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.