0.981492221
109 2.11292635451895E+104 0.981661924
110 2.07453140556488E+105 0.981828544
111 2.03717358124137E+106 0.981992163
112 2.00081586145092E+107 0.982152861
113 1.96542286328847E+108 0.982310717
114 1.93096075491773E+109 0.982465805
115 1.89739717468717E+110 0.982618197
116 1.86470115512453E+111 0.982767962
117 1.83284305147636E+112 0.982915169
118 1.80179447448478E+113 0.983059882
119 1.77152822711648E+114 0.983202164
120 1.74201824498087E+115 0.983342076
121 1.71323954019337E+116 0.983479677
122 1.68516814845820E+117 0.983615022
123 1.65778107916121E+118 0.983748168
124 1.63105626827879E+119 0.983879168
125 1.60497253392264E+120 0.984008072
126 1.57950953435311E+121 0.984134931
127 1.55464772830571E+122 0.984259794
128 1.53036833748627E+123 0.984382706
129 1.50665331110034E+124 0.984503713
130 1.48348529229167E+125 0.98462286
131 1.46084758637320E+126 0.984740189
132 1.43872413074216E+127 0.98485574
133 1.41709946637771E+128 0.984969555
134 1.39595871082698E+129 0.985081671
135 1.37528753259118E+130 0.985192128
136 1.35507212682951E+131 0.985300961
137 1.33529919230390E+132 0.985408205
138 1.31595590949281E+133 0.985513896
139 1.29702991980662E+134 0.985618067
140 1.27850930584193E+135 0.98572075
141 1.26038257261568E+136 0.985821978
142 1.24263862972390E+137 0.98592178
143 1.22526677437360E+138 0.986020187
144 1.20825667523898E+139 0.986117228
145 1.19159835709681E+140 0.986212931
146 1.17528218619813E+141 0.986307323
147 1.15929885633620E+142 0.986400432
148 1.14363937557321E+143 0.986492283
149 1.12829505359030E+144 0.986582902
150 1.11325748962761E+145 0.986672312
151 1.09851856098332E+146 0.986760539
152 1.08407041204193E+147 0.986847606
153 1.06990544380455E+148 0.986933535
154 1.05601630389470E+149 0.987018348
155 1.04239587701542E+150 0.987102068
156 1.02903727583434E+151 0.987184714
157 1.01593383227504E+152 0.987266308
158 1.00307908919407E+153 0.98734687
159 9.90466792424291E+153 0.987426418
160 9.78090883166089E+154 0.987504973
161 9.65945490709406E+155 0.987582552
162 9.54024925470054E+156 0.987659174
163 9.42323672325000E+157 0.987734856
164 9.30836384232007E+158 0.987809615
165 9.19557876119873E+159 0.987883469
166 9.08483119036197E+160 0.987956433
167 8.97607234540376E+161 0.988028523
168 8.86925489330133E+162 0.988099756
169 8.76433290090543E+163 0.988170146
170 8.66126178555075E+164 0.988239708
171 8.55999826768749E+165 0.988308457
172 8.46050032543997E+166 0.988376406
173 8.36272715100336E+167 0.988443571
174 8.26663910879381E+168 0.988509963
175 8.17219769527218E+169 0.988575597
176 8.07936550036496E+170 0.988640486
177 7.98810617041052E+171 0.988704641
178 7.89838437256197E+172 0.988768076
179 7.81016576058164E+173 0.988830803
180 7.72341694196532E+174 0.988892833
181 7.63810544633750E+175 0.988954177
182 7.55419969506180E+176 0.989014848
183 7.47166897201342E+177 0.989074856
184 7.39048339546316E+178 0.989134211
185 7.31061389102496E+179 0.989192926
186 7.23203216562116E+180 0.989251009
187 7.15471068242205E+181 0.989308471
188 7.07862263671818E+182 0.989365322
189 7.00374193268605E+183 0.989421571
