（C1^k)*10^n/ln^2(10^n)<T(10^n)< （C1^k+1)*10^n/ln^2(10^n),k是正整数

小草

表孪生素数对数T(10^n)则（C1^k)*10^n/ln^2(10^n)<T(10^n)<
（C1^k+1)*10^n/ln^2(10^n),k是正整数

                       文/施承忠


因为C1*x/lnx=π(x),C2*π(x)/lnx=T(x),C*x/ln^2(x)=T(x),因为C1>C2,所以C2≈C1^s*,(s*是正实数),C1^s=C.(s是正实数).所以存在C1^m<C<C1^m+1.
表孪生素数对数T(10^n)则（C1^k)*10^n/ln^2(10^n)<T(10^n)<
（C1^k+1)*10^n/ln^2(10^n),k是正整数



(1.1512925464970228420089957273422^3)*10^2/ln^2(10^2)=
0.71955784156063927625562232958889
T(10^2)=8
(1.1512925464970228420089957273422^4)*10^2/ln^2(10^2)=
8.2842157976224968915026037321788

(1.160502886868999024745067693161^3)*10^3/ln^2(10^3)=
32.754033478990628474404790571782
T(10^3)=35
(1.160502886868999024745067693161^4)*10^3/ln^2(10^3)=
38.011150408972467861210689406091

(1.1319508317158728582632445991228^4)*10^4/ln^2(10^4)=
193.53472138818782508591048421128
T(10^4)=205
(1.1319508317158728582632445991228^5)*10^4/ln^2(10^4)=
219.0717888412589363634217349278

(1.1043198105999443100550287016666^4)*10^/ln^2(10^5)=
1122.0392944746746807116439946672
T(10^5)=1224
(1.1043198105999443100550287016666^5)*10^/ln^2(10^5)=
1239.0902211599678835935332403664

(1.0844899477790795886242657592589^5)*10^6/ln^2(10^6)=
7859.4793303069015205175022997425
T(10^6)=8169
(1.0844899477790795886242657592589^6)*10^6/ln^2(10^6)=
8523.5263284952870471766660632951

(1.0711747889618229206472949200739^6)*10^7/ln^2(10^7)=
58147.975436763869463720066296406
T(10^7)=58980
(1.0711747889618229206472949200739^7)*10^7/ln^2(10^7)=
62286.645317032800843175405097292

(1.0612992317564807581131101565239^6)*10^8/ln^2(10^8)=
421129.58055039669414479517920099
T(10^8)=440312
(1.0612992317564807581131101565239^7)*10^8/ln^2(10^8)=
446944.50030806499259943350749876

(1.0537269642351710974409226939339^7)*10^9/ln^2(10^9)=
3358769.9452715615753372602709288
T(10^9)=3424506
(1.0537269642351710974409226939339^8)*10^9/ln^2(10^9)=
3539226.4579953343484002895325122

(1.0477971283581089965610995806307^8)*10^10/ln^2(10^10)=
27402251.848895606725084617731966
T(10^10)=27412679
(1.0477971283581089965610995806307^9)*10^10/ln^2(10^10)=
28712000.797818499611483689470413

(1.0430388787000800650353383577411^8)*10^11`/ln^2(10^11)=
218367110.194087858020268401527
T(10^11)=224376048
(1.0430388787000800650353383577411^9)*10^11/ln^2(10^11)=
227765385.761818222369139980105

(1.0391450110953410193890550162821^9)*10^12/ln^2(10^12)=
1850510453.4686406686783333896729
T(10^12)=1870585220
(1.0391450110953410193890550162821^10)*10^12/ln^2(10^12)=
1922948705.7017151488325967661743

(1.0358989502218021047112303278898^9)*10^13/ln^2(10^13)=
15329869386.365554436961147666783
T(10^13)=15834664872
(1.0358989502218021047112303278898^10)*10^13/ln^2(10^13)=
15880195604.373419452260030976233

(1.0331511539035291862176910483981^10)*10^14/ln^2(10^14)=
133337160436.57249840706102984945
T(10^14)=135780321665
(1.0331511539035291862176910483981^11)*10^14/ln^2(10^14)=
137757441163.26487615917982669191

