New estimates for the th prime number
C Axler - arXiv preprint arXiv:1706.03651, 2017 - arxiv.org
C Axler
arXiv preprint arXiv:1706.03651, 2017•arxiv.orgIn this paper we establish a new explicit upper and lower bound for the $ n $-th prime
number, which improve the currently best estimates given by Dusart in 2010. As the main
tool we use some recently obtained explicit estimates for the prime counting function. A
further main tool is the usage of estimates concerning the reciprocal of $\log p_n $. As an
application we derive refined estimates for $\vartheta (p_n) $ in terms of $ n $, where
$\vartheta (x) $ is Chebyshev's $\vartheta $-function.
number, which improve the currently best estimates given by Dusart in 2010. As the main
tool we use some recently obtained explicit estimates for the prime counting function. A
further main tool is the usage of estimates concerning the reciprocal of $\log p_n $. As an
application we derive refined estimates for $\vartheta (p_n) $ in terms of $ n $, where
$\vartheta (x) $ is Chebyshev's $\vartheta $-function.
In this paper we establish a new explicit upper and lower bound for the -th prime number, which improve the currently best estimates given by Dusart in 2010. As the main tool we use some recently obtained explicit estimates for the prime counting function. A further main tool is the usage of estimates concerning the reciprocal of . As an application we derive refined estimates for in terms of , where is Chebyshev's -function.
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