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.

2more comments