基于偶数哥猜哈-李素对计算公式改进的偶数素对计算式 Xi(M)≈ t1*c1*M/(logM)^2

愚工688 · 发表于 2020-8-26 14:49

区域下界计算值infS(m)=inf(M)/ k(m)，其特征是消除了受偶数M含有的素因子的影响，因而该值是近似于线性增大的，就是随着偶数M增大而单调增大，这就是随偶数增大，素对数量的低值不断增大的根本原因。
观察上面249#的infS(m)值，基本是隔4个偶数infS(m)值增大0.01；
就是隔400个偶数infS(m)值增大1，隔40000个偶数infS(m)值增大约100；
试计算一下偶数 20200787000 起的连续偶数的区域下界素对计算值的情况：

G(20200787000) = 34776905；
inf( 20200787000 )≈ 34755482.5 , jd ≈0.999384,infS(m) = 25874945.62 , k(m)= 1.34321
G(20200787002) = 29659180；
inf( 20200787002 )≈ 29638594.2 , jd ≈0.999306,infS(m) = 25874945.62 , k(m)= 1.14546
G(20200787004) = 54828526；
inf( 20200787004 )≈ 54794002.5 , jd ≈0.999370,infS(m) = 25874945.62 , k(m)= 2.11765
G(20200787006) = 25893139；
inf( 20200787006 )≈ 25875384.1 , jd ≈0.999314,infS(m) = 25874945.62 , k(m)= 1.00002
G(20200787008) = 26294142；
inf( 20200787008 )≈ 26276294.9 , jd ≈0.999321,infS(m) = 25874945.63 , k(m)= 1.01551
G(20200787010) = 69043324；
inf( 20200787010 )≈ 68999855.0 , jd ≈0.999370,infS(m) = 25874945.63 , k(m)= 2.66667
G(20200787012) = 31102466
inf( 20200787012 )≈ 31088315.4 , jd ≈0.999545,infS(m) = 25874945.63 , k(m)= 1.20148
time start =21:14:09 ,time end =21:16:38 ,time use =

很显然，偶数20200787000起始偶数的区域下界素对值infS(m)比偶数20200707000的区域下界素对值infS(m)增大了102，符合该值是近似于线性增大的分析结论。
那么偶数20208707000起始偶数的区域下界素对值infS(m)又比偶数20200707000的区域下界素对值infS(m)增大了多少呢？按照近似线性的推测，应该是增大10000左右，验证一下：

   G(20208707000) = 34533431；
inf( 20208707000 )≈  34512482.5 , jd ≈0.999393,infS(m) = 25884361.89 , k(m)= 1.33333
   G(20208707002) = 30470340；
inf( 20208707002 )≈  30452190.5 , jd ≈0.999404,infS(m) = 25884361.89 , k(m)= 1.17647
   G(20208707004) = 51879252；
inf( 20208707004 )≈  51849739.0 , jd ≈0.999431,infS(m) = 25884361.89 , k(m)= 2.00313
   G(20208707006) = 25916869；
inf( 20208707006 )≈  25897766.5 , jd ≈0.999263,infS(m) = 25884361.9 , k(m)= 1.00052
   G(20208707008) = 31402037；
inf( 20208707008 )≈  31375475.9 , jd ≈0.999154,infS(m) = 25884361.9 , k(m)= 1.21214
   G(20208707010) = 69074473；
inf( 20208707010 )≈  69024965.1 , jd ≈0.999283,infS(m) = 25884361.9 , k(m)= 2.66667
   G(20208707012) = 28250307；
inf( 20208707012 )≈  28237485.7 , jd ≈ 0.999546,infS(m) = 25884361.9 , k(m)= 1.09091

infS(20208707000) = 25884361.89；
infS（20200707000) = 25874843.15 ；
infS(20208707000) -infS（20200707000)= 25884361.89-25874843.15≈  9519
实际的偶数20208707000的区域下界计算值比偶数20200707000的区域下界计算值大了9519，略低于10000。
由此可见，随着偶数M增大，偶数的素对发生概率与素对区域下界值的发生率都是随√M内的最大素数r的增大而逐渐缓慢的降低的。
因此偶数素对区域下界值的增大只是接近线性的增大。

