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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 ${{\mathcal P} := \{2,3,5,\dots\}}$, the Cramér conjecture and the Hardy-Littlewood conjecture:

Conjecture 1 (Cramér conjecture) If ${x}$ is a large number, then the largest prime gap ${G_{\mathcal P}(x) := \sup_{p_n, p_{n+1} \leq x} p_{n+1}-p_n}$ in ${[1,x]}$ is of size ${\asymp \log^2 x}$. (Granville refines this conjecture to ${\gtrsim \xi \log^2 x}$, where ${\xi := 2e^{-\gamma} = 1.1229\dots}$. Here we use the asymptotic notation ${X \gtrsim Y}$ for ${X \geq (1-o(1)) Y}$, ${X \sim Y}$ for ${X \gtrsim Y \gtrsim X}$, ${X \gg Y}$ for ${X \geq C^{-1} Y}$, and ${X \asymp Y}$ for ${X \gg Y \gg X}$.)

Conjecture 2 (Hardy-Littlewood conjecture) If ${\mathcal{H} := \{h_1,\dots,h_k\}}$ are fixed distinct integers, then the number of numbers ${n \in [1,x]}$ with ${n+h_1,\dots,n+h_k}$ all prime is ${({\mathfrak S}(\mathcal{H}) +o(1)) \int_2^x \frac{dt}{\log^k t}}$ as ${x \rightarrow \infty}$, where the singular series ${{\mathfrak S}(\mathcal{H})}$ is defined by the formula

$\displaystyle {\mathfrak S}(\mathcal{H}) := \prod_p \left( 1 - \frac{|{\mathcal H} \hbox{ mod } p|}{p}\right) (1-\frac{1}{p})^{-k}.$

(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 ${p_{n+1}-p_n}$ with ${n \leq x}$ were asymptotically distributed according to an exponential distribution of mean ${\log x}$, in the sense that

$\displaystyle | \{ n: p_n \leq x, p_{n+1}-p_n \geq \lambda \log x \}| = (e^{-\lambda}+o(1)) \frac{x}{\log x} \ \ \ \ \ (1)$

as ${x \rightarrow \infty}$ for any fixed ${\lambda \geq 0}$. Roughly speaking, the way this is established is by using the Hardy-Littlewood conjecture to control the mean values of ${\binom{|{\mathcal P} \cap (p_n, p_n + \lambda \log x)|}{k}}$ for fixed ${k,\lambda}$, where ${p_n}$ ranges over the primes in ${[1,x]}$. The relevance of these quantities arises from the Bonferroni inequalities (or “Brun pure sieve“), which can be formulated as the assertion that

$\displaystyle 1_{N=0} \leq \sum_{k=0}^K (-1)^k \binom{N}{k}$

when ${K}$ is even and

$\displaystyle 1_{N=0} \geq \sum_{k=0}^K (-1)^k \binom{N}{k}$

when ${K}$ is odd, for any natural number ${N}$; setting ${N := |{\mathcal P} \cap (p_n, p_n + \lambda \log x)|}$ and taking means, one then gets upper and lower bounds for the probability that the interval ${(p_n, p_n + \lambda \log x)}$ is free of primes. The most difficult step is to control the mean values of the singular series ${{\mathfrak S}(\mathcal{H})}$ as ${{\mathcal H}}$ ranges over ${k}$-tuples in a fixed interval such as ${[0, \lambda \log x]}$.

Heuristically, if one extrapolates the asymptotic (1) to the regime ${\lambda \asymp \log x}$, one is then led to Cramér’s conjecture, since the right-hand side of (1) falls below ${1}$ when ${\lambda}$ is significantly larger than ${\log x}$. 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 ${\lambda}$, which is only enough to establish the far weaker bound ${G_{\mathcal P}(x) / \log x \rightarrow \infty}$, which was already known (see this previous paper for a discussion of the best known unconditional lower bounds on ${G_{\mathcal P}(x)}$). An inspection of the argument shows that if one wished to extend (1) to parameter choices ${\lambda}$ that were allowed to grow with ${x}$, then one would need as input a stronger version of the Hardy-Littlewood conjecture in which the length ${k}$ of the tuple ${{\mathcal H} = (h_1,\dots,h_k)}$, as well as the magnitudes of the shifts ${h_1,\dots,h_k}$, were also allowed to grow with ${x}$. 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 ${k}$-tuples of length ${k \ll \frac{\log x}{\log\log x}}$, and with shifts ${h_1,\dots,h_k}$ of size ${O( \log^2 x )}$, with a power savings in the error term, then ${G_{\mathcal P}(x) \gg \frac{\log^2 x}{\log\log x}}$.
• If the Hardy-Littlewood conjecture is “true on average” for ${k}$-tuples of length ${k \ll \frac{y}{\log x}}$ and shifts ${h_1,\dots,h_k}$ of size ${y}$ for all ${\log x \leq y \leq \log^2 x \log\log x}$, with a power savings in the error term, then ${G_{\mathcal P}(x) \gg \log^2 x}$.

