Rosser's Theorem
The prime number theorem shows that the 
th 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 
(Rosser 1938). 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 
.
REFERENCES:
Dusart, P. "The 
Prime is Greater than 
for 
." Math. Comput. 68, 411-415, 1999.
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 
-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.
