Editing Legendre's constant (section)

==Numerical values==
Using [[Prime-counting function#Table of π(x), x/log x , and li(x)|known values for <math>\pi(x)</math>]], we can compute <math>B(x) = \log x - \frac{x}{\pi(x)}</math> for values of <math>x</math> far beyond what was available to Legendre:

{|class=wikitable
|+ Legendre's constant asymptotically approaching 1 for large values of <math>x</math>
! ''x'' || ''B''(''x'')
|rowspan=15|
! ''x'' || ''B''(''x'')
|rowspan=15|
! ''x'' || ''B''(''x'')
|rowspan=15|
! ''x'' || ''B''(''x'')
|-
| {{val|e=2}}  || {{round|{{#expr:ln(1e2)  - 1e2 /25}}|6}}
| {{val|e=16}} || {{round|{{#expr:ln(1e16) - 1e16/279238341033925}}|6}}
| {{val|e=30}} || {{round|{{#expr:ln(1e30) - 1e30/1.4692398897720432717e28}}|6}}
| {{val|e=44}} || {{round|{{#expr:ln(1e44) - 1e44/9.9697350476876981763e41}}|6}}
|-
| {{val|e=3}}  || {{round|{{#expr:ln(1e3)  - 1e3 /168}}|6}}
| {{val|e=17}} || {{round|{{#expr:ln(1e17) - 1e17/2623557157654233}}|6}}
| {{val|e=31}} || {{round|{{#expr:ln(1e31) - 1e31/1.4211509734808088639e29}}|6}}
| {{val|e=45}} || {{round|{{#expr:ln(1e45) - 1e45/9.7459820466492860355e42}}|6}}
|-
| {{val|e=4}}  || {{round|{{#expr:ln(1e4)  - 1e4 /1229}}|6}}
| {{val|e=18}} || {{round|{{#expr:ln(1e18) - 1e18/24739954287740860}}|6}}
| {{val|e=32}} || {{round|{{#expr:ln(1e32) - 1e32/1.3761108669937658653e30}}|6}}
| {{val|e=46}} || {{round|{{#expr:ln(1e46) - 1e46/9.5320530117476458339e43}}|6}}
|-
| {{val|e=5}}  || {{round|{{#expr:ln(1e5)  - 1e5 /9592}}|6}}
| {{val|e=19}} || {{round|{{#expr:ln(1e19) - 1e19/234057667276344607}}|6}}
| {{val|e=33}} || {{round|{{#expr:ln(1e33) - 1e33/1.3338384833104449549e31}}|6}}
| {{val|e=47}} || {{round|{{#expr:ln(1e47) - 1e47/9.3273147934738153021e44}}|6}}
|-
| {{val|e=6}}  || {{round|{{#expr:ln(1e6)  - 1e6 /78498}}|6}}
| {{val|e=20}} || {{round|{{#expr:ln(1e20) - 1e20/2220819602560918840}}|6}}
| {{val|e=34}} || {{round|{{#expr:ln(1e34) - 1e34/1.2940862650515894138e32}}|6}}
| {{val|e=48}} || {{round|{{#expr:ln(1e48) - 1e48/9.1311875111614162331e45}}|6}}
|-
| {{val|e=7}}  || {{round|{{#expr:ln(1e7)  - 1e7 /664579}}|6}}
| {{val|e=21}} || {{round|{{#expr:ln(1e21) - 1e21/21127269486018731928}}|6}}
| {{val|e=35}} || {{round|{{#expr:ln(1e35) - 1e35/1.2566353288183164740e33}}|6}}
| {{val|e=49}} || {{round|{{#expr:ln(1e49) - 1e49/8.9431390658025913832e46}}|6}}
|-
| {{val|e=8}}  || {{round|{{#expr:ln(1e8)  - 1e8 /5761455}}|6}}
| {{val|e=22}} || {{round|{{#expr:ln(1e22) - 1e22/201467286689315906290}}|6}}
| {{val|e=36}} || {{round|{{#expr:ln(1e36) - 1e36/1.2212914297619365443e34}}|6}}
| {{val|e=50}} || {{round|{{#expr:ln(1e50) - 1e50/8.7626803175078416888e47}}|6}}
|-
| {{val|e=9}}  || {{round|{{#expr:ln(1e9)  - 1e9 /50847534}}|6}}
| {{val|e=23}} || {{round|{{#expr:ln(1e23) - 1e23/1925320391606803968923}}|6}}