6455640412 -- 6458211770 r=  80347    P(m)min= .0031811
6458211772 -- 6459176162 r=  80363    P(m)min= .003181
6459176164 -- 6462069770 r=  80369    P(m)min= .003181
6462069772 -- 6465285650 r=  80387    P(m)min= .0031809
6465285652 -- 6468824042 r=  80407    P(m)min= .0031808
6468824044 -- 6471719810 r=  80429    P(m)min= .0031807
6471719812 -- 6472041602 r=  80447    P(m)min= .0031806
6472041604 -- 6475581842 r=  80449    P(m)min= .0031806
6475581844 -- 6475903730 r=  80471    P(m)min= .0031805

愚工688 · 发表于 2020-8-27 21:03

以今天日期2020-08-27日的10倍起始的连续偶数的素对计算值的精度会怎么样呢？
看下面的计算值，精度后面再计算：

  Xi(M)=t2*c1*M/(logM)^2

  G( 202008270 ) = ?    ;Xi(M)≈ 1087481.87          jdz(M)≈?
  G( 202008272 ) = ?    ;Xi(M)≈ 408177.8          jdz(M)≈?
  G( 202008274 ) = ?    ;Xi(M)≈ 444878.94          jdz(M)≈?
  G( 202008276 ) = ?    ;Xi(M)≈ 843735.96          jdz(M)≈?
  G( 202008278 ) = ?    ;Xi(M)≈ 409321.72          jdz(M)≈?
  G( 202008280 ) = ?    ;Xi(M)≈ 582203.52          jdz(M)≈?
  G( 202008282 ) = ?    ;Xi(M)≈ 988765.25          jdz(M)≈?
  G( 202008284 ) = ?    ;Xi(M)≈ 412493.16          jdz(M)≈?
  G( 202008286 ) = ?    ;Xi(M)≈ 407890.45          jdz(M)≈?
  G( 202008288 ) = ?    ;Xi(M)≈ 821934.06          jdz(M)≈?
  G( 202008290 ) = ?    ;Xi(M)≈ 604156.65          jdz(M)≈?
  G( 202008292 ) = ?    ;Xi(M)≈ 411479.68          jdz(M)≈?
  G( 202008294 ) = ?    ;Xi(M)≈ 820381.18          jdz(M)≈?
  G( 202008296 ) = ?    ;Xi(M)≈ 494027.54          jdz(M)≈?
  G( 202008298 ) = ?    ;Xi(M)≈ 407932.59          jdz(M)≈?
  time start =21:04:13, time end =21:04:23

愚工688 · 发表于 2020-8-27 21:31

上面252#的素对计算值的精度实际数据：

  Xi(M)=t2*c1*M/(logM)^2

  G(202008270) = 1087782    ;Xi(M)≈ 1087481.87 jdz(M)≈0.999724;
  G(202008272) = 407467    ;Xi(M)≈ 408177.8       jdz(M)≈1.001744;
  G(202008274) = 444378    ;Xi(M)≈ 444878.94    jdz(M)≈1.001127;
  G(202008276) = 844428    ;Xi(M)≈ 843735.96    jdz(M)≈0.999180;
  G(202008278) = 409643    ;Xi(M)≈ 409321.72    jdz(M)≈0.999216;
  G(202008280) = 583033    ;Xi(M)≈ 582203.52    jdz(M)≈0.998577;
  G(202008282) = 988647    ;Xi(M)≈ 988765.25    jdz(M)≈1.000119;
  G(202008284) = 412461    ;Xi(M)≈ 412493.16    jdz(M)≈1.000078;
  G(202008286) = 408196    ;Xi(M)≈ 407890.45    jdz(M)≈0.999251;
  G(202008288) = 822000    ;Xi(M)≈ 821934.06    jdz(M)≈0.999920;
  G(202008290) = 603915    ;Xi(M)≈ 604156.65    jdz(M)≈1.000400;
  G(202008292) = 411834    ;Xi(M)≈ 411479.68    jdz(M)≈0.999140;
  G(202008294) = 821094    ;Xi(M)≈ 820381.18    jdz(M)≈0.999132;
  G(202008296) = 494211    ;Xi(M)≈ 494027.54    jdz(M)≈0.999629;
  G(202008298) = 408296    ;Xi(M)≈ 407932.59    jdz(M)≈0.999110;
  time start =21:04:13, time end =21:04:23（这是计算Xi(M)值的时间，手工计算精度不计时）

