On the average value of $\pi (t)-\text{li}(t)$

Daniel R. Johnston

We prove that the Riemann hypothesis is equivalent to the condition ${\int }_2^x\left(\pi (t)-\text{li}(t)\right)\mathrm{d}t<0$ for all $x>2$. Here, $\pi (t)$ is the prime-counting functi…