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This post is a continuation of the previous post on sieve theory, which is an ongoing part of the Polymath8 project. As the previous post was getting somewhat full, we are rolling the thread over to the current post.

In this post we will record a new truncation of the elementary Selberg sieve discussed in this previous post (and also analysed in the context of bounded prime gaps by Graham-Goldston-Pintz-Yildirim and Motohashi-Pintz) that was recently worked out by Janos Pintz, who has kindly given permission to share this new idea with the Polymath8 project. This new sieve decouples the ${\delta}$ parameter that was present in our previous analysis of Zhang’s argument into two parameters, a quantity ${\delta}$ that used to measure smoothness in the modulus, but now measures a weaker notion of “dense divisibility” which is what is really needed in the Elliott-Halberstam type estimates, and a second quantity ${\delta'}$ which still measures smoothness but is allowed to be substantially larger than ${\delta}$. Through this decoupling, it appears that the ${\kappa}$ type losses in the sieve theoretic part of the argument can be almost completely eliminated (they basically decay exponential in ${\delta'}$ and have only mild dependence on ${\delta}$, whereas the Elliott-Halberstam analysis is sensitive only to ${\delta}$, allowing one to set ${\delta}$ far smaller than previously by keeping ${\delta'}$ large). This should lead to noticeable gains in the ${k_0}$ quantity in our analysis.

To describe this new truncation we need to review some notation. As in all previous posts (in particular, the first post in this series), we have an asymptotic parameter ${x}$ going off to infinity, and all quantities here are implicitly understood to be allowed to depend on ${x}$ (or to range in a set that depends on ${x}$) unless they are explicitly declared to be fixed. We use the usual asymptotic notation ${O(), o(), \ll}$ relative to this parameter ${x}$. To be able to ignore local factors (such as the singular series ${{\mathfrak G}}$), we also use the “${W}$-trick” (as discussed in the first post in this series): we introduce a parameter ${w}$ that grows very slowly with ${x}$, and set ${W := \prod_{p.

For any fixed natural number ${k_0}$, define an admissible ${k_0}$-tuple to be a fixed tuple ${{\mathcal H}}$ of ${k_0}$ distinct integers which avoids at least one residue class modulo ${p}$ for each prime ${p}$. Our objective is to obtain the following conjecture ${DHL[k_0,2]}$ for as small a value of the parameter ${k_0}$ as possible:

Conjecture 1 (${DHL[k_0,2]}$) Let ${{\mathcal H}}$ be a fixed admissible ${k_0}$-tuple. Then there exist infinitely many translates ${n+{\mathcal H}}$ of ${{\mathcal H}}$ that contain at least two primes.

The twin prime conjecture asserts that ${DHL[k_0,2]}$ holds for ${k_0}$ as small as ${2}$, but currently we are only able to establish this result for ${k_0 \geq 6329}$ (see this comment). However, with the new truncated sieve of Pintz described in this post, we expect to be able to lower this threshold ${k_0 \geq 6329}$ somewhat.

In previous posts, we deduced ${DHL[k_0,2]}$ from a technical variant ${MPZ[\varpi,\delta]}$ of the Elliot-Halberstam conjecture for certain choices of parameters ${0 < \varpi < 1/4}$, ${0 < \delta < 1/4+\varpi}$. We will use the following formulation of ${MPZ[\varpi,\delta]}$:

Conjecture 2 (${MPZ[\varpi,\delta]}$) Let ${{\mathcal H}}$ be a fixed ${k_0}$-tuple (not necessarily admissible) for some fixed ${k_0 \geq 2}$, and let ${b\ (W)}$ be a primitive residue class. Then

$\displaystyle \sum_{q \in {\mathcal S}_I: q< x^{1/2+2\varpi}} \sum_{a \in C(q)} |\Delta_{b,W}(\Lambda; q,a)| = O( x \log^{-A} x) \ \ \ \ \ (1)$

for any fixed ${A>0}$, where ${I = (w,x^{\delta})}$, ${{\mathcal S}_I}$ are the square-free integers whose prime factors lie in ${I}$, and ${\Delta_{b,W}(\Lambda;q,a)}$ is the quantity

$\displaystyle \Delta_{b,W}(\Lambda;q,a) := | \sum_{x \leq n \leq 2x: n=b\ (W); n = a\ (q)} \Lambda(n) \ \ \ \ \ (2)$