(1.0307949444307296700585968367039^11)*10^15/ln^2(10^15)=
1170256826198.6300717242873080371
T(10^15)=1177209242304
(1.0307949444307296700585968367039^12)*10^15/ln^2(10^15)=
1206294820131.0989542921093525203

(1.0287520663313651823411474364617^11)*10^16/ln^2(10^16)=
10063442817547.270133399707048213
T(10^16)=10304195696798
(1.0287520663313651823411474364617^12)*10^16/ln^2(10^16)=
10352787592959.289766611305910187

(1.026963812311519265191973032908^12)*10^17/ln^2(10^17)=
89811606435813.880673821437494298
T(10^17)=90948889353159
(1.026963812311519265191973032908^13)*10^17/ln^2(10^17)=
92233269735145.201861896208318594

(1.0253852989975120928002258276784^13)*10^18/ln^2(10^18)=
806409692013960.95485181968404105
T(10^18)=808675888577435
(1.0253852989975120928002258276784^14)*10^/ln^2(10^18)=
826880643160226.99338632942108303

小草

我们可以估计

(1.0239816215857955655703197202188^13)*10^19/ln^2(10^19)=
7109835075167362.476904252402407
T(10^19)=
(1.0239816215857955655703197202188^15)*10^19/ln^2(10^19)=
7454934819152571.3604188726851944

小草

如果n>19
那么T(10^n)肯定大于C1^13*10^n/ln^2(10^n)。

小草

(1.0239816215857955655703197202188^13)*10^19/ln^2(10^19)=
7109835075167362.476904252402407

(1.0227252222171465767428331550436^13)*10^20/ln^2(10^20)=
63150267779201948.021538436580132

(1.0215940512576688179742020765786^13)*10^21/ln^2(10^21)=
564610113200609865.58279480095466

(1.0205702563249083622735362563577^13)*10^22/ln^2(10^22)=
5077863851436426545.7136104147604

(1.0196392275787955891545338557924^13)*10^23/ln^2(10^23)=
45911122328130367212.799482218006

(1.0187888929128663997794785749204^13)*10^24/ln^2(10^24)=
417100546712214686842.21318712558

(1.0180091894337030790118729957885^13)*10^25/ln^2(10^25)=
3805929012446805901818.76923198

(1.0172916618481345700697380334442^13)*10^26/ln^2(10^26)=
34866891454405268782598.077423501

(1.0166291535407589742648124253842^13)*10^27/ln^2(10^27)=
320593246877061392314943.30470245

(1.0160155663019066364130861570581^13)*10^28/ln^2(10^28)=
2957721397533519964280069.8375319

(1.0154456715279689899611432643734^13)*10^29/ln^2(10^29)=
27372195416618654585822944.007226

小草

如果n>19
那么T(10^n)肯定小于C1^n-4*10^n/ln^2(10^n)。

我们有：

(1.0239816215857955655703197202188^13)*10^19/ln^2(10^19)=
7109835075167362.476904252402407<
T(10^19)<
(1.0239816215857955655703197202188^15)*10^19/ln^2(10^19)=
7454934819152571.3604188726851944

(1.0227252222171465767428331550436^13)*10^20/ln^2(10^20)=
63150267779201948.021538436580132<
T(10^20)<
(1.0227252222171465767428331550436^16)*10^20/ln^2(10^20)=
67554159686111559.056875372306743

(1.0215940512576688179742020765786^13)*10^21/ln^2(10^21)=
564610113200609865.58279480095466<
T(10^21)<
(1.0215940512576688179742020765786^17)*10^21/ln^2(10^21)=
614981532451901673.00407273341117

(1.0205702563249083622735362563577^13)*10^22/ln^2(10^22)=
5077863851436426545.7136104147604<
T(10^22)<
(1.0205702563249083622735362563577^18)*10^22/ln^2(10^22)=
5622061440288239922.4361331643181

(1.0196392275787955891545338557924^13)*10^23/ln^2(10^23)=
45911122328130367212.799482218006<
T(10^23)<
(1.0196392275787955891545338557924^19)*10^23/ln^2(10^23)=
51593753135713409595.47086454443

