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 .
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 Bordeaux8, 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.
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