$\displaystyle - \frac{1}{\phi(q)} \sum_{x \leq n \leq 2x: n = b\ (W)} \Lambda(n)|.$

and ${C(q)}$ is the set of congruence classes

$\displaystyle C(q) := \{ a \in ({\bf Z}/q{\bf Z})^\times: P(a) = 0 \}$

and ${P}$ is the polynomial

$\displaystyle P(a) := \prod_{h \in {\mathcal H}} (a+h).$

The conjecture ${MPZ[\varpi,\delta]}$ is currently known to hold whenever ${87 \varpi + 17 \delta < \frac{1}{4}}$ (see this comment and this confirmation). Actually, we can prove a stronger result than ${MPZ[\varpi,\delta]}$ in this regime in a couple ways. Firstly, the congruence classes ${C(q)}$ can be replaced by a more general system of congruence classes obeying a certain controlled multiplicity axiom; see this post. Secondly, and more importantly for this post, the requirement that the modulus ${q}$ lies in ${{\mathcal S}_I}$ can be relaxed; see below.

To connect the two conjectures, the previously best known implication was the folowing (see Theorem 2 from this post):

Theorem 3 Let ${0 < \varpi < 1/4}$, ${0 < \delta < 1/4 + \varpi}$ and ${k_0 \geq 2}$ be such that we have the inequality

$\displaystyle (1 +4 \varpi) (1-\kappa') > \frac{j^2_{k_0-2}}{k_0(k_0-1)} (1+\kappa) \ \ \ \ \ (3)$

where ${j_{k_0-2} = j_{k_0-2,1}}$ is the first positive zero of the Bessel function ${J_{k_0-2}}$, and ${\kappa,\kappa'>0}$ are the quantities

$\displaystyle \kappa := \sum_{1 \leq n < \frac{1+4\varpi}{4\delta}} \frac{3^n+1}{2} \frac{k_0^n}{n!} (\int_{4\delta/(1+4\varpi)}^1 (1-t)^{k_0/2}\ \frac{dt}{t})^n$

and

$\displaystyle \kappa' := \sum_{2 \leq n < \frac{1+4\varpi}{4\delta}} \frac{3^n-1}{2} \frac{(k_0-1)^n}{n!}$

$\displaystyle (\int_{4\delta/(1+4\varpi)}^1 (1-t)^{(k_0-1)/2}\ \frac{dt}{t})^n.$

Then ${MPZ[\varpi,\delta]}$ implies ${DHL[k_0,2]}$.

Actually there have been some slight improvements to the quantities ${\kappa,\kappa'}$; see the comments to this previous post. However, the main error ${\kappa}$ remains roughly of the order ${\delta^{-1} \exp( - 2 k_0\delta )}$, which limits one from taking ${\delta}$ too small.

To improve beyond this, the first basic observation is that the smoothness condition ${q \in {\mathcal S}_I}$, which implies that all prime divisors of ${q}$ are less than ${x^\delta}$, can be relaxed in the proof of ${MPZ[\varpi,\delta]}$. Indeed, if one inspects the proof of this proposition (described in these three previous posts), one sees that the key property of ${q}$ needed is not so much the smoothness, but a weaker condition which we will call (for lack of a better term) dense divisibility:

Definition 4 Let ${y > 1}$. A positive integer ${q}$ is said to be ${y}$-densely divisible if for every ${1 \leq R \leq q}$, one can find a factor of ${q}$ in the interval ${[y^{-1} R, R]}$. We let ${{\mathcal D}_y}$ denote the set of positive integers that are ${y}$-densely divisible.

Certainly every integer which is ${y}$-smooth (i.e. has all prime factors at most ${y}$ is also ${y}$-densely divisible, as can be seen from the greedy algorithm; but the property of being ${y}$-densely divisible is strictly weaker than ${y}$-smoothness, which is a fact we shall exploit shortly.

We now define ${MPZ'[\varpi,\delta]}$ to be the same statement as ${MPZ[\varpi,\delta]}$, but with the condition ${q \in {\mathcal S}_I}$ replaced by the weaker condition ${q \in {\mathcal S}_{[w,+\infty)} \cap {\mathcal D}_{x^\delta}}$. The arguments in previous posts then also establish ${MPZ'[\varpi,\delta]}$ whenever ${87 \varpi + 17 \delta < \frac{1}{4}}$.

