Geodesic
On the average value of $\pi (t)-\operatorname {\textrm {li}}(t)$
Canadian mathematical bulletin, Tome 66 (2023) no. 1, pp. 185-195

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DOI

We prove that the Riemann hypothesis is equivalent to the condition $\int _{2}^x\left (\pi (t)-\operatorname {\textrm {li}}(t)\right )\textrm {d}t<0$ for all $x>2$. Here, $\pi (t)$ is the prime-counting function and $\operatorname {\textrm {li}}(t)$ is the logarithmic integral. This makes explicit a claim of Pintz. Moreover, we prove an analogous result for the Chebyshev function $\theta (t)$ and discuss the extent to which one can make related claims unconditionally.
DOI : 10.4153/S0008439522000212
Mots-clés : Distribution of primes, prime-counting function, logarithmic integral, Riemann hypothesis, Riemann zeta function 
Johnston, Daniel R. On the average value of $\pi (t)-\operatorname {\textrm {li}}(t)$. Canadian mathematical bulletin, Tome 66 (2023) no. 1, pp. 185-195. doi: 10.4153/S0008439522000212
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