  应该说计算的这些偶数的素对计算值的精度还是不错的！

愚工688 · 发表于 2020-8-28 13:03

今天是20220-08-28日，以今天日期的百倍起始的连续偶数素对数量的计算：

Xi(M)=t2*c1*M/(logM)^2

  G(2020082800) = 4580174    ;Xi(M)≈ 4579813.53          jdz(M)≈0.999921;
  G(2020082802) = 6414501    ;Xi(M)≈ 6415880.57          jdz(M)≈1.000215;
  G(2020082804) = 3207481    ;Xi(M)≈ 3207940.29          jdz(M)≈1.000143;
  G(2020082806) = 3848494    ;Xi(M)≈ 3849528.41          jdz(M)≈1.000269;
  G(2020082808) = 6415633    ;Xi(M)≈ 6415880.59          jdz(M)≈1.000039;
  G(2020082810) = 4509529    ;Xi(M)≈ 4507560.34          jdz(M)≈0.999563;
  G(2020082812) = 3959523    ;Xi(M)≈ 3959639.98          jdz(M)≈1.000030;
  G(2020082814) = 6414465    ;Xi(M)≈ 6415880.61          jdz(M)≈1.000221;
  G(2020082816) = 3207172    ;Xi(M)≈ 3207940.31          jdz(M)≈1.000239;
  G(2020082818) = 3509710    ;Xi(M)≈ 3506786.9          jdz(M)≈0.999167;
  G(2020082820) = 10372851    ;Xi(M)≈ 10370624.32       jdz(M)≈0.999785;
  G(2020082822) = 3319333    ;Xi(M)≈ 3318559.03          jdz(M)≈0.999767;
  G(2020082824) = 3361364    ;Xi(M)≈ 3360699.36          jdz(M)≈0.999802;
  G(2020082826) = 6415022    ;Xi(M)≈ 6415880.65          jdz(M)≈1.000134;
  G(2020082828) = 3290790    ;Xi(M)≈ 3290333.7          jdz(M)≈0.999861;
  time start =12:40:26, time end =12:41:11

愚工688 · 发表于 2020-8-29 08:46

本帖最后由愚工688 于 2020-8-29 00:50 编辑

使用基于哈代公式改进的Xi(M)计算式，能够得到比较高的计算精度，但是目前的式子还不能控制好计算值始终处于下限位，即计算值≤真值。
使用素数连乘式计算偶数的素对数量，不仅仅能够得到比较高精度的素对计算值，还能够比较好的控制计算值的属性，即始终保持在下界计算值≤真值。当然连乘式的方法的运算速度比较Xi(M)计算式的运算速度，要慢很多。
下面用素数连乘式计算今天日期的千倍20200829000起始的连续偶数的素对下界数量，看看精度怎么样：

G(20200829000) = 38357465；
inf( 20200829000 )≈ 38333332.5 , jd ≈0.999371,infS(m) = 25874999.41 , k(m)= 1.48148
G(20200829002) = 25886949；
inf( 20200829002 )≈ 25874999.4 , jd ≈0.999538,infS(m) = 25874999.42 , k(m)= 1
G(20200829004) = 51784442；
inf( 20200829004 )≈ 51757571.6 , jd ≈0.999481,infS(m) = 25874999.42 , k(m)= 2.00029
G(20200829006) = 26216550；
inf( 20200829006 )≈ 26199204.2 , jd ≈0.999338,infS(m) = 25874999.42 , k(m)= 1.01253
G(20200829008) = 26150127；
inf( 20200829008 )≈ 26136363.1 , jd ≈0.999474,infS(m) = 25874999.42 , k(m)= 1.0101
G(20200829010) = 69033256；
inf( 20200829010 )≈ 68999998.5 , jd ≈0.999518,infS(m) = 25874999.43 , k(m)= 2.66667
G(20200829012) = 33895114；
inf( 20200829012 )≈ 33872726.5 , jd ≈0.999340,infS(m) = 25874999.43 , k(m)= 1.30909
G(20200829014) = 25897945；
inf( 20200829014 )≈ 25882339.9 , jd ≈0.999397,infS(m) = 25874999.43 , k(m)= 1.00028
G(20200829016) = 52089711；
inf( 20200829016 )≈ 52063635.2 , jd ≈0.999499,infS(m) = 25874999.43 , k(m)= 2.01212
G(20200829018) = 25890575；
inf( 20200829018 )≈ 25874999.4 , jd ≈0.999398,infS(m) = 25874999.44 , k(m)= 1
G(20200829020) = 36908243；
inf( 20200829020 )≈ 36890416.9 , jd ≈0.999517,infS(m) = 25874999.44 , k(m)= 1.42572
G(20200829022) = 60477838；
inf( 20200829022 )≈ 60439779.2 , jd ≈0.999371,infS(m) = 25874999.44 , k(m)= 2.33584

