William Banks, Kevin Ford, and I have just uploaded to the arXiv our paper “Large prime gaps and probabilistic models“, submitted to Inventiones. In this paper we introduce a random model to help understand the connection between two well known conjectures regarding the primes , the Cramér conjecture and the Hardy-Littlewood conjecture:

Conjecture 1 (Cramér conjecture)If is a large number, then the largest prime gap in is of size . (Granville refines this conjecture to , where . Here we use the asymptotic notation for , for , for , and for .)

Conjecture 2 (Hardy-Littlewood conjecture)If are fixed distinct integers, then the number of numbers with all prime is as , where the singular series is defined by the formula

(One can view these conjectures as modern versions of two of the classical Landau problems, namely Legendre’s conjecture and the twin prime conjecture respectively.)

A well known connection between the Hardy-Littlewood conjecture and prime gaps was made by Gallagher. Among other things, Gallagher showed that if the Hardy-Littlewood conjecture was true, then the prime gaps with were asymptotically distributed according to an exponential distribution of mean , in the sense that

as for any fixed . Roughly speaking, the way this is established is by using the Hardy-Littlewood conjecture to control the mean values of for fixed , where ranges over the primes in . The relevance of these quantities arises from the Bonferroni inequalities (or “Brun pure sieve“), which can be formulated as the assertion that

when is even and

when is odd, for any natural number ; setting and taking means, one then gets upper and lower bounds for the probability that the interval is free of primes. The most difficult step is to control the mean values of the singular series as ranges over -tuples in a fixed interval such as .

Heuristically, if one extrapolates the asymptotic (1) to the regime , one is then led to Cramér’s conjecture, since the right-hand side of (1) falls below when is significantly larger than . However, this is not a rigorous derivation of Cramér’s conjecture from the Hardy-Littlewood conjecture, since Gallagher’s computations only establish (1) for *fixed* choices of , which is only enough to establish the far weaker bound , which was already known (see this previous paper for a discussion of the best known unconditional lower bounds on ). An inspection of the argument shows that if one wished to extend (1) to parameter choices that were allowed to grow with , then one would need as input a stronger version of the Hardy-Littlewood conjecture in which the length of the tuple , as well as the magnitudes of the shifts , were also allowed to grow with . Our initial objective in this project was then to quantify exactly what strengthening of the Hardy-Littlewood conjecture would be needed to rigorously imply Cramer’s conjecture. The precise results are technical, but roughly we show results of the following form:

Theorem 3 (Large gaps from Hardy-Littlewood, rough statement)

- If the Hardy-Littlewood conjecture is uniformly true for -tuples of length , and with shifts of size , with a power savings in the error term, then .
- If the Hardy-Littlewood conjecture is “true on average” for -tuples of length and shifts of size for all , with a power savings in the error term, then .

In particular, we can recover Cramer’s conjecture given a sufficiently powerful version of the Hardy-Littlewood conjecture “on the average”.

Our proof of this theorem proceeds more or less along the same lines as Gallagher’s calculation, but now with allowed to grow slowly with . Again, the main difficulty is to accurately estimate average values of the singular series . Here we found it useful to switch to a probabilistic interpretation of this series. For technical reasons it is convenient to work with a truncated, unnormalised version

of the singular series, for a suitable cutoff ; it turns out that when studying prime tuples of size , the most convenient cutoff is the “Pólya magic cutoff“, defined as the largest prime for which

(this is well defined for ); by Mertens’ theorem, we have . One can interpret probabilistically as

where is the randomly sifted set of integers formed by removing one residue class uniformly at random for each prime . The Hardy-Littlewood conjecture can be viewed as an assertion that the primes behave in some approximate statistical sense like the random sifted set , and one can prove the above theorem by using the Bonferroni inequalities both for the primes and for the random sifted set, and comparing the two (using an even for the sifted set and an odd for the primes in order to be able to combine the two together to get a useful bound).

The proof of Theorem 3 ended up not using any properties of the set of primes other than that this set obeyed some form of the Hardy-Littlewood conjectures; the theorem remains true (with suitable notational changes) if this set were replaced by any other set. In order to convince ourselves that our theorem was not vacuous due to our version of the Hardy-Littlewood conjecture being too strong to be true, we then started exploring the question of coming up with random models of which obeyed various versions of the Hardy-Littlewood and Cramér conjectures.

This line of inquiry was started by Cramér, who introduced what we now call the *Cramér random model* of the primes, in which each natural number is selected for membership in with an independent probability of . This model matches the primes well in some respects; for instance, it almost surely obeys the “Riemann hypothesis”

and Cramér also showed that the largest gap was almost surely . On the other hand, it does not obey the Hardy-Littlewood conjecture; more precisely, it obeys a simplified variant of that conjecture in which the singular series is absent.