The main result of this post is then the following implication, essentially due to Pintz:

Theorem 5 Let ${0 < \varpi < 1/4}$, ${0 < \delta \leq \delta' < 1/4 + \varpi}$, ${A \geq 0}$, and ${k_0 \geq 2}$ be such that

$\displaystyle (1 +4 \varpi) (1-2\kappa_1 - 2\kappa_2 - 2\kappa_3) > \frac{j^2_{k_0-2}}{k_0(k_0-1)}$

where

$\displaystyle \kappa_1 := \int_{\theta}^1 (1-t)^{(k_0-1)/2}\ \frac{dt}{t}$

$\displaystyle \kappa_2 := (k_0-1) \int_{\theta}^1 (1-t)^{k_0-1}\ \frac{dt}{t}$

$\displaystyle \kappa_3 := e^A \frac{G_{k_0-1,\tilde \theta}(0,0)}{G_{k_0-1}(0,0)} \sum_{0 \leq J \leq 1/\tilde \delta} \frac{(k_0-1)^J}{J!} (\int_{\tilde \delta}^\theta e^{-At} \frac{dt}{t})^J$

and

$\displaystyle \theta := \frac{\delta'}{1/4+\varpi}$

$\displaystyle \tilde \theta := \frac{(\delta' - \delta)/2 + \varpi}{1/4 + \varpi}$

$\displaystyle \tilde \delta := \frac{\delta}{1/4+\varpi}$

$\displaystyle G_{k_0-1}(0,0) := \int_0^1 f(t)^2 \frac{t^{k_0-2}}{(k_0-2)!}\ dt$

$\displaystyle G_{k_0-1,\tilde \theta}(0,0) := \int_0^{\tilde \theta} f(t)^2 \frac{t^{k_0-2}}{(k_0-2)!}\ dt$

and

$\displaystyle f(t) := t^{1-k_0/2} J_{k_0-2}( \sqrt{t} j_{k_0-2} ).$

Then ${MPZ'[\varpi,\delta]}$ implies ${DHL[k_0,2]}$.

This theorem has rather messy constants, but we can isolate some special cases which are a bit easier to compute with. Setting ${\delta' = \delta}$, we see that ${\kappa_3}$ vanishes (and the argument below will show that we only need ${MPZ[\varpi,\delta]}$ rather than ${MPZ'[\varpi,\delta]}$), and we obtain the following slight improvement of Theorem 3:

Theorem 6 Let ${0 < \varpi < 1/4}$, ${0 < \delta < 1/4 + \varpi}$ and ${k_0 \geq 2}$ be such that we have the inequality

$\displaystyle (1 +4 \varpi) (1-2\kappa_1-2\kappa_2) > \frac{j^2_{k_0-2}}{k_0(k_0-1)} \ \ \ \ \ (4)$

where

$\displaystyle \kappa_1 := \int_{4\delta/(1+4\varpi)}^1 (1-t)^{(k_0-1)/2}\ \frac{dt}{t}$

$\displaystyle \kappa_2 := (k_0-1) \int_{4\delta/(1+4\varpi)}^1 (1-t)^{k_0-1}\ \frac{dt}{t}.$

Then ${MPZ[\varpi,\delta]}$ implies ${DHL[k_0,2]}$.

This is a little better than Theorem 3, because the error ${2\kappa_1+2\kappa_2}$ has size about ${\frac{1}{2 k_0 \delta} \exp( - 2 k_0 \delta) + \frac{1}{2 \delta} \exp(-4 k_0 \delta)}$, which compares favorably with the error in Theorem 3 which is about ${\frac{1}{\delta} \exp(-2 k_0 \delta)}$. This should already give a “cheap” improvement to our current threshold ${k_0 \geq 6329}$, though it will fall short of what one would get if one fully optimised over all parameters in the above theorem.

