高精度计算大偶数表为两个素数和的表法数值的实例（以当天日期为随机数选择偶数）

重生888@

谢谢解答！祝贺您计算精度越来越高！

愚工688

（这是昨天计算的，这里也发一下）
偶数的表法数值的变化是有规律性的，因此是能够比较精确的进行计算的。
今天是2020-08-11日，以今天日期的千倍偶数为起始的连续偶数素对数量的计算，请看精度如何？
精度值： jdz =sp(m)/G(m)
G(20200811000) = 36818761；Sp( 20200811000 *)≈ 36806874.8 , jdz ≈0.999677
G(20200811002) = 26069591；Sp( 20200811002 *)≈ 26059409.0 , jdz ≈0.999609
G(20200811004) = 57672374；Sp( 20200811004 *)≈ 57647650.0 , jdz ≈0.999571
G(20200811006) = 25888718；Sp( 20200811006 *)≈ 25879998.3 , jdz ≈0.999663
G(20200811008) = 31318293；Sp( 20200811008 *)≈ 31304504.5 , jdz ≈0.999560
G(20200811010) = 71595400；Sp( 20200811010 *)≈ 71567899.3 , jdz ≈0.999616
G(20200811012) = 25890922；Sp( 20200811012 *)≈ 25881946.0 , jdz ≈0.999653
G(20200811014) = 26288124；Sp( 20200811014 *)≈ 26277609.2 , jdz ≈0.999600
G(20200811016) = 51783778；Sp( 20200811016 *)≈ 51758927.2 , jdz ≈0.999520
G(20200811018) = 25899963；Sp( 20200811018 *)≈ 25890287.3 , jdz ≈0.999626


计算式如下：
Sp( 20200811000 *) = 1/(1+ .1533 )*( 20200811000 /2 -2)*p(m) ≈ 36806874.8 , k(m)= 1.422243
Sp( 20200811002 *) = 1/(1+ .1533 )*( 20200811002 /2 -2)*p(m) ≈ 26059409 ,    k(m)= 1.006953
Sp( 20200811004 *) = 1/(1+ .1533 )*( 20200811004 /2 -2)*p(m) ≈ 57647650 ,    k(m)= 2.227544
Sp( 20200811006 *) = 1/(1+ .1533 )*( 20200811006 /2 -2)*p(m) ≈ 25879998.3 , k(m)= 1.000021
Sp( 20200811008 *) = 1/(1+ .1533 )*( 20200811008 /2 -2)*p(m) ≈ 31304504.5 , k(m)= 1.209627
Sp( 20200811010 *) = 1/(1+ .1533 )*( 20200811010 /2 -2)*p(m) ≈ 71567899.3 , k(m)= 2.765432
Sp( 20200811012 *) = 1/(1+ .1533 )*( 20200811012 /2 -2)*p(m) ≈ 25881946 ,    k(m)= 1.000096
Sp( 20200811014 *) = 1/(1+ .1533 )*( 20200811014 /2 -2)*p(m) ≈ 26277609.2 , k(m)= 1.015385
Sp( 20200811016 *) = 1/(1+ .1533 )*( 20200811016 /2 -2)*p(m) ≈ 51758927.2 , k(m)= 2
Sp( 20200811018 *) = 1/(1+ .1533 )*( 20200811018 /2 -2)*p(m) ≈ 25890287.3 , k(m)= 1.000418

重生888@

请问先生P（m）=?

