TITLE: Largest prime gap under $2^{64}$
QUESTION [6 upvotes]: Thanks to Tomás Oliveira e Silva's extensive calculations, it is known that the largest prime gap less than $4\cdot10^{18}\approx2^{61.8}$ is 1476. I'd like an upper bound for the largest prime gap less than $2^{64}$. I know it will be horrifically large, but I'm hoping that more can be done than the best I know:
$$
\left\lfloor\frac{2^{64}}{25\log^2(2^{64})}\right\rfloor=374946102691094
$$
due to [1]. As DanaJ suggested in the comments, this can be improved by a result of Axler [2]. In fact Büthe [3] improves on both for this range, reducing the bound to 672606194883.
Note: When I write "the largest prime gap less than $x$" I mean "the largest $q-p$ where $p$ and $q$ are consecutive primes and $p<x$".
[1] Pierre Dusart, Estimates of Some Functions Over Primes without R.H. (2010)
[2] Christian Axler, New bounds for the prime counting function π(x) (2014)
[3] Jan Büthe, Estimating π(x) and related functions under partial RH assumptions (2014)

REPLY [4 votes]: With the new computations by Prime Gap Search project (Mersenne forum) it seems we now have the exact answer: the largest prime gap below $2^{64}$ has the size 1550. This gap follows the 20-digit prime $$p=18361375334787046697.$$ We have $\log_2 p \approx63.9933$.
See also T.R.Nicely's page First Occurrence Prime Gaps.