| {{val|e=37}} || {{round|{{#expr:ln(1e37) - 1e37/1.1878815891216825171e35}}|6}}
| {{val|e=51}} || {{round|{{#expr:ln(1e51) - 1e51/8.5893608355366697274e48}}|6}}
|-
| {{val|e=10}} || {{round|{{#expr:ln(1e10) - 1e10/455052511}}|6}}
| {{val|e=24}} || {{round|{{#expr:ln(1e24) - 1e24/18435599767349200867866}}|6}}
| {{val|e=38}} || {{round|{{#expr:ln(1e38) - 1e38/1.1562512610265168980e36}}|6}}
| {{val|e=52}} || {{round|{{#expr:ln(1e52) - 1e52/8.4227651431212736771e49}}|6}}
|-
| {{val|e=11}} || {{round|{{#expr:ln(1e11) - 1e11/4118054813}}|6}}
| {{val|e=25}} || {{round|{{#expr:ln(1e25) - 1e25/176846309399143769411680}}|6}}
| {{val|e=39}} || {{round|{{#expr:ln(1e39) - 1e39/1.1262619405559203146e37}}|6}}
| {{val|e=53}} || {{round|{{#expr:ln(1e53) - 1e53/8.2625093911512648214e50}}|6}}
|-
| {{val|e=12}} || {{round|{{#expr:ln(1e12) - 1e12/37607912018}}|6}}
| {{val|e=26}} || {{round|{{#expr:ln(1e26) - 1e26/1699246750872437141327603}}|6}}
| {{val|e=40}} || {{round|{{#expr:ln(1e40) - 1e40/1.0977891348982830283e38}}|6}}
| {{val|e=54}} || {{round|{{#expr:ln(1e54) - 1e54/8.1082384046412221424e51}}|6}}
|-
| {{val|e=13}} || {{round|{{#expr:ln(1e13) - 1e13/346065536839}}|6}}
| {{val|e=27}} || {{round|{{#expr:ln(1e27) - 1e27/16352460426841680446427399}}|6}}
| {{val|e=41}} || {{round|{{#expr:ln(1e41) - 1e41/1.0707206348800354655e39}}|6}}
| {{val|e=55}} || {{round|{{#expr:ln(1e55) - 1e55/7.9596230541302091359e52}}|6}}
|-
| {{val|e=14}} || {{round|{{#expr:ln(1e14) - 1e14/3204941750802}}|6}}
| {{val|e=28}} || {{round|{{#expr:ln(1e28) - 1e28/157589269275973410412739598}}|6}}
| {{val|e=42}} || {{round|{{#expr:ln(1e42) - 1e42/1.0449550362264587535e40}}|6}}
| {{val|e=56}} || {{round|{{#expr:ln(1e56) - 1e56/7.8163579110561252124e53}}|6}}
|-
| {{val|e=15}} || {{round|{{#expr:ln(1e15) - 1e15/29844570422669}}|6}}
| {{val|e=29}} || {{round|{{#expr:ln(1e29) - 1e29/1520698109714272166094258063}}|6}}
| {{val|e=43}} || {{round|{{#expr:ln(1e43) - 1e43/1.0204004694436591088e41}}|6}}
| {{val|e=57}} || {{round|{{#expr:ln(1e57) - 1e57/7.6781591519438897256e54}}|6}}
|}
Values up to <math>\pi(10^{29})</math> (the first two columns) are known exactly; the values in the third and fourth columns are estimated using the [[Prime-counting function#Exact form|Riemann R function]].<!--Mathematica: ScientificForm[RiemannR[10^Range[29,57]],20].  Can be evaluated by Wolfram Alpha.  The subtraction causes some cancellation, but ln(1e57) = 131.24735..., so we only lose a factor of 131 (2.1 digits) of precision, and 9 digit estimates of π(x) are sufficient for 6-digit estimates of B.  Sanity checking against a known value
π(10^29) = 1520698109714272166094258063
R(10^29) = 1520698109714271830281953370.16
Difference =               335812304692.84
shows about 15.66 digits of accuracy, and the relative accuracy improves for larger arguments.-->