Returning to the full strength of Theorem 5, let us obtain a crude upper bound for ${\kappa_3}$ that is a little simpler to understand. Extending the ${J}$ summation to infinity and using the Taylor series for the exponential, we have

$\displaystyle \kappa_3 \leq \frac{G_{k_0-1,\tilde \theta}(0,0)}{G_{k_0-1}(0,0)} \exp( A + (k_0-1) \int_{\tilde \delta}^\theta e^{-At} \frac{dt}{t} ).$

We can crudely bound

$\displaystyle \int_{\tilde \delta}^\theta e^{-At} \frac{dt}{t} \leq \frac{1}{A \tilde \delta}$

and then optimise in ${A}$ to obtain

$\displaystyle \kappa_3 \leq \frac{G_{k_0-1,\tilde \theta}(0,0)}{G_{k_0-1}(0,0)} \exp( 2 (k_0-1)^{1/2} \tilde \delta^{-1/2} ).$

Because of the ${t^{k_0-2}}$ factor in the integrand for ${G_{k_0-1}}$ and ${G_{k_0-1,\tilde \theta}}$, we expect the ratio ${\frac{G_{k_0-1,\tilde \theta}(0,0)}{G_{k_0-1}(0,0)}}$ to be of the order of ${\tilde \theta^{k_0-1}}$, although one will need some theoretical or numerical estimates on Bessel functions to make this heuristic more precise. Setting ${\tilde \theta}$ to be something like ${1/2}$, we get a good bound here as long as ${\tilde \delta \gg 1/k_0}$, which at current values of ${\delta, k_0}$ is a mild condition.

Pintz’s argument uses the elementary Selberg sieve, discussed in this previous post, but with a more efficient estimation of the quantity ${\beta}$, in particular avoiding the truncation to moduli ${d}$ between ${x^{-\delta} R}$ and ${R}$ which was the main source of inefficiency in that previous post. The basic idea is to “linearise” the effect of the truncation of the sieve, so that this contribution can be dealt with by the union bound (basically, bounding the contribution of each large prime one at a time). This mostly avoids the more complicated combinatorial analysis that arose in the analytic Selberg sieve, as seen in this previous post.

This post is a continuation of the previous post on sieve theory, which is an ongoing part of the Polymath8 project to improve the various parameters in Zhang’s proof that bounded gaps between primes occur infinitely often. Given that the comments on that page are getting quite lengthy, this is also a good opportunity to “roll over” that thread.

We will continue the notation from the previous post, including the concept of an admissible tuple, the use of an asymptotic parameter ${x}$ going to infinity, and a quantity ${w}$ depending on ${x}$ that goes to infinity sufficiently slowly with ${x}$, and ${W := \prod_{p (the ${W}$-trick).

The objective of this portion of the Polymath8 project is to make as efficient as possible the connection between two types of results, which we call ${DHL[k_0,2]}$ and ${MPZ[\varpi,\delta]}$. Let us first state ${DHL[k_0,2]}$, which has an integer parameter ${k_0 \geq 2}$:

Conjecture 1 (${DHL[k_0,2]}$) Let ${{\mathcal H}}$ be a fixed admissible ${k_0}$-tuple. Then there are infinitely many translates ${n+{\mathcal H}}$ of ${{\mathcal H}}$ which contain at least two primes.

Zhang was the first to prove a result of this type with ${k_0 = 3,500,000}$. Since then the value of ${k_0}$ has been lowered substantially; at this time of writing, the current record is ${k_0 = 26,024}$.

There are two basic ways known currently to attain this conjecture. The first is to use the Elliott-Halberstam conjecture ${EH[\theta]}$ for some ${\theta>1/2}$:

Conjecture 2 (${EH[\theta]}$) One has

$\displaystyle \sum_{1 \leq q \leq x^\theta} \sup_{a \in ({\bf Z}/q{\bf Z})^\times} |\sum_{n < x: n = a\ (q)} \Lambda(n) - \frac{1}{\phi(q)} \sum_{n < x} \Lambda(n)|$

$\displaystyle = O( \frac{x}{\log^A x} )$

for all fixed ${A>0}$. Here we use the abbreviation ${n=a\ (q)}$ for ${n=a \hbox{ mod } q}$.

Here of course ${\Lambda}$ is the von Mangoldt function and ${\phi}$ the Euler totient function. It is conjectured that ${EH[\theta]}$ holds for all ${0 < \theta < 1}$, but this is currently only known for ${0 < \theta < 1/2}$, an important result known as the Bombieri-Vinogradov theorem.

