Editing Prime number theorem (section)

== Table of ''π''(''x''), ''x'' / log ''x'', and li(''x'') ==
The table compares exact values of {{math|''π''(''x'')}} to the two approximations {{math|''x'' / log ''x''}} and {{math|li(''x'')}}. The approximation difference columns are rounded to the nearest integer, but the "% error" columns are computed based on the unrounded approximations.  The last column, {{math|''x'' / ''π''(''x'')}}, is the average [[prime gap]] below&nbsp;{{mvar|x}}.
: {| class="wikitable col1left" style="text-align: right"
!rowspan=2 scope=col| {{mvar|x}}
!rowspan=2 scope=col| {{math|''π''(''x'')}}
!rowspan=2 scope=col| {{math|''π''(''x'') − {{sfrac|''x''|log(''x'')}}}}
!rowspan=2 scope=col| {{math|li(''x'') − ''π''(''x'')}}
!colspan=2 scope=colgroup| % error
!rowspan=2 scope=col| {{math|{{sfrac|''x''|''π''(''x'')}}}}
|-
!scope=col| {{math|{{sfrac|''x''|log(''x'')}}}}
!scope=col| {{math|li(''x'')}}
|-
| 10
| 4
| 0
| 2
|8.22%
|42.606%
| 2.500
|-
| 10<sup>2</sup>
| 25
| 3
| 5
|14.06%
|18.597%
| 4.000
|-
| 10<sup>3</sup>
| 168
| 23
| 10
|14.85%
|5.561%
| 5.952
|-
| 10<sup>4</sup>
| 1,229
| 143
| 17
|12.37%
|1.384%
| 8.137
|-
| 10<sup>5</sup>
| 9,592
| 906
| 38
|9.91%
|0.393%
| 10.425
|-
| 10<sup>6</sup>
| 78,498
| 6,116
| 130
|8.11%
|0.164%
| 12.739
|-
| 10<sup>7</sup>
| 664,579
| 44,158
| 339
|6.87%
|0.051%
| 15.047
|-
| 10<sup>8</sup>
| 5,761,455
| 332,774
| 754
|5.94%
|0.013%
| 17.357
|-
| 10<sup>9</sup>
| 50,847,534
| 2,592,592
| 1,701
|5.23%
|3.34{{e|−3}} %
| 19.667
|-
| 10<sup>10</sup>
| 455,052,511
| 20,758,029
| 3,104
|4.66%
|6.82{{e|−4}} %
| 21.975
|-
| 10<sup>11</sup>
| 4,118,054,813
| 169,923,159
| 11,588
|4.21%
|2.81{{e|−4}} %
| 24.283
|-
| 10<sup>12</sup>
| 37,607,912,018
| 1,416,705,193
| 38,263
|3.83%
|1.02{{e|−4}} %
| 26.590
|-
| 10<sup>13</sup>
| 346,065,536,839
| 11,992,858,452
| 108,971
|3.52%
|3.14{{e|−5}} %
| 28.896
|-
| 10<sup>14</sup>
| {{zwsp|3,|204,|941,|750,|802}}
| 102,838,308,636
| 314,890
|3.26%
|9.82{{e|−6}} %
| 31.202
|-
| 10<sup>15</sup>
| {{zwsp|29,|844,|570,|422,|669}}
| 891,604,962,452
| 1,052,619
|3.03%
|3.52{{e|−6}} %
| 33.507
|-
| 10<sup>16</sup>
| {{zwsp|279,|238,|341,|033,|925}}
| {{zwsp|7,|804,|289,|844,|393}}
| 3,214,632
|2.83%
|1.15{{e|−6}} %
| 35.812
|-
| 10<sup>17</sup>
| {{zwsp|2,|623,|557,|157,|654,|233}}
| {{zwsp|68,|883,|734,|693,|928}}
| 7,956,589
|2.66%
|3.03{{e|−7}} %
| 38.116
|-
| 10<sup>18</sup>
| {{zwsp|24,|739,|954,|287,|740,|860}}
| {{zwsp|612,|483,|070,|893,|536}}
| 21,949,555
|2.51%
|8.87{{e|−8}} %
| 40.420
|-
| 10<sup>19</sup>
| {{zwsp|234,|057,|667,|276,|344,|607}}
