An Asymptotic Formula for the Chebyshev Theta Function

Abstract

Let $\{p_n{\}}_{n\ge 1}$ be the sequence of primes and $\vartheta (x)={\sum }_{p\le x}\log p$, 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 $\vartheta (p_n)/n=\log p_{n+1}\left(1-\frac{1}{\log n}+\frac{\log \log n}{{\log }^{2}n}\left(1+o(1)\right)\right).$