Granville proposed a refinement to Cramér’s random model in which one first sieves out (in each dyadic interval ) all residue classes for for a certain threshold , and then places each surviving natural number in with an independent probability . One can verify that this model obeys the Hardy-Littlewood conjectures, and Granville showed that the largest gap in this model was almost surely , leading to his conjecture that this bound also was true for the primes. (Interestingly, this conjecture is not yet borne out by numerics; calculations of prime gaps up to , for instance, have shown that never exceeds in this range. This is not necessarily a conflict, however; Granville’s analysis relies on inspecting gaps in an extremely sparse region of natural numbers that are more devoid of primes than average, and this region is not well explored by existing numerics. See this previous blog post for more discussion of Granville’s argument.)

However, Granville’s model does not produce a power savings in the error term of the Hardy-Littlewood conjectures, mostly due to the need to truncate the singular series at the logarithmic cutoff . After some experimentation, we were able to produce a tractable random model for the primes which obeyed the Hardy-Littlewood conjectures with power savings, and which reproduced Granville’s gap prediction of (we also get an upper bound of for both models, though we expect the lower bound to be closer to the truth); to us, this strengthens the case for Granville’s version of Cramér’s conjecture. The model can be described as follows. We select one residue class uniformly at random for each prime , and as before we let be the sifted set of integers formed by deleting the residue classes with . We then set

with Pólya’s magic cutoff (this is the cutoff that gives a density consistent with the prime number theorem or the Riemann hypothesis). As stated above, we are able to show that almost surely one has

and that the Hardy-Littlewood conjectures hold with power savings for up to for any fixed and for shifts of size . This is unfortunately a tiny bit weaker than what Theorem 3 requires (which more or less corresponds to the endpoint ), although there is a variant of Theorem 3 that can use this input to produce a lower bound on gaps in the model (but it is weaker than the one in (3)). In fact we prove a more precise almost sure asymptotic formula for that involves the optimal bounds for the *linear sieve* (or *interval sieve*), in which one deletes one residue class modulo from an interval for all primes up to a given threshold. The lower bound in (3) relates to the case of deleting the residue classes from ; the upper bound comes from the delicate analysis of the linear sieve by Iwaniec. Improving on either of the two bounds looks to be quite a difficult problem.

The probabilistic analysis of is somewhat more complicated than of or as there is now non-trivial coupling between the events as varies, although moment methods such as the second moment method are still viable and allow one to verify the Hardy-Littlewood conjectures by a lengthy but fairly straightforward calculation. To analyse large gaps, one has to understand the statistical behaviour of a random linear sieve in which one starts with an interval and randomly deletes a residue class for each prime up to a given threshold. For very small this is handled by the deterministic theory of the linear sieve as discussed above. For medium sized , it turns out that there is good concentration of measure thanks to tools such as Bennett’s inequality or Azuma’s inequality, as one can view the sieving process as a martingale or (approximately) as a sum of independent random variables. For larger primes , in which only a small number of survivors are expected to be sieved out by each residue class, a direct combinatorial calculation of all possible outcomes (involving the random graph that connects interval elements to primes if falls in the random residue class ) turns out to give the best results.

## 30 comments

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26 August, 2019 at 11:17 am

thebirdreaderWhy do you reveal the journal you submitted it to? More generally, what’s the policy around this?

26 August, 2019 at 11:33 am

domotorpIt’s nice of William Banks and Kevin Ford that they’ve helped you in uploading your own paper ;)

[Corrected, thanks – T.]26 August, 2019 at 11:49 am

AnonymousIn the very first sentence, “my paper” should be “our paper”.

26 August, 2019 at 1:05 pm

AnonymousIs it possible to improve the upper bound in (3) by optimizing “Polya magic cutoff” (perhaps by a slight modification of (2))?

26 August, 2019 at 2:12 pm

Terence TaoVarying the cutoff significantly would vary the density of the set , which would then also vary the lower and upper bounds we obtain for the largest gap (but the ratio between the upper and lower bounds would be essentially unchanged). The more precise version of (3) given in the paper involves the function , defined as the largest for which it is possible to delete one residue class mod from for each prime and end up with at most survivors. We basically show that the largest gap is almost surely asymptotic to . On the other hand, the best known bounds on are

which is where the bounds (3) are coming from. We believe the lower bound here is closer to the truth, but improving the upper bound requires improving Iwaniec’s bound on the linear sieve, which is likely to be a very difficult problem (relating to breaking the notorious “parity barrier”). In an appendix to our paper we elaborate on the link with the parity problem by recording the folklore observation that a sufficiently strong Siegel zero can lead to a significant improvement in the lower bound for and hence for . So showing that the lower bound is sharp is at least as hard as ruling out Siegel zeroes.