In a breakthrough paper, Goldston, Yildirim, and Pintz established an implication of the form

$\displaystyle EH[\theta] \implies DHL[k_0,2] \ \ \ \ \ (1)$

for any ${1/2 < \theta < 1}$, where ${k_0 = k_0(\theta)}$ depends on ${\theta}$. This deduction was very recently optimised by Farkas, Pintz, and Revesz and also independently in the comments to the previous blog post, leading to the following implication:

Theorem 3 (EH implies DHL) Let ${1/2 < \theta < 1}$ be a real number, and let ${k_0 \geq 2}$ be an integer obeying the inequality

$\displaystyle 2\theta > \frac{j_{k_0-2}^2}{k_0(k_0-1)}, \ \ \ \ \ (2)$

where ${j_n}$ is the first positive zero of the Bessel function ${J_n(x)}$. Then ${EH[\theta]}$ implies ${DHL[k_0,2]}$.

Note that the right-hand side of (2) is larger than ${1}$, but tends asymptotically to ${1}$ as ${k_0 \rightarrow \infty}$. We give an alternate proof of Theorem 3 below the fold.

Implications of the form Theorem 3 were modified by Motohashi and Pintz, which in our notation replaces ${EH[\theta]}$ by an easier conjecture ${MPZ[\varpi,\delta]}$ for some ${0 < \varpi < 1/4}$ and ${0 < \delta < 1/4+\varpi}$, at the cost of degrading the sufficient condition (2) slightly. In our notation, this conjecture takes the following form for each choice of parameters ${\varpi,\delta}$:

Conjecture 4 (${MPZ[\varpi,\delta]}$) Let ${{\mathcal H}}$ be a fixed ${k_0}$-tuple (not necessarily admissible) for some fixed ${k_0 \geq 2}$, and let ${b\ (W)}$ be a primitive residue class. Then

$\displaystyle \sum_{q \in {\mathcal S}_I: q< x^{1/2+2\varpi}} \sum_{a \in C(q)} |\Delta_{b,W}(\Lambda; q,a)| = O( x \log^{-A} x) \ \ \ \ \ (3)$

for any fixed ${A>0}$, where ${I = (w,x^{\delta})}$, ${{\mathcal S}_I}$ are the square-free integers whose prime factors lie in ${I}$, and ${\Delta_{b,W}(\Lambda;q,a)}$ is the quantity

$\displaystyle \Delta_{b,W}(\Lambda;q,a) := | \sum_{x \leq n \leq 2x: n=b\ (W); n = a\ (q)} \Lambda(n) \ \ \ \ \ (4)$

$\displaystyle - \frac{1}{\phi(q)} \sum_{x \leq n \leq 2x: n = b\ (W)} \Lambda(n)|.$

and ${C(q)}$ is the set of congruence classes

$\displaystyle C(q) := \{ a \in ({\bf Z}/q{\bf Z})^\times: P(a) = 0 \}$

and ${P}$ is the polynomial

$\displaystyle P(a) := \prod_{h \in {\mathcal H}} (a+h).$

This is a weakened version of the Elliott-Halberstam conjecture:

Proposition 5 (EH implies MPZ) Let ${0 < \varpi < 1/4}$ and ${0 < \delta < 1/4+\varpi}$. Then ${EH[1/2+2\varpi+\epsilon]}$ implies ${MPZ[\varpi,\delta]}$ for any ${\epsilon>0}$. (In abbreviated form: ${EH[1/2+2\varpi+]}$ implies ${MPZ[\varpi,\delta]}$.)

In particular, since ${EH[\theta]}$ is conjecturally true for all ${0 < \theta < 1/2}$, we conjecture ${MPZ[\varpi,\delta]}$ to be true for all ${0 < \varpi < 1/4}$ and ${0<\delta<1/4+\varpi}$.

Proof: Define

$\displaystyle E(q) := \sup_{a \in ({\bf Z}/q{\bf Z})^\times} |\sum_{x \leq n \leq 2x: n = a\ (q)} \Lambda(n) - \frac{1}{\phi(q)} \sum_{x \leq n \leq 2x} \Lambda(n)|$

then the hypothesis ${EH[1/2+2\varpi+\epsilon]}$ (applied to ${x}$ and ${2x}$ and then subtracting) tells us that

$\displaystyle \sum_{1 \leq q \leq Wx^{1/2+2\varpi}} E(q) \ll x \log^{-A} x$

for any fixed ${A>0}$. From the Chinese remainder theorem and the Siegel-Walfisz theorem we have

