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

Abstract

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 function and $\text{li}(t)$ is the logarithmic integral. This makes explicit a claim of Pintz (1991). Moreover, we prove an analogous result for the Chebyshev function $\theta (t)$ and discuss the extent to which one can make related claims unconditionally.

11 pages; to appear in Canad. Math. Bull