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

Abstract

We prove that the Riemann hypothesis is equivalent to the condition x 2 π(t) − li(t) dt < 0 for all x > 2. Here, π(t) is the prime-counting function and li(t) is the logarithmic integral. This makes explicit a claim of Pintz (1991). Moreover, we prove an analogous result for the Chebyshev function θ(t) and discuss the extent to which one can make related claims unconditionally.

Large gaps between primes in arithmetic progressions -- an empirical approach

“…Yet, using the upper bounding prime, ρ * (p, q) is similarly inconclusive-although more appropriate-for very small p + , as the respective gap is linked to the asymptotic density of primes only at the point where the upper bounding prime p + is located. Moreover, any such conventional measures transfer the "flaw" of the logarithmic integral li(x) overestimating π(x) for most x (see [11] for a very recent paper on this issue). To adequately process the results of the exhaustive computation, we demand a measure that captures the purpose of the conventional measure, the Cramér-Shanks-Granville (CSG) ratio g/ log 2 p, while taking into account the exact average local distribution of primes in the ranges of concern.…”