愚工688

以今天日期的百倍为随机偶数计算的连续偶数素对数量计算精度：
  Xi(M)=t2*c1*M/(logM)^2   

  G(2020081700) = 5599584      ;Xi(M)≈ 5599310.93           jdz(M)≈0.999951
  G(2020081702) = 3209497      ;Xi(M)≈ 3207938.54           jdz(M)≈0.999514
  G(2020081704) = 6450811      ;Xi(M)≈ 6449823.58           jdz(M)≈0.999847
  G(2020081706) = 3322379      ;Xi(M)≈ 3321478.46           jdz(M)≈0.999729
  G(2020081708) = 3208371      ;Xi(M)≈ 3207938.55           jdz(M)≈0.999865
  G(2020081710) = 9393256      ;Xi(M)≈ 9393179.37           jdz(M)≈0.999992
  G(2020081712) = 3800959      ;Xi(M)≈ 3802001.11           jdz(M)≈1.000274
  G(2020081714) = 3963303      ;Xi(M)≈ 3964396.83           jdz(M)≈1.000276
  G(2020081716) = 6428036      ;Xi(M)≈ 6429854.95           jdz(M)≈1.000283
  G(2020081718) = 3228243      ;Xi(M)≈ 3231354.19           jdz(M)≈1.000964
  G(2020081720) = 4480428      ;Xi(M)≈ 4480929.93           jdz(M)≈1.000112
  G(2020081722) = 6494114      ;Xi(M)≈ 6495085.64           jdz(M)≈1.000150
  G(2020081724) = 3210417      ;Xi(M)≈ 3211311.67           jdz(M)≈1.000279
  G(2020081726) = 3590668      ;Xi(M)≈ 3589302.01           jdz(M)≈0.999620
  G(2020081728) = 7700392      ;Xi(M)≈ 7699052.71           jdz(M)≈0.999826
  time start =14:49:16, time end =14:50:01

愚工688

2020-08-20日的10倍起的连续偶数的素对计算值以及计算精度：
  Xi(M)=t2*c1*M/(logM)^2   

  G(202008200) = 563926       ;Xi(M)≈ 564349.59            jdz(M)≈1.000750;
  G(202008202) = 454402       ;Xi(M)≈ 454043.89            jdz(M)≈0.999212;
  G(202008204) = 819135       ;Xi(M)≈ 819141.9             jdz(M)≈1.000008;
  G(202008206) = 413748       ;Xi(M)≈ 413715.78            jdz(M)≈0.999922;
  G(202008208) = 410217       ;Xi(M)≈ 409618.05            jdz(M)≈0.998540;
  G(202008210) = 1114890      ;Xi(M)≈ 1114304.3            jdz(M)≈0.999474;
  G(202008212) = 530566       ;Xi(M)≈ 531222.73            jdz(M)≈1.001238;
  G(202008214) = 427380       ;Xi(M)≈ 427346.65            jdz(M)≈0.999922;
  G(202008216) = 828631       ;Xi(M)≈ 828159.01            jdz(M)≈0.999430;
  G(202008218) = 427105       ;Xi(M)≈ 427224.9             jdz(M)≈1.000281;
  G(202008220) = 544084       ;Xi(M)≈ 544194.24            jdz(M)≈1.000202;
  G(202008222) = 900599       ;Xi(M)≈ 901002.3             jdz(M)≈1.000447;
  G(202008224) = 453456       ;Xi(M)≈ 453117.34            jdz(M)≈0.999252;
  G(202008226) = 489656       ;Xi(M)≈ 489366.73            jdz(M)≈0.999409;
  G(202008228) = 897847       ;Xi(M)≈ 896856.73            jdz(M)≈0.998897;

愚工688

今天是2020-08-24日，以今天日期的千倍为随机数的连续偶数的素对下界数量的计算，看精度如何？

素对下界计算值inf(M)精度 jd(M)=inf(M)/G(M)
素对下界计算值inf(M)与素对区域下界值infS(m)的关系：
    infS(m) = inf(M)/k(m) ；
    k(m)——偶数含有的奇素因子形成的系数，反映了偶数素对数量的波动幅度。


