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New estimates for the $n$th prime number

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arxiv 1706.03651 v2 pith:M5FOLC2L submitted 2017-06-12 math.NT

Christian Axler This is my paper · ORCID
classification math.NT
keywords estimatesprimevarthetaexplicitfunctionmainnumbertool
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abstract

In 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.

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  1. Behaviour of the sequence $\vartheta_n = \vartheta(p_n)$

    math.NT 2025-07 reject novelty 5.0 of 10

    Replacing the n-th prime by the sum of logarithms of the first n primes makes analogues of Cramer, Andrica, Legendre, Oppermann, Brocard, Firoozbakht, Fourges, Nicholson, and Farhadian conjectures provable theorems.

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