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 ${k}$ allowed to grow slowly with ${x}$. Again, the main difficulty is to accurately estimate average values of the singular series ${{\mathfrak S}({\mathfrak H})}$. 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

$\displaystyle V_{\mathcal H}(z) := \prod_{p \leq z} \left( 1 - \frac{|{\mathcal H} \hbox{ mod } p|}{p} \right)$

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

$\displaystyle \prod_{p \leq z(t)}(1-\frac{1}{p}) \geq \frac{1}{\log t} \ \ \ \ \ (2)$

(this is well defined for ${t \geq e^2}$); by Mertens’ theorem, we have ${z(t) \sim t^{1/e^\gamma}}$. One can interpret ${V_{\mathcal Z}(z)}$ probabilistically as

$\displaystyle V_{\mathcal Z}(z) = \mathbf{P}( {\mathcal H} \subset \mathcal{S}_z )$

where ${\mathcal{S}_z \subset {\bf Z}}$ is the randomly sifted set of integers formed by removing one residue class ${a_p \hbox{ mod } p}$ uniformly at random for each prime ${p \leq z}$. The Hardy-Littlewood conjecture can be viewed as an assertion that the primes ${{\mathcal P}}$ behave in some approximate statistical sense like the random sifted set ${\mathcal{S}_z}$, and one can prove the above theorem by using the Bonferroni inequalities both for the primes ${{\mathcal P}}$ and for the random sifted set, and comparing the two (using an even ${K}$ for the sifted set and an odd ${K}$ 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 ${{\mathcal P}}$ 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 ${{\mathcal P}}$ 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 ${{\mathcal C}}$ of the primes, in which each natural number ${n \geq 3}$ is selected for membership in ${{\mathcal C}}$ with an independent probability of ${1/\log n}$. This model matches the primes well in some respects; for instance, it almost surely obeys the “Riemann hypothesis”

$\displaystyle | {\mathcal C} \cap [1,x] | = \int_2^x \frac{dt}{\log t} + O( x^{1/2+o(1)})$

and Cramér also showed that the largest gap ${G_{\mathcal C}(x)}$ was almost surely ${\sim \log^2 x}$. 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 ${{\mathfrak S}({\mathcal H})}$ is absent.

Granville proposed a refinement ${{\mathcal G}}$ to Cramér’s random model ${{\mathcal C}}$ in which one first sieves out (in each dyadic interval ${[x,2x]}$) all residue classes ${0 \hbox{ mod } p}$ for ${p \leq A}$ for a certain threshold ${A = \log^{1-o(1)} x = o(\log x)}$, and then places each surviving natural number ${n}$ in ${{\mathcal G}}$ with an independent probability ${\frac{1}{\log n} \prod_{p \leq A} (1-\frac{1}{p})^{-1}}$. One can verify that this model obeys the Hardy-Littlewood conjectures, and Granville showed that the largest gap ${G_{\mathcal G}(x)}$ in this model was almost surely ${\gtrsim \xi \log^2 x}$, 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 ${10^{18}}$, for instance, have shown that ${G_{\mathcal P}(x)}$ never exceeds ${0.9206 \log^2 x}$ 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 ${A}$. After some experimentation, we were able to produce a tractable random model ${{\mathcal R}}$ for the primes which obeyed the Hardy-Littlewood conjectures with power savings, and which reproduced Granville’s gap prediction of ${\gtrsim \xi \log^2 x}$ (we also get an upper bound of ${\lesssim \xi \log^2 x \frac{\log\log x}{2 \log\log\log x}}$ 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 ${a_p \hbox{ mod } p}$ uniformly at random for each prime ${p}$, and as before we let ${S_z}$ be the sifted set of integers formed by deleting the residue classes ${a_p \hbox{ mod } p}$ with ${p \leq z}$. We then set

$\displaystyle {\mathcal R} := \{ n \geq e^2: n \in S_{z(t)}\}$

with ${z(t)}$ Pólya’s magic cutoff (this is the cutoff that gives ${{\mathcal R}}$ 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