G(20200824000) = 83727124；
inf( 20200824000 )≈  83675464.7 , jd(M)≈0.999383,infS(m) = 25874993.01 , k(m)= 3.23384
G(20200824002) = 27696303；
inf( 20200824002 )≈  27677158   , jd(M)≈0.999309,infS(m) = 25874993.01 , k(m)= 1.06965
G(20200824004) = 26797139；
inf( 20200824004 )≈  26779324.4 , jd(M)≈0.999335,infS(m) = 25874993.01 , k(m)= 1.03495
G(20200824006) = 58109113；
inf( 20200824006 )≈  58072107.3 , jd(M)≈0.999363,infS(m) = 25874993.02 , k(m)= 2.24433
G(20200824008) = 27513680；
inf( 20200824008 )≈  27499992.6 , jd(M)≈0.999503,infS(m) = 25874993.02 , k(m)= 1.0628
G(20200824010) = 34521725；
inf( 20200824010 )≈  34499990.7 , jd(M)≈0.999370,infS(m) = 25874993.02 , k(m)= 1.33333
G(20200824012) = 52697883；
inf( 20200824012 )≈  52662583.6 , jd(M)≈0.999330,infS(m) = 25874993.03 , k(m)= 2.03527
G(20200824014) = 31077877；
inf( 20200824014 )≈  31062211.2 , jd(M)≈0.999496,infS(m) = 25874993.03 , k(m)= 1.20047
G(20200824016) = 27428870；
inf( 20200824016 )≈  27407738.6 , jd(M)≈0.999230,infS(m) = 25874993.03 , k(m)= 1.05924
G(20200824018) = 51782483；
inf( 20200824018 )≈  51749986.1 , jd(M)≈0.999372,infS(m) = 25874993.03 , k(m)= 2
G(20200824020) = 37719610；
inf( 20200824020 )≈  37699477.1 , jd(M)≈0.999466,infS(m) = 25874993.04 , k(m)= 1.45699
G(20200824022) = 27614625；
inf( 20200824022 )≈  27602549.5 , jd(M)≈0.999563,infS(m) = 25874993.04 , k(m)= 1.06677
G(20200824024) = 51803519；
inf( 20200824024 )≈  51769551.3 , jd(M)≈0.999344,infS(m) = 25874993.04 , k(m)= 2.00076
G(20200824026) = 26439478；
inf( 20200824026 )≈  26424313.4 , jd(M)≈0.999426,infS(m) = 25874993.04 , k(m)= 1.02123
G(20200824028) = 34520710；
inf( 20200824028 )≈  34499990.7 , jd(M)≈0.999400,infS(m) = 25874993.05 , k(m)= 1.33333
G(20200824030) = 69044481；
inf( 20200824030 )≈  68999981.5 , jd(M)≈0.999355,infS(m) = 25874993.05 , k(m)= 2.66667
G(20200824032) = 26385457；
inf( 20200824032 )≈  26373955.5 , jd(M)≈0.999564,infS(m) = 25874993.05 , k(m)= 1.01928

可以看到，
1，各个偶数的素对下界计算值inf(M)与素对真值G（M）都是很接近的，计算值精度都很高；
2，排除了波动系数的素对区域下界值,infS(m)是随偶数M增大而缓慢的线性增大的；

下界计算式附上：
inf( 20200824000 ) = 1/(1+ .1535 )*( 20200824000 /2 -2)*p(m) ≈ 83675464.7
inf( 20200824002 ) = 1/(1+ .1535 )*( 20200824002 /2 -2)*p(m) ≈ 27677158
inf( 20200824004 ) = 1/(1+ .1535 )*( 20200824004 /2 -2)*p(m) ≈ 26779324.4
inf( 20200824006 ) = 1/(1+ .1535 )*( 20200824006 /2 -2)*p(m) ≈ 58072107.3
inf( 20200824008 ) = 1/(1+ .1535 )*( 20200824008 /2 -2)*p(m) ≈ 27499992.6
inf( 20200824010 ) = 1/(1+ .1535 )*( 20200824010 /2 -2)*p(m) ≈ 34499990.7
inf( 20200824012 ) = 1/(1+ .1535 )*( 20200824012 /2 -2)*p(m) ≈ 52662583.6
inf( 20200824014 ) = 1/(1+ .1535 )*( 20200824014 /2 -2)*p(m) ≈ 31062211.2
inf( 20200824016 ) = 1/(1+ .1535 )*( 20200824016 /2 -2)*p(m) ≈ 27407738.6
inf( 20200824018 ) = 1/(1+ .1535 )*( 20200824018 /2 -2)*p(m) ≈ 51749986.1
inf( 20200824020 ) = 1/(1+ .1535 )*( 20200824020 /2 -2)*p(m) ≈ 37699477.1
inf( 20200824022 ) = 1/(1+ .1535 )*( 20200824022 /2 -2)*p(m) ≈ 27602549.5
inf( 20200824024 ) = 1/(1+ .1535 )*( 20200824024 /2 -2)*p(m) ≈ 51769551.3
inf( 20200824026 ) = 1/(1+ .1535 )*( 20200824026 /2 -2)*p(m) ≈ 26424313.4
inf( 20200824028 ) = 1/(1+ .1535 )*( 20200824028 /2 -2)*p(m) ≈ 34499990.7
inf( 20200824030 ) = 1/(1+ .1535 )*( 20200824030 /2 -2)*p(m) ≈ 68999981.5
inf( 20200824032 ) = 1/(1+ .1535 )*( 20200824032 /2 -2)*p(m) ≈ 26373955.5

