Updating the error term in the prime number theorem

An improved estimate is given for $\mathrm{\mid }\theta (x)-x\mathrm{\mid }$, where $\theta(x) = \sum_{p\leq x} \log p$. Three applications are given: the first to arithmetic progressions that have points in common, the second to primes in short intervals, and the third to a conjecture by C. Pomerance.