$\displaystyle \sup_{a \in ({\bf Z}/q{\bf Z})^\times} \Delta_{b,W}(\Lambda;q,a) \ll E(qW) + \frac{1}{\phi(q)} x \log^{-A} x$

for any ${q}$ coprime to ${W}$ (and in particular for ${q \in {\mathcal S}_I}$). Since ${|C(q)| \leq k_0^{\Omega(q)}}$, where ${\Omega(q)}$ is the number of prime divisors of ${q}$, we can thus bound the left-hand side of (3) by

$\displaystyle \ll \sum_{q \in {\mathcal S}_I: q< x^{1/2+2\varpi}} k_0^{\Omega(q)} E(qW) + k_0^{\Omega(q)} \frac{1}{\phi(q)} x \log^{-A} x.$

The contribution of the second term is ${O(x \log^{-A+O(1)} x)}$ by standard estimates (see Proposition 8 below). Using the very crude bound

$\displaystyle E(q) \ll \frac{1}{\phi(q)} x \log x$

and standard estimates we also have

$\displaystyle \sum_{q \in {\mathcal S}_I: q< x^{1/2+2\varpi}} k_0^{2\Omega(q)} E(qW) \ll x \log^{O(1)} A$

and the claim now follows from the Cauchy-Schwarz inequality. $\Box$

In practice, the conjecture ${MPZ[\varpi,\delta]}$ is easier to prove than ${EH[1/2+2\varpi+]}$ due to the restriction of the residue classes ${a}$ to ${C(q)}$, and also the restriction of the modulus ${q}$ to ${x^\delta}$-smooth numbers. Zhang proved ${MPZ[\varpi,\varpi]}$ for any ${0 < \varpi < 1/1168}$. More recently, our Polymath8 group has analysed Zhang’s argument (using in part a corrected version of the analysis of a recent preprint of Pintz) to obtain ${MPZ[\varpi,\delta]}$ whenever ${\delta, \varpi > 0}$ are such that

$\displaystyle 207\varpi + 43\delta < \frac{1}{4}.$

The work of Motohashi and Pintz, and later Zhang, implicitly describe arguments that allow one to deduce ${DHL[k_0,2]}$ from ${MPZ[\varpi,\delta]}$ provided that ${k_0}$ is sufficiently large depending on ${\varpi,\delta}$. The best implication of this sort that we have been able to verify thus far is the following result, established in the previous post:

Theorem 6 (MPZ implies DHL) Let ${0 < \varpi < 1/4}$, ${0 < \delta < 1/4+\varpi}$, and let ${k_0 \geq 2}$ be an integer obeying the constraint

$\displaystyle 1+4\varpi > \frac{j_{k_0-2}^2}{k_0(k_0-1)} (1+\kappa) \ \ \ \ \ (5)$

where ${\kappa}$ is the quantity

$\displaystyle \kappa := \sum_{1 \leq n < \frac{1+4\varpi}{2\delta}} (1 - \frac{2n \delta}{1 + 4\varpi})^{k_0/2} \prod_{j=1}^{n} (1 + 3k_0 \log(1+\frac{1}{j})) ).$

Then ${MPZ[\varpi,\delta]}$ implies ${DHL[k_0,2]}$.

This complicated version of ${\kappa}$ is roughly of size ${3 \log(2) k_0 \exp( - k_0 \delta)}$. It is unlikely to be optimal; the work of Motohashi-Pintz and Pintz suggests that it can essentially be improved to ${\frac{1}{\delta} \exp(-k_0 \delta)}$, but currently we are unable to verify this claim. One of the aims of this post is to encourage further discussion as to how to improve the ${\kappa}$ term in results such as Theorem 6.

We remark that as (5) is an open condition, it is unaffected by infinitesimal modifications to ${\varpi,\delta}$, and so we do not ascribe much importance to such modifications (e.g. replacing ${\varpi}$ by ${\varpi-\epsilon}$ for some arbitrarily small ${\epsilon>0}$).

