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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.

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I think a big reason for the interest in gaps is that there are many interesting questions on the local distribution of prime numbers to which gaps are the first step. Most notably there's the Hardy-Littlewood tuples conjecture, but there are probably tuples versions of the other problems. And these tuples statements are the most closely connected to other fields like zeroes of L-functions. But we have to handle prime gaps first... – Will Sawin Mar 20 at 19:13
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Anyone who does not believe in the sacred meaning of reducing 70 million to 246 is a heretic and should be [censored]. Such is my belief. – Nikita Sidorov Mar 20 at 19:59
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@NikitaSidorov I could believe in its deep interest, surely not in its sacred meaning ! There is no place for authority argument or universally admitted research program as justification; but if it's the truth you will probably be able to convince me quite easily ;) – Desiderius Severus Mar 20 at 21:59
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I think there may be a selection bias in effect here, in that results whose statement is appealing and elementary are more likely to be publicised outside of the specialist field the statement arises from, whereas highly active, but more technical, work may not reach the awareness of the lay person or even the general mathematician. For instance, I would venture that there is significantly more work on the gaps between zeroes of zeta than gaps between primes, but these don't tend to get nearly as much exposure in the popular science press. – Terry Tao Mar 21 at 15:42
    
It is tempting to respond to the title question with motivations and applications using coprime gaps, of which large prime gaps are just one area of application. However, this question seems to focus on wanting applications specific to the realm of analytic number theory. Is that still the main intent, or are broader responses also welcome? Gerhard "Getting Into Practice For Writing" Paseman, 2016.03.22. – Gerhard Paseman Mar 22 at 23:08
up vote 25 down vote accepted

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.

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Is the claim "Large gaps may break some implementations." a hypothetical in the sense "One could imagine somebody wrote some code that relies on an unproved bound for the size of primes." or do you know of an actual example of an implementation (in relevant use) that would break? If so, in which way would it break? – user9072 Mar 20 at 17:37
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"Larger gap will give you nontrivial zero off the critical line, disproving RH." While true that gap is so huge relative to what is expected that I'd be quite surprised such large gaps were to be found even if RH were wrong. – user9072 Mar 20 at 17:40
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@quid I meant the implementation may malfunction, in the sense that it will fail to find prime due to hypothetically large gap (very unlikely to me), no need to write any code. As for RH, it is not strong enough for Legendre conjecture. I believe Cramer's conjecture (or maybe something slightly weaker, but still polylog), so the answer is purely hypothetical as the question is. Since current implementations work on large scale, large gaps don't seem to exist for current key size ;) – joro Mar 20 at 18:44
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@DesideriusSeverus "Even if larger gaps would disprove RH, it would be the case only for asymptotically larger gaps." That's false. There are completely explicit bounds on prime gaps under RH, and so one example could disprove it. (It's unrealistic as I commented, but it is possible.) Generally, you seem to severely underestimate the relevance of numerical statistics for the development of the theory. The very Prime Number Theorem was conjecture based on numerical statistics. – user9072 Mar 20 at 22:30
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I guess it is plausible somebody increments for one random n, but I am not quite sure if this were good practice. In such a scenario I would further agree that a very large gap would result it in the algo not terminatining; of course you cannot test everything over a sqaurerootish gap for 2000 bit prime generation. Although I would not describe this as "break." And I'd assume a decent implementation to have some fallback if it takes too long. (Also AFAIK in practice often not actual prime test are used and the thing is "probabilistic" in other ways.) – user9072 Mar 21 at 11:01

There is also a recent paper "Unexpected biases in the distribution of consecutive primes" by Robert J. Lemke Oliver and Kannan Soundararajan, which resulted in some sensational headlines. The relation to prime gaps is currently under discussion on math.SE.

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A new (strong) result may affect proving Legendre's conjecture

The first thing you can read there is "prime gaps".

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This conjecture, which I consider as historical artifact and not of much relevance today, basically is just a conjecture on prime gaps; to give this as motivation is almost circular. – user9072 Mar 20 at 19:19

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