蔡家雄

愚工688

今天是2020-08-31日，使用素数连乘式计算以今天日期千倍起始的连续偶数的素对下界，看计算值精度如何？

G(20200831000) = 43400560；
inf( 20200831000 )≈  43371777.6 ,   jd ≈0.999337  ,infS(m) = 25875001.98 , k(m)= 1.6762
G(20200831002) = 57650047；
inf( 20200831002 )≈  57618423.2 ,   jd ≈0.999451  ,infS(m) = 25875001.98 , k(m)= 2.2268
G(20200831004) = 26326773；
inf( 20200831004 )≈  26313561.3 ,   jd ≈0.999498  ,infS(m) = 25875001.98 , k(m)= 1.01695
G(20200831006) = 25889837；
inf( 20200831006 )≈  25875002 ,     jd ≈0.999427  ,infS(m) = 25875001.98 , k(m)= 1
G(20200831008) = 55325389；
inf( 20200831008 )≈  55288326.7 ,  jd ≈0.999330  ,infS(m) = 25875001.99 , k(m)= 2.13675
G(20200831010) = 35803007；
inf( 20200831010 )≈  35770586.9 ,  jd ≈0.999094  ,infS(m) = 25875001.99 , k(m)= 1.38244
G(20200831012) = 25888186
inf( 20200831012 )≈  25875002 ,     jd ≈0.999491 ,infS(m) = 25875001.99 , k(m)= 1
G(20200831014) = 67776860
inf( 20200831014 )≈  67745459.8 ,  jd ≈0.999537 ,infS(m) = 25875001.99 , k(m)= 2.61818
G(20200831016) = 25887011；
inf( 20200831016 )≈  25875002 ,    jd ≈0.999536 ,infS(m) = 25875002 ,      k(m)= 1
G(20200831018) = 25935715；
inf( 20200831018 )≈  25916938.8 , jd ≈0.999276  ,infS(m) = 25875002 ,     k(m)= 1.00162
G(20200831020) = 72035347；
inf( 20200831020 )≈  71992338 ,    jd ≈0.999403  ,infS(m) = 25875002 ,     k(m)= 2.78231
G(20200831022) = 26630652；
inf( 20200831022 )≈  26614287.8 , jd ≈0.999386  ,infS(m) = 25875002 ,      k(m)= 1.02857
time start =10:56:05  ,time end =11:00:24   ,time use =

素对下界计算式：
inf( 20200831000 ) = 1/(1+ .1535 )*( 20200831000 /2 -2)*p(m) ≈ 43371777.6
inf( 20200831002 ) = 1/(1+ .1535 )*( 20200831002 /2 -2)*p(m) ≈ 57618423.2
inf( 20200831004 ) = 1/(1+ .1535 )*( 20200831004 /2 -2)*p(m) ≈ 26313561.3
inf( 20200831006 ) = 1/(1+ .1535 )*( 20200831006 /2 -2)*p(m) ≈ 25875002
inf( 20200831008 ) = 1/(1+ .1535 )*( 20200831008 /2 -2)*p(m) ≈ 55288326.7
inf( 20200831010 ) = 1/(1+ .1535 )*( 20200831010 /2 -2)*p(m) ≈ 35770586.9
inf( 20200831012 ) = 1/(1+ .1535 )*( 20200831012 /2 -2)*p(m) ≈ 25875002
inf( 20200831014 ) = 1/(1+ .1535 )*( 20200831014 /2 -2)*p(m) ≈ 67745459.8
inf( 20200831016 ) = 1/(1+ .1535 )*( 20200831016 /2 -2)*p(m) ≈ 25875002
inf( 20200831018 ) = 1/(1+ .1535 )*( 20200831018 /2 -2)*p(m) ≈ 25916938.8
inf( 20200831020 ) = 1/(1+ .1535 )*( 20200831020 /2 -2)*p(m) ≈ 71992338
inf( 20200831022 ) = 1/(1+ .1535 )*( 20200831022 /2 -2)*p(m) ≈ 26614287.8