28 August, 2019 at 5:27 am

AnonymousIs it necessary to use the same cutoff for both (upper and lower) bounds? Is it possible to optimize the cutoff for the lower bound and use another optimized cutoff for the upper bound?

26 August, 2019 at 3:03 pm

Two conjectures in one – The nth Root[…] William Banks, Kevin Ford, and I have just uploaded to the arXiv my paper “Large prime gaps and probabilistic models“, submitted to Inventiones. In this paper we introduce a random model to help understand the connection between two well known conjectures regarding the primes {{mathcal P} := {2,3,5,dots}}, the Cramér conjecture and the Hardy-Littlewood conjecture: … (Terence Tao) […]

26 August, 2019 at 6:23 pm

Zachary Kyle KnutsonResearch gate my sample 2 that was my idea…

27 August, 2019 at 6:19 am

Zachary Kyle Knutsonhttps://www.researchgate.net/publication/327477654_Sample_2_Update_J_Equation

27 August, 2019 at 6:28 am

Zachary Kyle Knutsonyall misunderstand that was to comment not lol to Mr.Terrance…

27 August, 2019 at 3:33 pm

Some Math News | Not Even Wrong[…] too bad Pat didn’t live to see the latest from Terry Tao, who describes recent results which are related to old work of Gallagher’s by “Our […]

28 August, 2019 at 1:34 am

VincentSorry can I ask a very basic question about the constant S(H) in the Hardy-Littlewood conjecture? I would have looked it up on Wikipedia but curiously the first HL-conjecture doesn’t have a page (while the probably-false second HL-conjecture does). As written here it seems to me that the constant equals 0 when all $h_i$ lie in the same congruence class mod $p$ for some $p$, e.g. when they are all even. That seems a bit strange in light of the twin prime conjecture. Conversely I would expect the constant to become 0 when for some $p$ we have$|H mod p| = p$, i.e. the $h_i$ together cover all congruence classes mod $p$, but in this case the contribution of the prime $p$ to the constant seems to be non-zero. Am I missing something obvious here? What is the rationale behind the product making up $S(H)$?

[Oops, there was a typo in the definition of the singular series in the blog post, which has now been fixed. -T]28 August, 2019 at 8:26 pm

年轻的数学家Dear Prof. Tao,

first of all, there seems to be a typo in the conclusion of thm. 1.5 (or do they also use the lower index for iteration?). Then, I wonder whether even stronger Hardy‒Littlewood-type estimates would imply even stronger bounds on . Also, what about upper bounds? Without having read the paper, the Bonferroni inequalities as stated above seem pretty symmetrical.

30 August, 2019 at 7:52 am

Terence TaoThe lower index is indeed used for iteration (see the top of page 2).

Unfortunately, it appears that Hardy-Littlewood type conjectures can only be used to impose lower bounds on gaps, not upper bounds. To show that there is a prime gap of size at least H, it suffices to show that the number of in a given range such that the tuple is free of primes is positive; and this can in principle be done by the lower bound Bonferroni inequality and Hardy-Littlewood. But to show that all prime gaps have size at most H, one has to show that the number of for which the tuple is

zero. One could only achieve this from an upper bound Bonferroni inequality if the upper bound was absolutely tight (no error term allowed at all), which is basically impossible from analytic number theory techniques (for instance, the Hardy-Littlewood conjectures contain an error term and are thus not usable for such a strategy).One can also see this using the “redacted primes” test. One can “plant” a large gap in the primes by deleting an interval from the set of primes. For much smaller than the range (e.g. ), such a deletion will hardly make an impact on any of the Hardy-Littlewood conjectures (it will easily get absorbed into the error term). Thus these conjectures cannot prevent the creation of an extremely large gap.

28 August, 2019 at 8:38 pm

年轻的数学家I’d also be interested in estimates for that would prove stronger bounds on .

28 August, 2019 at 8:51 pm

年轻的数学家I realise that according to the above, one would need to disprove the existence of Siegel zeroes. So I’m in fact asking for upper bounds in the event that both the Riemann hypothesis and strong Hardy‒Littlewood-type estimates are true.

