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arXiv · 2201.06184

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

Daniel R. Johnston
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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.

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Daniel R. Johnston. 2022-01-17. On the average value of $\pi(t)-\text{li}(t)$. https://arxiv.org/abs/2201.06184

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