施承忠素数个数π(x)分段系数法

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施承忠素数个数π(x)分段系数法

素数定理虽然证明了π(x)≈x/lnx.若将π(x)≈x/lnx改写成π(x)=bx/lnx,则b作为密度系数，它在区间是波动的，不是恒为1.
为解决此问题我用解析方法得到了不同区间的密度系数bk.用这样的方法可以得到更精密的估计。下面我提供了b1到b70的数据，供大家使用。

(pk)^2≤x<(pk+1)^2；π(x)≈bk*x/lnx

b1=1.0397207708
b2=1.4648163849
b3=1.4163053629
b4=1.4296482728&#160;&#160;
b5=1.1494043456
b6=1.2748860712
b7=1.1568137529
b8=1.2723891433
b9=1.1972964681
b10=1.0410189249
b11=1.1506179811
b12=1.0445021866
b13=1.0559711171
b14=1.1472779152
b15=1.1468524771
b16=1.0798515564
b17=1.0331479533
b18=1.1091957430
b19=1.0659256406
b20=1.0823706095
b21=1.1480944198
b22=1.1089898090
b23=1.1225099526
b24=1.0925502993
b25=1.0317288443
b26=1.0514204569
b27=1.1052751758
b28=1.1199443027
b29=1.1695793643
b30=1.1802735035
b31=1.0337709685
b32=1.0522539995
b33=1.0427664832
b34=1.0869582868
b35=1.0264389657
b36=1.0685457167
b37=1.0605213523
b38=1.0536810231
b39=1.0698807389
b40=1.0634076945
b41=1.0578439766
b42=1.0942533966
b43=1.0478343798
b44=1.0828010848
b45=1.0969634787
b46=1.1302791747
b47=1.06722 21313
b48=1.0138243198
b49=1.0294234561
b50=1.1088974825
b51=1.0745641572
b52=1.0718800881
b53=1.10128444480
b54=1.0668340114
b55=1.0651362868
b56=1.0636950345
b57=1.0624856453
b58=1.0895911717
b59=1.0875832053
b60=1.0996651753
b61=1.1254380148
b62=1.0951579753
b63=1.0430568859
b64=1.0556077411
b65=1.0800414983
b66=1.0916175998
b67=1.0437998300
b68=1.0446210714
b69=1.0239439553
b70=1.0467898144


比如π(10)=4，(p2)^2=9,(p3)^2=25.9<10<25,取密度系数
b2=1.4648163849.
10/ln10=4.3429448190
4.3429448190*1.4648163849=6.3616167296

比如π(100)=25，(p4)^2=49，(p5)^2=121.49<100<121,取密度系数
b4=1.4296482728&#160;&#160;
100/ln100=21.7147240951
21.7147240951*1.4296482728=31.0444177969&#160;&#160;

比如π(1000)=168，(p11)^2=961，(p12)^2=1369.961<1000<1369,取密度系数b12=1.0445021866
1000/ln1000=144.7648273007
144.7648273007*1.0445021866=151.2071786584

比如π(10000)=1229，(p25)^2=9409，(p26)^2=10201.
9409<10000<10201,取密度系数b25=1.0317288443
10000/ln10000=1085.7362047553
1085.7362047553*1.0317288443=1120.1853597469

比如π(100000)=9592，(p65)^2=97969,(p66)^2=100489.
97969<10<100489,取密度系数b65=1.0800414983.
100000/ln100000=8685.8896380426
8685.8896380426*1.0800414983=9381.1212587400