30 August, 2019 at 5:54 am

MefesGood connection, but seems that landau’s problems are not provable though

30 August, 2019 at 2:03 pm

AnonymousDr.Tao, we can deduce the Hardy-Littlewood conjectures without using probabilistic models. You can see my article: https://lagrida.com/conjecture_k_uple.html

Where i prove

With : and and is the largest prime verify

30 August, 2019 at 2:06 pm

AnonymousAnd f course

With is the number of distinct residues $\pmod p$ in $\mathcal{H}_k$.

30 August, 2019 at 3:25 pm

AnonymousHmm. On the largest gap or space, G(X), between consecutive primes less than or equal to large real X, I believe G(x) ≤ (log X)^2 is true according to an analysis involving the Prime Number Theorem and the average spacing between consecutive primes less than or equal to large real X.

Relevant Reference Link:

“LARGE GAPS BETWEEN CONSECUTIVE PRIME NUMBERS”,

https://www.math10.com/forum/viewtopic.php?f=63&t=8263

David Cole.

30 August, 2019 at 3:33 pm

AnonymousOf course, the calculations are estimates/approximations within the known bounds of error associated with PNT/Riemann Prime-Counting Function for all large X… Our model/proof appears to be sound. David Cole.

30 August, 2019 at 3:38 pm

AnonymousRemark: Riemann Hypothesis is true! David Cole. :-)

31 August, 2019 at 1:36 pm

AnonymousNow I wonder why I even bother to visit this website/webpage… I always receive no positive feedback or worse. I will stop visiting this site in the future and this is my last comment and visit too. However, I do appreciate the information I gain here in the past. Thank you and goodbye. David Cole.

30 August, 2019 at 6:06 pm

anonymousA possibly naive question: do you know if the constant 3/2 is optimal your Theorem 1.4, for the random model?

Also, does this model make predictions for the number of primes in short intervals much larger than log(x), along the lines of this paper of Montgomery and Soundararajan https://arxiv.org/pdf/math/0409258.pdf (e.g. Corollary 1 there)? I guess Theorem 1.3 of your paper allows one to directly reproduce Montgomery and Soundararajan’s computation for intervals of length up to exp(log(x)^{1/2}/loglog(x)), but for this model is there a more direct route that doesn’t pass through the computation of that paper, and is it feasible to consider short intervals larger than this?

30 August, 2019 at 6:32 pm

Terence TaoWe didn’t seriously try to optimise the exponent in Theorem 1.4, which arises basically from the second moment method and the Borel-Cantelli lemma. One could perhaps do a bit better by exploiting higher moments, but they become more difficult to compute.

I was planning to ask a graduate student to look into how what this model predicts for primes in short intervals; as we remark in Section 2.5, one does have the analogue of the Maier fluctuations in this model, but we didn’t quantify the size of these fluctuations precisely.

2 September, 2019 at 7:07 am

MarkIf you assume a Heath-Brown-type “conjecture” that the Hardy-Littlewood asymptotic holds with O(1) error term, can you deduce anything more about gaps?

2 September, 2019 at 7:42 am

MarkI just noticed that the Heath-Brown’s “conjecture” already precludes the qualitative Hardy-Littlewood conjecture, as well as the Granville-Cramer gap prediction, so I’m not sure if there is much sense to be made from my question.

8 September, 2019 at 10:58 pm

BuzzmanLet , then there exists such that for one obtains , with the implied constant depending on .

11 September, 2019 at 7:34 am

humble suggestionI wonder if it might be worthwhile for Terry to hire an undergrad to periodically delete the nonsense in the comment sections. The posts about high profile results (and to a lesser extent, many other posts) tend to attract a lot of personal or crackpottish comments that are either off topic or nonsensical.

11 September, 2019 at 9:49 am

KipperockI think that this type of moderation would be welcome, both to the writer and the serious visitors.

This blog’s comment section is quite important, since, for example, in the comments about Prof. Tao’s books there are both errata and many super important explanations – I’m working through his book on measure theory rn and I just feel how invaluable a neat comment section is.

When reading other posts, expository or research – it would be very cool to see a neat comment section where serious discussions take place and are not spoiled by – sometimes dozens or more! – comments that are clearly.. not in the right neighborhood.

I guess I understand why the author refrains from flat out full time moderation – he’s got important research to do, teaching duties, two kids and a wife [this list is not written by order of importance :)))], and there probably are things I simply didn’t think of.

From my experience(see above, and take my word for it that it was me), it is also very, very tiring – since I would like to see certain comments completely gone, but others deserve at least an attempt at an answer.

I don’t even read the comments on other posts, in order to not get dragged down into the same mess of time and nerve consuming keyboard kung-fu..

Honestly I think this comment was just a very long way of saying “I agree”, since this comment section already had it’s way with me. I hope Prof. Tao would eventually accept your suggestion.