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Date: 27-2-2020
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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.
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