The hunt for Skewes' number

We study the regions where the function $\pi(x)-\li(x)$ is positive, the first such point being known as Skewes' number. We prove a new theorem which, after extensive numerical calculations, allows us to obtain a new lowest value where $\pi(x)-\li(x)$ is positive, under the assumption of the Riemann Hypothesis. This new lowest value is $1.397166161527 \times 10^{316}$. Our new theorem builds on previous work, but is different in that it does not estimate a particular constant, instead keeping it exact. This simplifies some of the calculations, permitting the error terms to be analysed more easily.