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(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

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发表于 2024-5-4 21:09 | 显示全部楼层 |阅读模式
本帖最后由 小草 于 2024-5-5 02:37 编辑

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

 楼主| 发表于 2024-5-5 20:28 | 显示全部楼层
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
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 楼主| 发表于 2024-5-13 14:34 | 显示全部楼层
本帖最后由 小草 于 2024-5-13 06:38 编辑

如果n>19
那么T(10^n)肯定大于C1^13*10^n/ln^2(10^n)。
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 楼主| 发表于 2024-5-14 09:56 | 显示全部楼层
本帖最后由 小草 于 2024-5-16 05:37 编辑

(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

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 楼主| 发表于 2024-5-16 13:26 | 显示全部楼层
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

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 楼主| 发表于 2024-5-20 11:47 | 显示全部楼层
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
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 楼主| 发表于 2024-5-29 20:29 | 显示全部楼层
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。
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\frac{\square}{\square}\sqrt{\square}\square_{\baguet}^{\baguet}\overarc{\square}\ \dot{\baguet}\left(\square\right)\binom{\square}{\square}\begin{cases}\square\\\square\end{cases}\ \begin{bmatrix}\square&\square\\\square&\square\end{bmatrix}\to\Rightarrow\mapsto\alpha\ \theta\ \pi\times\div\pm\because\angle\ \infty
\frac{\square}{\square}\sqrt{\square}\sqrt[\baguet]{\square}\square_{\baguet}\square^{\baguet}\square_{\baguet}^{\baguet}\sum_{\baguet}^{\baguet}\prod_{\baguet}^{\baguet}\coprod_{\baguet}^{\baguet}\int_{\baguet}^{\baguet}\lim_{\baguet}\lim_{\baguet}^{\baguet}\bigcup_{\baguet}^{\baguet}\bigcap_{\baguet}^{\baguet}\bigwedge_{\baguet}^{\baguet}\bigvee_{\baguet}^{\baguet}
\underline{\square}\overline{\square}\overrightarrow{\square}\overleftarrow{\square}\overleftrightarrow{\square}\underrightarrow{\square}\underleftarrow{\square}\underleftrightarrow{\square}\dot{\baguet}\hat{\baguet}\vec{\baguet}\tilde{\baguet}
\left(\square\right)\left[\square\right]\left\{\square\right\}\left|\square\right|\left\langle\square\right\rangle\left\lVert\square\right\rVert\left\lfloor\square\right\rfloor\left\lceil\square\right\rceil\binom{\square}{\square}\boxed{\square}
\begin{cases}\square\\\square\end{cases}\begin{matrix}\square&\square\\\square&\square\end{matrix}\begin{pmatrix}\square&\square\\\square&\square\end{pmatrix}\begin{bmatrix}\square&\square\\\square&\square\end{bmatrix}\begin{Bmatrix}\square&\square\\\square&\square\end{Bmatrix}\begin{vmatrix}\square&\square\\\square&\square\end{vmatrix}\begin{Vmatrix}\square&\square\\\square&\square\end{Vmatrix}\begin{array}{l|l}\square&\square\\\hline\square&\square\end{array}
\to\gets\leftrightarrow\nearrow\searrow\downarrow\uparrow\updownarrow\swarrow\nwarrow\Leftarrow\Rightarrow\Leftrightarrow\rightharpoonup\rightharpoondown\impliedby\implies\Longleftrightarrow\leftharpoonup\leftharpoondown\longleftarrow\longrightarrow\longleftrightarrow\Uparrow\Downarrow\Updownarrow\hookleftarrow\hookrightarrow\mapsto
\alpha\beta\gamma\Gamma\delta\Delta\epsilon\varepsilon\zeta\eta\theta\Theta\iota\kappa\varkappa\lambda\Lambda\mu\nu\xi\Xi\pi\Pi\varpi\rho\varrho\sigma\Sigma\tau\upsilon\Upsilon\phi\Phi\varphi\chi\psi\Psi\omega\Omega\digamma\vartheta\varsigma\mathbb{C}\mathbb{H}\mathbb{N}\mathbb{P}\mathbb{Q}\mathbb{R}\mathbb{Z}\Re\Im\aleph\partial\nabla
\times\cdot\ast\div\pm\mp\circ\backslash\oplus\ominus\otimes\odot\bullet\varnothing\neq\equiv\not\equiv\sim\approx\simeq\cong\geq\leq\ll\gg\succ\prec\in\ni\cup\cap\subset\supset\not\subset\not\supset\notin\not\ni\subseteq\supseteq\nsubseteq\nsupseteq\sqsubset\sqsupset\sqsubseteq\sqsupseteq\sqcap\sqcup\wedge\vee\neg\forall\exists\nexists\uplus\bigsqcup\bigodot\bigotimes\bigoplus\biguplus\bigcap\bigcup\bigvee\bigwedge
\because\therefore\angle\parallel\perp\top\nparallel\measuredangle\sphericalangle\diamond\diamondsuit\doteq\propto\infty\bowtie\square\smile\frown\bigtriangledown\triangle\triangleleft\triangleright\bigcirc \wr\amalg\models\preceq\mid\nmid\vdash\dashv\nless\ngtr\ldots\cdots\vdots\ddots\surd\ell\flat\sharp\natural\wp\clubsuit\heartsuit\spadesuit\oint\lfloor\rfloor\lceil\rceil\lbrace\rbrace\lbrack\rbrack\vert\hbar\aleph\dagger\ddagger

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