time start =08:19:45 ,time end =08:23:51 ,time use =
下界计算式显示如下：
inf( 20200829000 ) = 1/(1+ .1535 )*( 20200829000 /2 -2)*p(m) ≈ 38333332.5
inf( 20200829002 ) = 1/(1+ .1535 )*( 20200829002 /2 -2)*p(m) ≈ 25874999.4
inf( 20200829004 ) = 1/(1+ .1535 )*( 20200829004 /2 -2)*p(m) ≈ 51757571.6
inf( 20200829006 ) = 1/(1+ .1535 )*( 20200829006 /2 -2)*p(m) ≈ 26199204.2
inf( 20200829008 ) = 1/(1+ .1535 )*( 20200829008 /2 -2)*p(m) ≈ 26136363.1
inf( 20200829010 ) = 1/(1+ .1535 )*( 20200829010 /2 -2)*p(m) ≈ 68999998.5
inf( 20200829012 ) = 1/(1+ .1535 )*( 20200829012 /2 -2)*p(m) ≈ 33872726.5
inf( 20200829014 ) = 1/(1+ .1535 )*( 20200829014 /2 -2)*p(m) ≈ 25882339.9
inf( 20200829016 ) = 1/(1+ .1535 )*( 20200829016 /2 -2)*p(m) ≈ 52063635.2
inf( 20200829018 ) = 1/(1+ .1535 )*( 20200829018 /2 -2)*p(m) ≈ 25874999.4
inf( 20200829020 ) = 1/(1+ .1535 )*( 20200829020 /2 -2)*p(m) ≈ 36890416.9
inf( 20200829022 ) = 1/(1+ .1535 )*( 20200829022 /2 -2)*p(m) ≈ 60439779.2

愚工688 · 发表于 2020-9-21 11:06

今天是2020-09-21日，计算以日期的千倍起始的连续偶数的素对数量，看看精度如何？

  Xi(M)=t2*c1*M/(logM)^2


  G(20200921000) = 37719257    ;Xi(M)≈ 37647784.52       jdz(M)≈0.998900;
  G(20200921002) = 51784335    ;Xi(M)≈ 51692328.12       jdz(M)≈0.998223;
  G(20200921004) = 28838337    ;Xi(M)≈ 28785163.66       jdz(M)≈0.998156;
  G(20200921006) = 31955193    ;Xi(M)≈ 31895457.27       jdz(M)≈0.998131;
  G(20200921008) = 51778467    ;Xi(M)≈ 51682454.32       jdz(M)≈0.998146;
  G(20200921010) = 35287693    ;Xi(M)≈ 35220634.83       jdz(M)≈0.998100;
  G(20200921012) = 25890610    ;Xi(M)≈ 25841227.16       jdz(M)≈0.998093;
  G(20200921014) = 51785957    ;Xi(M)≈ 51682454.33       jdz(M)≈0.998001;
  G(20200921016) = 25893239    ;Xi(M)≈ 25841227.17       jdz(M)≈0.997991;
  G(20200921018) = 25949913    ;Xi(M)≈ 25903197.54       jdz(M)≈0.998200;
  G(20200921020) = 82848474    ;Xi(M)≈ 82691925.09       jdz(M)≈0.998110;
  G(20200921022) = 27129681    ;Xi(M)≈ 27077372.77       jdz(M)≈0.998072;
  G(20200921024) = 27989600    ;Xi(M)≈ 27934426.12       jdz(M)≈0.998029;
  G(20200921026) = 62817891    ;Xi(M)≈ 62697090.16       jdz(M)≈0.998077;
  G(20200921028) = 26749966    ;Xi(M)≈ 26699113.47       jdz(M)≈0.998099;
  time start =10:33:26, time end =10:37:00