愚工688

今天是2020-09-02日，以日期数加上2000后的百倍作为随机偶数，计算一下连续偶数的素对数量，看看计算值的精度如何？

使用的计算式 ：Xi(M)=t2*c1*M/(logM)^2


G(4020090200) = 9910188 ;     Xi(M)≈ 9907327.87       jd(M)≈0.999711；
G(4020090202) = 5981525 ;     Xi(M)≈ 5980902.05       jd(M)≈0.999896；
G(4020090204) = 13034047 ;   Xi(M)≈ 13027492.41     jd(M)≈0.999497；
G(4020090206) = 6702457 ;     Xi(M)≈ 6700057.85       jd(M)≈0.999642；
G(4020090208) = 5976334 ;     Xi(M)≈ 5973832.31       jd(M)≈0.999581；
G(4020090210) = 16322015 ;   Xi(M)≈ 16310844.47     jd(M)≈0.999316；
G(4020090212) = 5975435 ;     Xi(M)≈ 5973709.94       jd(M)≈0.999711；
G(4020090214) = 7215064 ;     Xi(M)≈ 7213209.26       jd(M)≈0.999743；
G(4020090216) = 11945641 ;   Xi(M)≈ 11941868.25     jd(M)≈0.999684；
G(4020090218) = 5974769 ;     Xi(M)≈ 5972621.51       jd(M)≈0.999641；
G(4020090220) = 7963352 ;     Xi(M)≈ 7961245.69       jd(M)≈0.999736；
G(4020090222) = 12102483 ;   Xi(M)≈ 12096957.44     jd(M)≈0.999543；
G(4020090224) = 6321553 ;     Xi(M)≈ 6322165.43       jd(M)≈1.000097；
G(4020090226) = 6050813 ;     Xi(M)≈ 6047941.78       jd(M)≈0.999525；
G(4020090228) = 15928966 ;   Xi(M)≈ 15922491.41     jd(M)≈0.999594；
time start =10:31:13, time end =10:32:24

可以看到，这些偶数的素对计算值的精度都是不错的！

愚工688

今天是2020-09-15日，以日期的千倍为起点的连续偶数的素对数量的计算，看看精度怎么样？

  Xi(M)=t2*c1*M/(logM)^2   

  G(20200915000) = 41427291     ;Xi(M)≈ 41345954.71          jdz(M)≈0.99804;
  G(20200915002) = 51800143     ;Xi(M)≈ 51704152.03          jdz(M)≈0.998147;
  G(20200915004) = 27540336     ;Xi(M)≈ 27485439.65          jdz(M)≈0.998007;
  G(20200915006) = 27622291     ;Xi(M)≈ 27563970.59          jdz(M)≈0.997889;
  G(20200915008) = 51866032     ;Xi(M)≈ 51764272.22          jdz(M)≈0.998038;
  G(20200915010) = 35128365     ;Xi(M)≈ 35059436.67          jdz(M)≈0.998038;
  G(20200915012) = 26635068     ;Xi(M)≈ 26584815.21          jdz(M)≈0.998113;
  G(20200915014) = 62132038     ;Xi(M)≈ 62018934.44          jdz(M)≈0.998180;
  G(20200915016) = 25963205     ;Xi(M)≈ 25918360.35          jdz(M)≈0.998273;
  G(20200915018) = 25944949     ;Xi(M)≈ 25897278.16          jdz(M)≈0.998163;
  G(20200915020) = 88335709     ;Xi(M)≈ 88163256.44          jdz(M)≈0.998048;
  G(20200915022) = 26129217     ;Xi(M)≈ 26076658.79          jdz(M)≈0.997989;
  G(20200915024) = 25922147     ;Xi(M)≈ 25873727.52          jdz(M)≈0.998132;
  G(20200915026) = 54825635     ;Xi(M)≈ 54722587.32          jdz(M)≈0.998120;
  G(20200915028) = 31168635     ;Xi(M)≈ 31108578.74          jdz(M)≈0.998073;

