Updating the error term in the prime number theorem

Abstract

An improved estimate is given for $|θ(x) -x|$, where $θ(x) = \sum_{p\leq x} \log p$. 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.

I wish to thank MG Johnson, RJ Harris, PM Siddle, and NM Lyon, all of whom enabled me to work two extra days on this article; 13 pages; to appear in Ramanujan J