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arXiv · 1902.09231

An Asymptotic Formula for the Chebyshev Theta Function

Aditya Ghosh
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Abstract

Let $\{p_n\}_{n\ge 1}$ be the sequence of primes and $\vartheta(x) = \sum_{p \leq x} \log p$, where $p$ runs over the primes not exceeding $x$, be the Chebyshev $\vartheta$-function. In this note we derive lower and upper bounds for $\vartheta(p_n)/n$ by comparing it with $\log p_{n+1}$ and deduce that $\vartheta(p_n)/n=\log p_{n+1}\left(1-\frac{1}{\log n}+\frac{\log\log n}{\log^2 n}\left(1+o(1)\right)\right).$

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Aditya Ghosh. 2019-02-25. An Asymptotic Formula for the Chebyshev Theta Function. https://doi.org/10.7546/nntdm.2019.25.4.1-7

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