应该说这些偶数的素对计算值的精度都不错，相对误差全部小于0.01；

当然采用埃拉脱色尼筛法的素数连乘式的素对下界计算值也能达到高精度：

   G(20200915000) = 41427291；
inf( 20200915000 )≈  41400175.3 , Δ≈-0.0006545,infS(m) = 25875109.57
   G(20200915002) = 51800143；
inf( 20200915002 )≈  51771953.8 , Δ≈-0.0005442,infS(m) = 25875109.57 ,
   G(20200915004) = 27540336；
inf( 20200915004 )≈  27521484.1 , Δ≈-0.0006845,infS(m) = 25875109.58 ,
   G(20200915006) = 27622291；
inf( 20200915006 )≈  27600116.9 , Δ≈-0.0008028,infS(m) = 25875109.58 ,
   G(20200915008) = 51866032；
inf( 20200915008 )≈  51832152.8 , Δ≈-0.0006532,infS(m) = 25875109.58 ,
   G(20200915010) = 35128365；
inf( 20200915010 )≈  35105411.8 , Δ≈-0.0006534,infS(m) = 25875109.58 ,
G(20200915012) = 26635068；
inf( 20200915012 )≈  26619678.3 , Δ≈-0.0005778,infS(m) = 25875109.59 ,
   G(20200915014) = 62132038；
inf( 20200915014 )≈  62100263 ,   Δ≈-0.0005114,infS(m) = 25875109.59 ,
   G(20200915016) = 25963205；
inf( 20200915016 )≈  25952348.7 , Δ≈-0.0004181,infS(m) = 25875109.59 ,
   G(20200915018) = 25944949；
inf( 20200915018 )≈  25931237.8 , Δ≈-0.0005285,infS(m) = 25875109.59 ,
   G(20200915020) = 88335709；
inf( 20200915020 )≈  88278875.1 , Δ≈-0.0006434,infS(m) = 25875109.6 ,
   G(20200915022) = 26129217；
inf( 20200915022 )≈  26110855.8 , Δ≈-0.0007027,infS(m) = 25875109.6 ,

time start =10:28:31  ,time end =10:32:48   ,time use =
计算式如下：
inf( 20200915000 ) = 1/(1+ .1535 )*( 20200915000 /2 -2)*p(m) ≈ 41400175.3
inf( 20200915002 ) = 1/(1+ .1535 )*( 20200915002 /2 -2)*p(m) ≈ 51771953.8
inf( 20200915004 ) = 1/(1+ .1535 )*( 20200915004 /2 -2)*p(m) ≈ 27521484.1
inf( 20200915006 ) = 1/(1+ .1535 )*( 20200915006 /2 -2)*p(m) ≈ 27600116.9
inf( 20200915008 ) = 1/(1+ .1535 )*( 20200915008 /2 -2)*p(m) ≈ 51832152.8
inf( 20200915010 ) = 1/(1+ .1535 )*( 20200915010 /2 -2)*p(m) ≈ 35105411.8
inf( 20200915012 ) = 1/(1+ .1535 )*( 20200915012 /2 -2)*p(m) ≈ 26619678.3
inf( 20200915014 ) = 1/(1+ .1535 )*( 20200915014 /2 -2)*p(m) ≈ 62100263
inf( 20200915016 ) = 1/(1+ .1535 )*( 20200915016 /2 -2)*p(m) ≈ 25952348.7
inf( 20200915018 ) = 1/(1+ .1535 )*( 20200915018 /2 -2)*p(m) ≈ 25931237.8
inf( 20200915020 ) = 1/(1+ .1535 )*( 20200915020 /2 -2)*p(m) ≈ 88278875.09999999
inf( 20200915022 ) = 1/(1+ .1535 )*( 20200915022 /2 -2)*p(m) ≈ 26110855.8