Editing Prime-counting function (section)

==Table of {{math|''π''(''x'')}}, {{math|{{sfrac|''x''|log ''x'' }}}}, and {{math|li(''x'')}}==

The table shows how the three functions {{math|''π''(''x'')}}, {{math|{{sfrac|''x''|log ''x''}}}}, and {{math|li(''x'')}} compared at powers of 10. See also,<ref name="Caldwell" /><ref name="Silva">{{cite web |title=Tables of values of {{math|''π''(''x'')}} and of {{math|''π''<sub>2</sub>(''x'')}} |url=https://sweet.ua.pt/tos/primes.html |publisher=Tomás Oliveira e Silva |access-date=2024-03-31}}</ref> and<ref name="Gourdon">{{cite web |title=A table of values of {{math|''π''(''x'')}} |url=http://numbers.computation.free.fr/Constants/Primes/pixtable.html |publisher=Xavier Gourdon, Pascal Sebah, Patrick Demichel |access-date=2008-09-14}}</ref>

:{| class="wikitable" style="text-align: right"
! {{mvar|x}}
! {{math|''π''(''x'')}}
! {{math|''π''(''x'') − {{sfrac|''x''|log ''x''}}}}
! {{math|li(''x'') − ''π''(''x'')}}
! {{math|{{sfrac|''x''|''π''(''x'')}}}}
!{{math|{{sfrac|''x''|log ''x''}}}}<br> % error
|-
| 10
| 4
| 0
| 2
| 2.500
| −8.57%
|-
| 10<sup>2</sup>
| 25
| 3
| 5
| 4.000
| +13.14%
|-
| 10<sup>3</sup>
| 168
| 23
| 10
| 5.952
| +13.83%
|-
| 10<sup>4</sup>
| 1,229
| 143
| 17
| 8.137
| +11.66%
|-
| 10<sup>5</sup>
| 9,592
| 906
| 38
| 10.425
| +9.45%
|-
| 10<sup>6</sup>
| 78,498
| 6,116
| 130
| 12.739
| +7.79%
|-
| 10<sup>7</sup>
| 664,579
| 44,158
| 339
| 15.047
| +6.64%
|-
| 10<sup>8</sup>
| 5,761,455
| 332,774
| 754
| 17.357
| +5.78%
|-
| 10<sup>9</sup>
| 50,847,534
| 2,592,592
| 1,701
| 19.667
| +5.10%
|-
| 10<sup>10</sup>
| 455,052,511
| 20,758,029
| 3,104
| 21.975
| +4.56%
|-
| 10<sup>11</sup>
| 4,118,054,813
| 169,923,159
| 11,588
| 24.283
| +4.13%
|-
| 10<sup>12</sup>
| 37,607,912,018
| 1,416,705,193
| 38,263
| 26.590
| +3.77%
|-
| 10<sup>13</sup>
| 346,065,536,839
| 11,992,858,452
| 108,971
| 28.896
| +3.47%
|-
| 10<sup>14</sup>
| 3,204,941,750,802
| 102,838,308,636
| 314,890
| 31.202
| +3.21%
|-
| 10<sup>15</sup>
| 29,844,570,422,669
| 891,604,962,452
| 1,052,619
| 33.507
| +2.99%
|-
| 10<sup>16</sup>
| 279,238,341,033,925
| 7,804,289,844,393
| 3,214,632
| 35.812
| +2.79%
|-
| 10<sup>17</sup>
| 2,623,557,157,654,233
| 68,883,734,693,928
| 7,956,589
| 38.116
| +2.63%
|-
| 10<sup>18</sup>
| 24,739,954,287,740,860
| 612,483,070,893,536
| 21,949,555
| 40.420
| +2.48%
|-
| 10<sup>19</sup>
| 234,057,667,276,344,607
| 5,481,624,169,369,961
| 99,877,775
| 42.725
| +2.34%
|-
| 10<sup>20</sup>
| 2,220,819,602,560,918,840
| 49,347,193,044,659,702
| 222,744,644
| 45.028
| +2.22%
|-
| 10<sup>21</sup>
| 21,127,269,486,018,731,928
| 446,579,871,578,168,707
| 597,394,254
| 47.332
| +2.11%
|-
| 10<sup>22</sup>
| 201,467,286,689,315,906,290
| 4,060,704,006,019,620,994
| 1,932,355,208
| 49.636
| +2.02%
|-
| 10<sup>23</sup>
| 1,925,320,391,606,803,968,923
| 37,083,513,766,578,631,309
| 7,250,186,216
| 51.939
| +1.93%
|-
| 10<sup>24</sup>
| 18,435,599,767,349,200,867,866
| 339,996,354,713,708,049,069
| 17,146,907,278
| 54.243
| +1.84%
|-
| 10<sup>25</sup>
| 176,846,309,399,143,769,411,680
| 3,128,516,637,843,038,351,228
| 55,160,980,939
| 56.546
| +1.77%
|-
| 10<sup>26</sup>
| 1,699,246,750,872,437,141,327,603
| 28,883,358,936,853,188,823,261
| 155,891,678,121
| 58.850
| +1.70%
|-
| 10<sup>27</sup>
| 16,352,460,426,841,680,446,427,399
| 267,479,615,610,131,274,163,365
| 508,666,658,006
| 61.153
| +1.64%
|-
| 10<sup>28</sup>
| 157,589,269,275,973,410,412,739,598
| 2,484,097,167,669,186,251,622,127
| 1,427,745,660,374
| 63.456
| +1.58%
|-
| 10<sup>29</sup>
| 1,520,698,109,714,272,166,094,258,063
| 23,130,930,737,541,725,917,951,446
| 4,551,193,622,464
| 65.759
| +1.52%
|}
[[File:Prime number theorem ratio convergence.svg|thumb|300px|Graph showing ratio of the prime-counting function {{math|''π''(''x'')}} to two of its approximations, {{math|{{sfrac|''x''|log ''x''}}}} and {{math|Li(''x'')}}. As {{mvar|x}} increases (note {{mvar|x}}-axis is logarithmic), both ratios tend towards 1. The ratio for {{math|{{sfrac|''x''|log ''x''}}}} converges from above very slowly, while the ratio for {{math|Li(''x'')}} converges more quickly from below.]]
In the [[On-Line Encyclopedia of Integer Sequences]], the {{math|''π''(''x'')}} column is sequence {{OEIS2C|id=A006880}}, {{math| ''π''(''x'') − {{sfrac|''x''|log ''x''}}}} is sequence {{OEIS2C|id=A057835}}, and {{math|li(''x'') − ''π''(''x'')}} is sequence {{OEIS2C|id=A057752}}.

