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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 logarithmicintegral. This makes explicit a claim of Pintz (1991). Moreover, we prove ananalogous result for the Chebyshev function $\theta(t)$ and discuss the extentto which one can make related claims unconditionally.

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