愚工688 · 发表于 2020-10-13 22:39

今天是2020-10-13日，计算以日期的十倍起始的连续偶数的素对数量，看看精度如何？
Xi(M)=t2*c1*M/(logM)^2

  G(202010130) = 1338845    ;Xi(M)≈ 1336997.23          jdz(M)≈0.99862；
  G(202010132) = 407762    ;Xi(M)≈ 407809.04          jdz(M)≈1.00012；
  G(202010134) = 410426    ;Xi(M)≈ 409760.29          jdz(M)≈0.99838；
  G(202010136) = 888645    ;Xi(M)≈ 887623.08          jdz(M)≈0.99885；
  G(202010138) = 455742    ;Xi(M)≈ 455935.59          jdz(M)≈1.00042；
  G(202010140) = 543606    ;Xi(M)≈ 543745.43          jdz(M)≈1.00026；
  G(202010142) = 816249    ;Xi(M)≈ 815618.13          jdz(M)≈0.99923；
  G(202010144) = 489413    ;Xi(M)≈ 489370.89          jdz(M)≈0.99991；
  G(202010146) = 445247    ;Xi(M)≈ 445386.9          jdz(M)≈1.00031；
  G(202010148) = 817221    ;Xi(M)≈ 816793.52          jdz(M)≈0.99948；
  G(202010150) = 607548    ;Xi(M)≈ 607672.76          jdz(M)≈1.00021；
  G(202010152) = 409228    ;Xi(M)≈ 409103.71          jdz(M)≈0.99970；
  time start =22:25:30, time end =22:25:38

愚工688 · 发表于 2020-10-18 11:35

愚工688 发表于 2020-10-13 14:39
今天是2020-10-13日，计算以日期的十倍起始的连续偶数的素对数量，看看精度如何？
Xi(M)=t2*c1*M/(logM)^ ...

在使用 Xi(M)=t2*c1*M/(logM)^2  计算式计算偶数的素对时，基于哈代公式基础上用系数t2进行误差修正。因为t2是个递减函数，（t2=1.358-(log(M))^(.5)*0.05484）其只能在一定范围内满足对哈代公式误差的修正要求。在更大偶数范围就不适用了（M≥2^62，即t2＜0时）。
在10^8-10^9范围，相对误差绝对值比较小，但是计算值不能始终处于下限位，即计算值≤真值。
在≥10^10范围，则相对误差绝对值逐渐增大，但是计算值始终处于下限位，即计算值≤真值。

例：该式在一万亿范围内的，相比哈代公式的计算值的相对误差，误差修正效果还是不错的。
Xi(M)=t2*c1*M/(logM)^2 ,t2=1.358-log(M)^(0.5)*.05484;

  S( 10000 ) = 127       ;Xi(M)≈ 123.87    δxi( 10000 )≈-0.024646
  S( 10002 ) = 197       ;Xi(M)≈ 185.83    δxi( 10002 )≈-0.056701
  S( 10004 ) = 99       ;Xi(M)≈ 96.93       δxi( 10004 )≈-0.020909

  S( 100000 ) = 810    ;Xi(M)≈ 778.61    δxi( 100000 )≈-0.038765
  S( 100002 ) = 1423    ;Xi(M)≈ 1401.51    δxi( 100002 )≈-0.015109
  S( 100004 ) = 627    ;Xi(M)≈ 611.78    δxi( 100004 )≈-0.024274

  S( 1000000 ) = 5402    ;Xi(M)≈ 5323.26    δxi( 1000000 )≈-0.014576
  S( 1000002 ) = 8200    ;Xi(M)≈ 7984.91    δxi( 1000002 )≈-0.026232
  S( 1000004 ) = 4160    ;Xi(M)≈ 4117.53    δxi( 1000004 )≈-0.010216

  S( 10000000 ) = 38807    ;Xi(M)≈ 38552.75    δxi( 10000000 )≈-0.006552
  S( 10000002 ) = 59624    ;Xi(M)≈ 59114.23    δxi( 10000002 )≈-0.008550
  S( 10000004 ) = 36850    ;Xi(M)≈ 36738.51    δxi( 10000004 )≈-0.003026

