Rosser's theorem states that the 
th prime number 
satisfies
 |
(1)
|
for 
(Rosser 1938). The prime number theorem gives
the asymptotic value
 |
(2)
|
as 
(Havil 2003, p. 182), and Rosser's theorem makes this a rigorous lower bound.
This result was subsequently improved to
 |
(3)
|
where 
(Rosser and Schoenfeld 1975). The constant 
was subsequently reduced to 
(Robin 1983). Massias and Robin (1996) then showed
that 
was admissible for 
and 
.
Finally, Dusart (1999) showed that 
holds for all 
(Havil 2003, p. 183). The plots above show 
(black), 
(blue), and 
(red).
The difference between 
and 
is plotted above. The slope of the difference taken out to 
is approximately 
.
Prime Is Greater than 
for 
." Math. Comput. 68, 411-415, 1999.Havil,
J. 
sur le 
-iéme nombre premier et grandes valeurs de la fonction
, nombres de diviseurs premiers
de 
."
Acta Arith. 42, 367-389, 1983.Robin, G. "Permanence
de relations de récurrence dans certains développements asymptotiques."
Publ. Inst. Math., Nouv. Sér. 43, 17-25, 1988.Rosser,
J. B. "The 
th
Prime Is Greater than 
."
Proc. London Math. Soc. 45, 21-44, 1938.Rosser, J. B.
and Schoenfeld, L. "Sharper Bounds for Chebyshev Functions 
and 
." Math. Comput. 29, 243-269, 1975.Salvy,
B. "Fast Computation of Some Asymptotic Functional Inverses." J. Symb.
Comput. 17, 227-236, 1994.