The value for {{math|''π''(10<sup>24</sup>)}} was originally computed by J. Buethe, [[Jens Franke|J. Franke]], A. Jost, and T. Kleinjung assuming the [[Riemann hypothesis]].<ref name="Franke">{{cite web |title=Conditional Calculation of π(10<sup>24</sup>) |first=Jens |last=Franke |author-link=Jens Franke |date=2010-07-29 |url=https://t5k.org/notes/pi(10to24).html |publisher=Chris K. Caldwell |access-date=2024-03-30}}</ref>
It was later verified unconditionally in a computation by D. J. Platt.<ref name="Platt2012">{{cite journal |title=Computing {{math|''π''(''x'')}} Analytically |arxiv=1203.5712 |last1=Platt |first1=David J. |journal=Mathematics of Computation |volume=84 |issue=293 |date=May 2015 |orig-date=March 2012 |pages=1521–1535 |doi=10.1090/S0025-5718-2014-02884-6 |doi-access=free}}</ref>
The value for {{math|''π''(10<sup>25</sup>)}} is by the same four authors.<ref name="Buethe">{{cite web |title=Analytic Computation of the prime-counting Function |url=http://www.math.uni-bonn.de/people/jbuethe/topics/AnalyticPiX.html |publisher=J. Buethe |date=27 May 2014 |access-date=2015-09-01}}  Includes 600,000 value of {{math|''π''(''x'')}} for {{math|10<sup>14</sup> ≤ ''x'' ≤ 1.6×10<sup>18</sup>}}</ref> 
The value for {{math|''π''(10<sup>26</sup>)}} was computed by D. B. Staple.<ref name="Staple">{{cite thesis |title=The combinatorial algorithm for computing π(x) |date=19 August 2015 |url=http://dalspace.library.dal.ca/handle/10222/60524 |publisher=Dalhousie University |access-date=2015-09-01|type=Thesis |last1=Staple |first1=Douglas }}</ref> All other prior entries in this table were also verified as part of that work.

The values for 10<sup>27</sup>, 10<sup>28</sup>, and 10<sup>29</sup> were announced by David Baugh and Kim Walisch in 2015,<ref>{{cite web|website=Mersenne Forum|first=Kim |last=Walisch|title=New confirmed π(10<sup>27</sup>) prime counting function record |date=September 6, 2015|url=http://www.mersenneforum.org/showthread.php?t=20473}}</ref> 2020,<ref>{{cite web |last=Baugh |first=David |date=August 30, 2020 |title=New prime counting function record, pi(10^28) |url=https://www.mersenneforum.org/showpost.php?p=555434&postcount=28 |website=Mersenne Forum}}</ref> and 2022,<ref>{{cite web |first=Kim |last=Walisch |date=March 4, 2022 |title=New prime counting function record: PrimePi(10^29) |url=https://www.mersenneforum.org/showpost.php?p=601061&postcount=38 |website=Mersenne Forum}}</ref> respectively.