$\displaystyle \xi \log^2 x \lesssim {\mathcal G}_{\mathcal R}(x) \lesssim \xi \log^2 x \frac{\log\log x}{2 \log\log\log x} \ \ \ \ \ (3)$

and that the Hardy-Littlewood conjectures hold with power savings for ${k}$ up to ${\log^c x}$ for any fixed ${c < 1}$ and for shifts ${h_1,\dots,h_k}$ of size ${O(\log^c x)}$. This is unfortunately a tiny bit weaker than what Theorem 3 requires (which more or less corresponds to the endpoint ${c=1}$), although there is a variant of Theorem 3 that can use this input to produce a lower bound on gaps in the model ${{\mathcal R}}$ (but it is weaker than the one in (3)). In fact we prove a more precise almost sure asymptotic formula for ${{\mathcal G}_{\mathcal R}(x) }$ that involves the optimal bounds for the linear sieve (or interval sieve), in which one deletes one residue class modulo ${p}$ from an interval ${[0,y]}$ for all primes ${p}$ up to a given threshold. The lower bound in (3) relates to the case of deleting the ${0 \hbox{ mod } p}$ residue classes from ${[0,y]}$; 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 ${{\mathcal R}}$ is somewhat more complicated than of ${{\mathcal C}}$ or ${{\mathcal G}}$ as there is now non-trivial coupling between the events ${n \in {\mathcal R}}$ as ${n}$ 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 ${[0,y]}$ and randomly deletes a residue class ${a_p \hbox{ mod } p}$ for each prime ${p}$ up to a given threshold. For very small ${p}$ this is handled by the deterministic theory of the linear sieve as discussed above. For medium sized ${p}$, 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 ${p}$, 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 ${n \in [0,y]}$ to primes ${p}$ if ${n}$ falls in the random residue class ${a_p \hbox{ mod } p}$) turns out to give the best results.

We now move away from the world of multiplicative prime number theory covered in Notes 1 and Notes 2, and enter the wider, and complementary, world of non-multiplicative prime number theory, in which one studies statistics related to non-multiplicative patterns, such as twins ${n,n+2}$. This creates a major jump in difficulty; for instance, even the most basic multiplicative result about the primes, namely Euclid’s theorem that there are infinitely many of them, remains unproven for twin primes. Of course, the situation is even worse for stronger results, such as Euler’s theorem, Dirichlet’s theorem, or the prime number theorem. Finally, even many multiplicative questions about the primes remain open. The most famous of these is the Riemann hypothesis, which gives the asymptotic ${\sum_{n \leq x} \Lambda(n) = x + O( \sqrt{x} \log^2 x )}$ (see Proposition 24 from Notes 2). But even if one assumes the Riemann hypothesis, the precise distribution of the error term ${O( \sqrt{x} \log^2 x )}$ in the above asymptotic (or in related asymptotics, such as for the sum ${\sum_{x \leq n < x+y} \Lambda(n)}$ that measures the distribution of primes in short intervals) is not entirely clear.

Despite this, we do have a number of extremely convincing and well supported models for the primes (and related objects) that let us predict what the answer to many prime number theory questions (both multiplicative and non-multiplicative) should be, particularly in asymptotic regimes where one can work with aggregate statistics about the primes, rather than with a small number of individual primes. These models are based on taking some statistical distribution related to the primes (e.g. the primality properties of a randomly selected ${k}$-tuple), and replacing that distribution by a model distribution that is easy to compute with (e.g. a distribution with strong joint independence properties). One can then predict the asymptotic value of various (normalised) statistics about the primes by replacing the relevant statistical distributions of the primes with their simplified models. In this non-rigorous setting, many difficult conjectures on the primes reduce to relatively simple calculations; for instance, all four of the (still unsolved) Landau problems may now be justified in the affirmative by one or more of these models. Indeed, the models are so effective at this task that analytic number theory is in the curious position of being able to confidently predict the answer to a large proportion of the open problems in the subject, whilst not possessing a clear way forward to rigorously confirm these answers!

As it turns out, the models for primes that have turned out to be the most accurate in practice are random models, which involve (either explicitly or implicitly) one or more random variables. This is despite the prime numbers being obviously deterministic in nature; no coins are flipped or dice rolled to create the set of primes. The point is that while the primes have a lot of obvious multiplicative structure (for instance, the product of two primes is never another prime), they do not appear to exhibit much discernible non-multiplicative structure asymptotically, in the sense that they rarely exhibit statistical anomalies in the asymptotic limit that cannot be easily explained in terms of the multiplicative properties of the primes. As such, when considering non-multiplicative statistics of the primes, the primes appear to behave pseudorandomly, and can thus be modeled with reasonable accuracy by a random model. And even for multiplicative problems, which are in principle controlled by the zeroes of the Riemann zeta function, one can obtain good predictions by positing various pseudorandomness properties of these zeroes, so that the distribution of these zeroes can be modeled by a random model.

