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




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== References ==


[1] *James Maynard*, [**Small gaps between primes**](http://dx.doi.org/10.4007/annals.2015.181.1.7), *Ann. of Math. (2)* **181** (2015), no. 1, 383--413.


[2] *Yitang Zhang*, [**Bounded gaps between primes**](http://dx.doi.org/10.4007/annals.2014.179.3.7), *Ann. of Math. (2)* **179** (2014), no. 3, 1121--1174.


[3] *Andrew Odlyzko, Michael Rubinstein, and Marek Wolf*, [**Jumping champions**](http://projecteuclid.org/euclid.em/1047477055), *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**](http://dx.doi.org/10.1112/S0024611500012211), *Proc. London Math. Soc. (3)* **80** (2000), no. 1, 31--49.



Why such an interest in studying prime gaps?
Since you ask about zeta zeros, [Riemann hypothesis implies](https://en.wikipedia.org/wiki/Prime_gap) 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](http://michaelnielsen.org/polymath1/index.php?title=Finding_primes).


For practical purposes, some cryptographic algorithms need to find prime efficiently. Large gaps may break some implementations.


