Estimates for $\pi(x)$ for large values of $x$ and Ramanujan's prime counting inequality

Christian Axler

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arXiv · 1703.02407

Estimates for $\pi(x)$ for large values of $x$ and Ramanujan's prime counting inequality

Christian Axler

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Abstract

In this paper we use refined approximations for Chebyshev's $\vartheta$-function to establish new explicit estimates for the prime counting function $\pi(x)$, which improve the current best estimates for large values of $x$. As an application we find an upper bound for the number $H_0$ which is defined to be the smallest positive integer so that Ramanujan's prime counting inequality holds for every $x \geq H_0$.

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Christian Axler. 2017-03-07. Estimates for $\pi(x)$ for large values of $x$ and Ramanujan's prime counting inequality. https://arxiv.org/abs/1703.02407

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