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

TL;DR

This paper establishes a new equivalence for the Riemann hypothesis involving the integral of the difference between the prime-counting function and the logarithmic integral, and explores similar results for related functions.

Contribution

It proves that the Riemann hypothesis is equivalent to the negativity of an integral involving $\pi(t)- ext{li}(t)$ for all $x>2$, extending previous claims.

Findings

Riemann hypothesis is equivalent to $\int_{2}^x (\pi(t)- ext{li}(t) ext{)} dt<0$ for all $x>2$

An analogous result is proved for the Chebyshev function $ heta(t)$

Discussion on making related claims unconditional

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.