Average value of $\pi(t) - li(t)$

Central to comparative number theory is the study of the difference $\Delta(t) = \pi(t) − li(t)$, where $\pi(t)$ is the prime counting function and $li(t)$ is the logarithmic integral. Prior to a celebrated 1914 paper of Littlewood, it was believed that $\Delta < 0$ for all $t > 2$. We now know however that $\Delta(t)$ changes sign infinitely often, with the first sign change occuring before 10320. Despite this, it still appears that $\Delta(t)$ is negative “on average”, in that integrating $\Delta (t)$ from $t = 2$ onwards yields a negative value. In this talk, we will explore this idea in detail, discussing links with the Riemann hypothesis and also extending such ideas to other differences involving arithmetic functions.