The prime number theorem shows that the nth prime number 
has the asymptotic value
 |
(1) |
as 
(Havil 2003, p. 182). Rosser's theorem makes this a rigorous lower bound by stating that
 |
(2) |
for n > 1 (Rosser 1938). This result was subsequently improved to
 |
(3) |
where 
(Rosser and Schoenfeld 1975). The constant c was subsequently reduced to 
(Robin 1983). Robin and Massias (1996) then showed that c = 1 was admissible for 
and 
. (black), 
(blue), and 
(red).
The difference between 
and is plotted above. The slope of the difference taken out to 
is approximately 
.
Prime Formulas, Prime Number, Prime Number Theorem
Dusart, P. "The 
Prime is Greater than 
for 
.
Havil, J. Gamma: Exploring Euler's Constant. Princeton, NJ: Princeton University Press, 2003.
Massias, J.-P. and Robin, G. "Bornes effectives pour certaines fonctions concernant les nombres premiers." J. Théor. Nombres Bordeaux 8, 215-242, 1996.
Riesel, H. Prime Numbers and Computer Methods for Factorization, 2nd ed. Boston, MA: Birkhäuser, pp. 56-57, 1994.
Robin, G. "Estimation de la fonction de Tschebychef 
sur le k-iéme nombre premier et grandes valeurs de la fonction 
,
Robin, G. "Permanence de relations de récurrence dans certains développements asymptotiques." Publ. Inst. Math., Nouv. Sér. 43, 17-25, 1988.
Rosser, J. B. "The nth Prime is Greater than 
.
Rosser, J. B. and Schoenfeld, L. "Sharper Bounds for Chebyshev Functions 
and 
.
Salvy, B. "Fast Computation of Some Asymptotic Functional Inverses." J. Symb. Comput. 17, 227-236, 1994.