# An Asymptotic Formula for the Chebyshev Theta Function

## TL;DR

This paper derives an asymptotic formula for the Chebyshev theta function evaluated at prime indices, providing bounds and an approximation involving prime logarithms, advancing understanding of prime distribution asymptotics.

## Contribution

It introduces a new asymptotic formula for the Chebyshev theta function at prime indices, refining previous bounds and approximations.

## Key findings

- Derived bounds for $ heta(p_n)/n$ using $ ext{log } p_{n+1}$
- Established an asymptotic expression for $ heta(p_n)/n$ involving $ ext{log } p_{n+1}$
- Provided insights into the distribution of primes through asymptotic analysis.

## 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 $\vartheta(p_n)/n=\log p_{n+1}\left(1-\frac{1}{\log n}+\frac{\log\log n}{\log^2 n}\left(1+o(1)\right)\right).$