An asymptotic formula for the Chebyshev theta function
An asymptotic formula for the Chebyshev theta function
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 …