Prime gaps studies seems to be one of the most fertile topics in analytic number theory, for long and in lots of directions :
- lower bounds (recent works by Maynard, Tao et al. [1])
- upper bounds (recent works by Zhang and the whole Polymath 8 project [2])
- statistics on most frequent gaps ("jumping champions" [3])
- mean gaps (prime number theorem)
- median gaps (Erdös-Kac and related conjectures)
I keep wondering about why so many efforts ? indeed it can be for pure knowledge of prime number distributions and properties for themselves, and that would already be a sufficient motivation, but is there any hope for further applications and consequences ?
What I am thinking about is the following. Since Weil's works on explicit formulae, prime distribution knowledge is useful to deduce properties on operator's spectrum or zeroes of L-functions. For instance, all the works since Montgomery around pair correlation of zeroes and $n$-densities estimates, and their relations with primes properties (underlined by Montgomery and Goldston [4]).
So my question is, mainly related to the jumping champions problem [3] : could we, by the mean of explicit formulae or whatever else, deduce from prime gaps properties some properties out of this apparently very specific field (zeroes of zeta functions, spectral informations, families statistics, etc?) ?
Hoping the question will not be an affront to those for who the answers will be obvious and trivial, I keep impatiently waiting for possible motivations and external relations for this prime gap world 😉
Best regards
== References ==
[1] James Maynard, Small gaps between primes, Ann. of Math. (2) 181 (2015), no. 1, 383–413.
[2] Yitang Zhang, Bounded gaps between primes, Ann. of Math. (2) 179 (2014), no. 3, 1121–1174.
[3] Andrew Odlyzko, Michael Rubinstein, and Marek Wolf, Jumping champions, Experiment. Math. 8 (1999), no. 2, 107–118.
[4] D. A. Goldston, S. M. Gonek, A. E. Özlük, and C. Snyder, On the pair correlation of zeros of the Riemann zeta-function, Proc. London Math. Soc. (3) 80 (2000), no. 1, 31–49.
Best Answer
Since you ask about zeta zeros, Riemann hypothesis implies the gap is $O(\sqrt{p_n} \log p_n)$.
Larger gap will give you nontrivial zero off the critical line, disproving RH.
On the other hand, bounding the gap by $O(polylog(p_n))$ will solve the open problem for deterministically finding primes in polynomial time.
For practical purposes, some cryptographic algorithms need to find prime efficiently. Large gaps may break some implementations.