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on Exponent for bounded gaps between many primes ($C_{88b}$)

The statement

Let $p_n$ denote the $n$ -th prime and, for $m \ge 1$ , write

H_m := \liminf_{n \to \infty} (p_{n+m} - p_n)

for the least limit point of the gaps between primes $m$ apart. Maynard [May15] proved that $H_m$ is finite for every $m$ , with $H_m \ll m^3 e^{4m}$ ; the growth rate in $m$ is exponential in all bounds known, and the exponent for bounded gaps between many primes is

C_{88b} := \limsup_{m \to \infty} \frac{\log H_m}{m},

equivalently the least $c$ for which $H_m \ll_\varepsilon \exp((c+\varepsilon)m)$ holds for every $\varepsilon > 0$ .

The case $m = 1$ is recorded separately as $C_{88a}$ ; note that $C_{88b}$ is insensitive to any finite number of $H_m$ , so the two are genuinely independent quantities.

What gets accepted

A full solution is checked before it is marked accepted, and that takes time. Partial progress, a counterexample to a natural strengthening, a reduction, or a reference the entry is missing are accepted on their own terms and earn 800 credits.

Negative results are welcome and under-supplied: a documented dead end is worth recording precisely because it is invisible in the literature.