(1.0187888929128663997794785749204^13)*10^24/ln^2(10^24)=
417100546712214686842.21318712558<
T(10^24)<
(1.0187888929128663997794785749204^20)*10^24/ln^2(10^24)=
475149383183005405124.40769787917

(1.0180091894337030790118729957885^13)*10^25/ln^2(10^25)=
3805929012446805901818.76923198<
T(10^25)<
(1.0180091894337030790118729957885^21)*10^25/ln^2(10^25)=
4390098558438405923144.9503443485

(1.0172916618481345700697380334442^13)*10^26/ln^2(10^26)=
34866891454405268782598.077423501<
T(10^26)<
(1.0172916618481345700697380334442^22)*10^26/ln^2(10^26)=
40683901427144832025660.868956041

(1.0166291535407589742648124253842^13)*10^27/ln^2(10^27)=
320593246877061392314943.30470245<
T(10^27)<
(1.0166291535407589742648124253842^23)*10^27/ln^2(10^27)=
378076746121077513856240.84503898

(1.0160155663019066364130861570581^13)*10^28/ln^2(10^28)=
2957721397533519964280069.8375319<
T(10^28)<
(1.0160155663019066364130861570581^24)*10^28/ln^2(10^28)=
3522583065729462093468477.8484618

(1.0154456715279689899611432643734^13)*10^29/ln^2(10^29)=
27372195416618654585822944.007226
T(10^29)<
(1.0154456715279689899611432643734^25)*10^29/ln^2(10^29)=
32899548979017739300654212.525374

小草

缩小包围圈

(1.0239816215857955655703197202188^3)*10^19/ln^2(10^19)=
5609677658499311.1822971508347971
(1.0239816215857955655703197202188^4)*10^19/ln^2(10^19)=
5744206825323733.3884598538543548
(1.0239816215857955655703197202188^5)*10^19/ln^2(10^19)=
5881962219719191.250880350980905
(1.0239816215857955655703197202188^6)*10^19/ln^2(10^19)=
6023021211854443.0070491363021165
(1.0239816215857955655703197202188^7)*10^19/ln^2(10^19)=
6167463027360356.0836505949287825
(1.0239816215857955655703197202188^8)*10^19/ln^2(10^19)=
6315368791826897.2659203438316361
(1.0239816215857955655703197202188^9)*10^19/ln^2(10^19)=
6466821576367232.8469509657621164
(1.0239816215857955655703197202188^10)*10^19/ln^2(10^19)=
6621906444274529.7846756845138121
(1.0239816215857955655703197202188^11)*10^19/ln^2(10^19)=
6780710498797662.6086025314086751
(1.0239816215857955655703197202188^12)*10^19/ln^2(10^19)=
6943322932062659.2505796901596029
【】【】【】1913.9852178827016818127615662824
(1.0239816215857955655703197202188^13)*10^19/ln^2(10^19)=
7109835075167362.476904252402407<
T(10^19)
(1.0253852989975120928002258276784^14)*10^18/ln^2(10^18)=
7421311035565472.1844091615631832
T(10^19)
(1.0239816215857955655703197202188^15)*10^19/ln^2(10^19)=
7454934819152571.3604188726851944
【】【】【】10000000000000000000
(1.0239816215857955655703197202188^16)*10^19/ln^2(10^19)=
7633716244932259.6266352421927656
(1.0239816215857955655703197202188^17)*10^19/ln^2(10^19)=
7816785139211565.3727782404040269
(1.0239816215857955655703197202188^18)*10^19/ln^2(10^19)=
8004244322437607.4439310411881298
(1.0239816215857955655703197202188^19)*10^19/ln^2(10^19)=
8196199080858558.7716235329804403
(1.0239816215857955655703197202188^20)*10^19/ln^2(10^19)=
8392757225657554.1587893081904199
(1.0239816215857955655703197202188^21)*10^19/ln^2(10^19)=
8594029153504725.0641108215860621

小草

这比哈代系数更完美，利用哈代系数只能得出一个近似值。用我的方法对于任意偶数都能得到一个上下界值。