190 6.93004316100945E+184 0.989477229
191 6.85750157732082E+185 0.989532304
192 6.78609308142820E+186 0.989586806
193 6.71579419729548E+187 0.989640743
194 6.64658205374475E+188 0.989694124
195 6.57843436585119E+189 0.989746958
196 6.51132941700215E+190 0.989799252
197 6.44524604159362E+191 0.989851016
198 6.38016360833819E+192 0.989902258
199 6.31606200416000E+193 0.989952984
200 6.25292161865322E+194 0.990003204
201 6.19072332908157E+195 0.990052924
202 6.12944848589754E+196 0.990102151
203 6.06907889876077E+197 0.990150894
204 6.00959682303605E+198 0.990199159
205 5.95098494675224E+199 0.990246954
206 5.89322637800418E+200 0.990294284
207 5.83630463278055E+201 0.990341158
208 5.78020362320114E+202 0.99038758
209 5.72490764614811E+203 0.990433559
210 5.67040137227597E+204 0.9904791
211 5.61666983538615E+205 0.990524209
212 5.56369842215221E+206 0.990568893
213 5.51147286218276E+207 0.990613158
214 5.45997921840916E+208 0.990657009
215 5.40920387778624E+209 0.990700452
216 5.35913354229420E+210 0.990743493
217 5.30975522023075E+211 0.990786137
218 5.26105621778275E+212 0.99082839
219 5.21302413086715E+213 0.990870258
220 5.16564683723157E+214 0.990911745
221 5.11891248880484E+215 0.990952856
222 5.07280950428885E+216 0.990993598
223 5.02732656198280E+217 0.991033974
224 4.98245259283158E+218 0.99107399
225 4.93817677369046E+219 0.99111365
226 4.89448852079831E+220 0.991152959
227 4.85137748345196E+221 0.991191922
228 4.80883353787481E+222 0.991230543
229 4.76684678127279E+223 0.991268827
230 4.72540752607112E+224 0.991306778
231 4.68450629432572E+225 0.991344401
232 4.64413381230322E+226 0.9913817
233 4.60428100522367E+227 0.991418678
234 4.56493899216057E+228 0.991455341
235 4.52609908109271E+229 0.991491691
236 4.48775276410278E+230 0.991527734
237 4.44989171271774E+231 0.991563472
238 4.41250777338618E+232 0.99159891
239 4.37559296308818E+233 0.991634052
240 4.33913946507306E+234 0.991668901
241 4.30313962472103E+235 0.991703461
242 4.26758594552444E+236 0.991737735
243 4.23247108518478E+237 0.991771727
244 4.19778785182159E+238 0.991805441
245 4.16352920028975E+239 0.99183888
246 4.12968822860146E+240 0.991872047
247 4.09625817444959E+241 0.991904945
248 4.06323241182918E+242 0.991937578
249 4.03060444775390E+243 0.991969949
250 3.99836791906435E+244 0.992002061
251 3.96651658932545E+245 0.992033917
252 3.93504434580993E+246 0.992065521
253 3.90394519656526E+247 0.992096875
254 3.87321326756141E+248 0.992127981
255 3.84284279991692E+249 0.992158844
256 3.81282814720072E+250 0.992189466
这是根据熊一兵先生提供的系数对自然对数的平方倒数进行积分获得的理论值。可与实际数据进行比对。
另外给了个增速值=10^(n+1)的素数对数量/10^n的素数对数量,然后除10(与自然数的增速比较),增速值的极限是1.任何k生素数的数量增速极限都为1. M(X)也是积分值? 在10^256时,增速已达到99.2189%以上。 白新岭 发表于 2021-6-20 09:47
2C=1.32032236316937 (施承忠),以此系数给出积分式的值。
用 1.320322363169373914,百度下,能找到施承忠给的资料 njzz_yy 发表于 2021-6-20 13:46
用 1.320322363169373914,百度下,能找到施承忠给的资料
我打开那个连接了。但是那种处理方式是不正确的。它掩盖了事实的真像。对于k生素数的数量来说,一定是自然对数的k次方倒数的积分(为了精确可以从2开始积分,为了简便可以从1开始)。决定它的精确度的关在在于系数,其实系数是一个变量,而不是孪生素数常数的2倍,开始大,逐步缩小,有极限。(当然有统一表达式)。最大为2,素数3时1.5,素数5时,1.40625;素数7时,C=1.3671875;.......;可见开始变化很快,直到2C(孪生素数常数的2倍)。
页:
[1]