Journal of Integer Sequences, Vol. 22 (2019), Article 19.4.2

New Estimates for the nth Prime Number

Christian Axler
Department of Mathematics
40225 Düsseldorf


In this paper we establish new upper and lower bounds for the nth prime number pn, which improve several existing bounds of similar shape. As the main tool, we use some explicit estimates recently obtained for the prime counting function. A further main tool is the use of estimates concerning the reciprocal of log pn. As an application, we derive new estimates for ϑ(pn), where ϑ(x) is Chebyshev's ϑ-function.

Full version:  pdf,    dvi,    ps,    latex    

(Concerned with sequences A000040 A006988.)

Received December 12 2017; revised versions received March 15 2018; February 18 2019; February 25 2019; May 21 2019. Published in Journal of Integer Sequences, May 23 2019.

Return to Journal of Integer Sequences home page