| {{zwsp|5,|481,|624,|169,|369,|961}}
| 99,877,775
|2.36%
|4.26{{e|−8}} %
| 42.725
|-
| 10<sup>20</sup>
| {{zwsp|2,|220,|819,|602,|560,|918,|840}}
| {{zwsp|49,|347,|193,|044,|659,|702}}
| 222,744,644
|2.24%
|1.01{{e|−8}} %
| 45.028
|-
| 10<sup>21</sup>
| {{zwsp|21,|127,|269,|486,|018,|731,|928}}
| {{zwsp|446,|579,|871,|578,|168,|707}}
| 597,394,254
|2.13%
|2.82{{e|−9}} %
| 47.332
|-
| 10<sup>22</sup>
| {{zwsp|201,|467,|286,|689,|315,|906,|290}}
| {{zwsp|4,|060,|704,|006,|019,|620,|994}}
| 1,932,355,208
|2.03%
|9.59{{e|−10}} %
| 49.636
|-
| 10<sup>23</sup>
| {{zwsp|1,|925,|320,|391,|606,|803,|968,|923}}
| {{zwsp|37,|083,|513,|766,|578,|631,|309}}
| 7,250,186,216
|1.94%
|3.76{{e|−10}} %
| 51.939
|-
| 10<sup>24</sup>
| {{zwsp|18,|435,|599,|767,|349,|200,|867,|866}}
| {{zwsp|339,|996,|354,|713,|708,|049,|069}}
| 17,146,907,278
|1.86%
|9.31{{e|−11}} %
| 54.243
|-
| 10<sup>25</sup>
| {{zwsp|176,|846,|309,|399,|143,|769,|411,|680}}
| {{zwsp|3,|128,|516,|637,|843,|038,|351,|228}}
| 55,160,980,939
|1.78%
|3.21{{e|−11}} %
| 56.546
|-
| 10<sup>26</sup>
| {{zwsp|1,|699,|246,|750,|872,|437,|141,|327,|603}}
| {{zwsp|28,|883,|358,|936,|853,|188,|823,|261}}
| 155,891,678,121
|1.71%
|9.17{{e|−12}} %
| 58.850
|-
| 10<sup>27</sup>
| {{zwsp|16,|352,|460,|426,|841,|680,|446,|427,|399}}
| {{zwsp|267,|479,|615,|610,|131,|274,|163,|365}}
| 508,666,658,006
|1.64%
|3.11{{e|−12}} %
| 61.153
|-
| 10<sup>28</sup>
| {{zwsp|157,|589,|269,|275,|973,|410,|412,|739,|598}}
| {{zwsp|2,|484,|097,|167,|669,|186,|251,|622,|127}}
| {{zwsp|1,|427,|745,|660,|374}}
|1.58%
|9.05{{e|−13}} %
| 63.456
|-
| 10<sup>29</sup>
| {{zwsp|1,|520,|698,|109,|714,|272,|166,|094,|258,|063}}
| {{zwsp|23,|130,|930,|737,|541,|725,|917,|951,|446}}
| {{zwsp|4,|551,|193,|622,|464}}
|1.53%
|2.99{{e|−13}} %
| 65.759
|}

The value for {{math|''π''(10<sup>24</sup>)}} was originally computed assuming the [[Riemann hypothesis]];<ref name="Franke">{{cite web |title=Conditional Calculation of {{math|''π''(10<sup>24</sup>)}} |url=http://primes.utm.edu/notes/pi(10%5E24).html |publisher=Chris K. Caldwell |archiveurl=https://web.archive.org/web/20100804004746/http://primes.utm.edu/notes/pi(10%5E24).html |archivedate=2010-08-04 |access-date=2010-08-03}}</ref> it has since been verified unconditionally.<ref name="PlattARXIV2012">{{cite journal | last=Platt | first=David | title=Computing {{math|''π''(''x'')}} analytically | year=2015 | journal=[[Mathematics of Computation]] | volume=84 | issue=293 | pages=1521–1535 | mr=3315519 | arxiv=1203.5712 | doi=10.1090/S0025-5718-2014-02884-6 | s2cid=119174627 }}</ref>