  S( 100000000 ) = 291400 ;Xi(M)≈ 291217.74 δxi( 100000000 )≈-0.000625
  S( 100000002 ) = 464621 ;Xi(M)≈ 463540.71 δxi( 100000002 )≈-0.002325
  S( 100000004 ) = 247582 ;Xi(M)≈ 247142.31 δxi( 100000004 )≈-0.001776

  S( 1000000000 ) = 2274205 ;Xi(M)≈ 2271715.94 δxi( 1000000000 )≈-0.001094
  S( 1000000002 ) = 3496205 ;Xi(M)≈ 3495130.33 δxi( 1000000002 )≈-0.000307
  S( 1000000004 ) = 1747858 ;Xi(M)≈ 1747473.79 δxi( 1000000004 )≈-0.000220

  S( 10000000000 ) = 18200488  ;Xi(M)≈ 18176704.15  δxi( 10000000000 )≈-0.001307
  S( 10000000002 ) = 27302893  ;Xi(M)≈ 27265055.61  δxi( 10000000002 )≈-0.001386
  S( 10000000004 ) = 13655366  ;Xi(M)≈ 13632527.81  δxi( 10000000004 )≈-0.001672

  S( 100000000000 ) = 149091160 ;Xi(M)≈ 148458403.95 δxi( 100000000000 )≈-0.004244
  S( 100000000002 ) = 268556111 ;Xi(M)≈ 267398538.1  δxi( 100000000002 )≈-0.00431
  S( 100000000004 ) = 111836359 ;Xi(M)≈ 111350133.8  δxi( 100000000004 )≈-0.00435

  S( 1000000000000 ) = 1243722370;Xi(M)≈ 1233313937.7  δxi( 1000000000000 )≈-0.008369
  S( 1000000000002 ) = 1865594604;Xi(M)≈ 1849970864.8  δxi( 1000000000002 )≈-0.008375
  S( 1000000000004 ) = 1006929938;Xi(M)≈ 998459235.3 δxi( 1000000000004 )≈-0.008412
  time start =11:46:17    end time =11:57:09

愚工688 · 发表于 2020-10-23 10:03

今天是2020-10-23日，以20201023000为起点的连续偶数的素对计算，能否在小于真值的情况下保持比较好的精度呢？

  Xi(M)=t2*c1*M/(logM)^2

  G( 20201023000 ) = ?    ;Xi(M)≈ 36527740.24       δxi(M)≈?
  G( 20201023002 ) = ?    ;Xi(M)≈ 52477815.53       δxi(M)≈?
  G( 20201023004 ) = ?    ;Xi(M)≈ 27119466.94       δxi(M)≈?
  G( 20201023006 ) = ?    ;Xi(M)≈ 25883366.84       δxi(M)≈?
  G( 20201023008 ) = ?    ;Xi(M)≈ 52634066.25       δxi(M)≈?
  G( 20201023010 ) = ?    ;Xi(M)≈ 41380301.09       δxi(M)≈?
  G( 20201023012 ) = ?    ;Xi(M)≈ 25892319.12       δxi(M)≈?
  G( 20201023014 ) = ?    ;Xi(M)≈ 51682698.49       δxi(M)≈?
  G( 20201023016 ) = ?    ;Xi(M)≈ 25906115.01       δxi(M)≈?
  G( 20201023018 ) = ?    ;Xi(M)≈ 28720792.87       δxi(M)≈?
  G( 20201023020 ) = ?    ;Xi(M)≈ 68910266.22       δxi(M)≈?
  G( 20201023022 ) = ?    ;Xi(M)≈ 27514286.38       δxi(M)≈?
  G( 20201023024 ) = ?    ;Xi(M)≈ 37338123.12       δxi(M)≈?
  G( 20201023026 ) = ?    ;Xi(M)≈ 51682698.52       δxi(M)≈?
  G( 20201023028 ) = ?    ;Xi(M)≈ 25843495.75       δxi(M)≈?
  time start =09:44:19, time end =09:47:52

任在深 · 发表于 2020-10-23 11:24

啊！
这么多的无用功？？？

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基于偶数哥猜哈-李素对计算公式改进的偶数素对计算式 Xi(M)≈ t1c1M/(logM)^2

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