# Question

What is the best known *effective* upper bound on the prime gap following x?

# Motivation

Suppose you needed to show a good bound for the gap between a fixed large constant, say $G=10^{10^{100}}$, and the following prime. Bertrand's postulate gives $G$, but we know unconditionally that the prime gap is $O(G^\theta)$ for $\theta$ near 1/2, so $G^\theta$ seems more reasonable. But it seems that the best that can be proved at present is much larger: $G/k\log^2G$.

# Background

Bounds on large prime gaps seem to fall into three categories. We expect that the maximal prime gap below x is polylogarithmic: Cramér conjectured that it is $O(\log^2 x)$. (Maier suggests that the constant should be about $2e^{-\gamma}$.)

With the Riemann hypothesis, the gap falls to $O(\sqrt x\log x)$ thanks to Cramér. The unconditional result, due to Baker, Harman, & Pintz (extending Ingham's method), is nearly as good: $O(n^{21/40})$.

Schoenfeld's result allows an effective version of Cramér's conditional result with a constant near $1/4\pi$. But for an unconditional effective result, I know of nothing better than Dusart's $x/25\log^2x$.

That is, the results fall into three categories: those with exponent near 0 (Cramér's conjecture, Maier's theorem, etc.); those with exponent near 1/2 (Baker-Harman-Pintz, Cramér); and those with exponent near 1 (Rosser & Schoenfeld, Dusart, Chebyshev).

# Further

If, as I suppose, there are no further results known, I raise this "soft" question:

Why is "the best we can (effectively) prove" in the same neighborhood as Bertrand's postulate, even though we can show much more (and expect quite a bit more)? It might be too much to expect an effective version for $\theta=0.525$, but we lack such a result for Chudakov's $\theta=3/4+\varepsilon$ and even Hoheisel's $\theta=32999/33000$.

# References

On request. Most of the papers I gave are well-known: Maier 1985, Baker-Harman-Pintz 2001, Schoenfeld 1976, etc. The Dusart preprint is at http://arxiv.org/abs/1002.0442 .