Of course, one cannot expect perfect accuracy when replicating a deterministic set such as the primes by a probabilistic model of that set, and each of the heuristic models we discuss below have some limitations to the range of statistics about the primes that they can expect to track with reasonable accuracy. For instance, many of the models about the primes do not fully take into account the multiplicative structure of primes, such as the connection with a zeta function with a meromorphic continuation to the entire complex plane; at the opposite extreme, we have the GUE hypothesis which appears to accurately model the zeta function, but does not capture such basic properties of the primes as the fact that the primes are all natural numbers. Nevertheless, each of the models described below, when deployed within their sphere of reasonable application, has (possibly after some fine-tuning) given predictions that are in remarkable agreement with numerical computation and with known rigorous theoretical results, as well as with other models in overlapping spheres of application; they are also broadly compatible with the general heuristic (discussed in this previous post) that in the absence of any exploitable structure, asymptotic statistics should default to the most “uniform”, “pseudorandom”, or “independent” distribution allowable.

As hinted at above, we do not have a single unified model for the prime numbers (other than the primes themselves, of course), but instead have an overlapping family of useful models that each appear to accurately describe some, but not all, aspects of the prime numbers. In this set of notes, we will discuss four such models:

1. The Cramér random model and its refinements, which model the set ${{\mathcal P}}$ of prime numbers by a random set.
2. The Möbius pseudorandomness principle, which predicts that the Möbius function ${\mu}$ does not correlate with any genuinely different arithmetic sequence of reasonable “complexity”.
3. The equidistribution of residues principle, which predicts that the residue classes of a large number ${n}$ modulo a small or medium-sized prime ${p}$ behave as if they are independently and uniformly distributed as ${p}$ varies.
4. The GUE hypothesis, which asserts that the zeroes of the Riemann zeta function are distributed (at microscopic and mesoscopic scales) like the zeroes of a GUE random matrix, and which generalises the pair correlation conjecture regarding pairs of such zeroes.

This is not an exhaustive list of models for the primes and related objects; for instance, there is also the model in which the major arc contribution in the Hardy-Littlewood circle method is predicted to always dominate, and with regards to various finite groups of number-theoretic importance, such as the class groups discussed in Supplement 1, there are also heuristics of Cohen-Lenstra type. Historically, the first heuristic discussion of the primes along these lines was by Sylvester, who worked informally with a model somewhat related to the equidistribution of residues principle. However, we will not discuss any of these models here.

A word of warning: the discussion of the above four models will inevitably be largely informal, and “fuzzy” in nature. While one can certainly make precise formalisations of at least some aspects of these models, one should not be inflexibly wedded to a specific such formalisation as being “the” correct way to pin down the model rigorously. (To quote the statistician George Box: “all models are wrong, but some are useful”.) Indeed, we will see some examples below the fold in which some finer structure in the prime numbers leads to a correction term being added to a “naive” implementation of one of the above models to make it more accurate, and it is perfectly conceivable that some further such fine-tuning will be applied to one or more of these models in the future. These sorts of mathematical models are in some ways closer in nature to the scientific theories used to model the physical world, than they are to the axiomatic theories one is accustomed to in rigorous mathematics, and one should approach the discussion below accordingly. In particular, and in contrast to the other notes in this course, the material here is not directly used for proving further theorems, which is why we have marked it as “optional” material. Nevertheless, the heuristics and models here are still used indirectly for such purposes, for instance by

• giving a clearer indication of what results one expects to be true, thus guiding one to fruitful conjectures;
• providing a quick way to scan for possible errors in a mathematical claim (e.g. by finding that the main term is off from what a model predicts, or an error term is too small);
• gauging the relative strength of various assertions (e.g. classifying some results as “unsurprising”, others as “potential breakthroughs” or “powerful new estimates”, others as “unexpected new phenomena”, and yet others as “way too good to be true”); or
• setting up heuristic barriers (such as the parity barrier) that one has to resolve before resolving certain key problems (e.g. the twin prime conjecture).

See also my previous essay on the distinction between “rigorous” and “post-rigorous” mathematics, or Thurston’s essay discussing, among other things, the “definition-theorem-proof” model of mathematics and its limitations.

Remark 1 The material in this set of notes presumes some prior exposure to probability theory. See for instance this previous post for a quick review of the relevant concepts.

In the third Marker lecture, I would like to discuss the recent progress, particularly by Goldston, Pintz, and Yıldırım, on finding small gaps $p_{n+1}-p_n$ between consecutive primes.  (See also the surveys by Goldston-Pintz-Yıldırım, by Green, and by Soundararajan on the subject; the material here is based to some extent on these prior surveys.)