The known deductions of ${DHL[k_0,2]}$ from claims such as ${EH[\theta]}$ or ${MPZ[\varpi,\delta]}$ rely on the following elementary observation of Goldston, Pintz, and Yildirim (essentially a weighted pigeonhole principle), which we have placed in “${W}$-tricked form”:

Lemma 7 (Criterion for DHL) Let ${k_0 \geq 2}$. Suppose that for each fixed admissible ${k_0}$-tuple ${{\mathcal H}}$ and each congruence class ${b\ (W)}$ such that ${b+h}$ is coprime to ${W}$ for all ${h \in {\mathcal H}}$, one can find a non-negative weight function ${\nu: {\bf N} \rightarrow {\bf R}^+}$, fixed quantities ${\alpha,\beta > 0}$, a quantity ${A>0}$, and a fixed positive power ${R}$ of ${x}$ such that one has the upper bound

$\displaystyle \sum_{x \leq n \leq 2x: n = b\ (W)} \nu(n) \leq (\alpha+o(1)) A\frac{x}{W}, \ \ \ \ \ (6)$

the lower bound

$\displaystyle \sum_{x \leq n \leq 2x: n = b\ (W)} \nu(n) \theta(n+h_i) \geq (\beta-o(1)) A\frac{x}{W} \log R \ \ \ \ \ (7)$

for all ${h_i \in {\mathcal H}}$, and the key inequality

$\displaystyle \frac{\log R}{\log x} > \frac{1}{k_0} \frac{\alpha}{\beta} \ \ \ \ \ (8)$

holds. Then ${DHL[k_0,2]}$ holds. Here ${\theta(n)}$ is defined to equal ${\log n}$ when ${n}$ is prime and ${0}$ otherwise.

Proof: Consider the quantity

$\displaystyle \sum_{x \leq n \leq 2x: n = b\ (W)} \nu(n) (\sum_{h \in {\mathcal H}} \theta(n+h) - \log(3x)). \ \ \ \ \ (9)$

By (6), (7), this quantity is at least

$\displaystyle k_0 \beta A\frac{x}{W} \log R - \alpha \log(3x) A\frac{x}{W} - o(A\frac{x}{W} \log x).$

By (8), this expression is positive for all sufficiently large ${x}$. On the other hand, (9) can only be positive if at least one summand is positive, which only can happen when ${n+{\mathcal H}}$ contains at least two primes for some ${x \leq n \leq 2x}$ with ${n=b\ (W)}$. Letting ${x \rightarrow \infty}$ we obtain ${DHL[k_0,2]}$ as claimed. $\Box$

In practice, the quantity ${R}$ (referred to as the sieve level) is a power of ${x}$ such as ${x^{\theta/2}}$ or ${x^{1/4+\varpi}}$, and reflects the strength of the distribution hypothesis ${EH[\theta]}$ or ${MPZ[\varpi,\delta]}$ that is available; the quantity ${R}$ will also be a key parameter in the definition of the sieve weight ${\nu}$. The factor ${A}$ reflects the order of magnitude of the expected density of ${\nu}$ in the residue class ${b\ (W)}$; it could be absorbed into the sieve weight ${\nu}$ by dividing that weight by ${A}$, but it is convenient to not enforce such a normalisation so as not to clutter up the formulae. In practice, ${A}$ will some combination of ${\frac{\phi(W)}{W}}$ and ${\log R}$.

Once one has decided to rely on Lemma 7, the next main task is to select a good weight ${\nu}$ for which the ratio ${\alpha/\beta}$ is as small as possible (and for which the sieve level ${R}$ is as large as possible. To ensure non-negativity, we use the Selberg sieve

$\displaystyle \nu = \lambda^2, \ \ \ \ \ (10)$

where ${\lambda(n)}$ takes the form

$\displaystyle \lambda(n) = \sum_{d \in {\mathcal S}_I: d|P(n)} \mu(d) a_d$

for some weights ${a_d \in {\bf R}}$ vanishing for ${d>R}$ that are to be chosen, where ${I \subset (w,+\infty)}$ is an interval and ${P}$ is the polynomial ${P(n) := \prod_{h \in {\mathcal H}} (n+h)}$. If the distribution hypothesis is ${EH[\theta]}$, one takes ${R := x^{\theta/2}}$ and ${I := (w,+\infty)}$; if the distribution hypothesis is instead ${MPZ[\varpi,\delta]}$, one takes ${R := x^{1/4+\varpi}}$ and ${I := (w,x^\delta)}$.

One has a useful amount of flexibility in selecting the weights ${a_d}$ for the Selberg sieve. The original work of Goldston, Pintz, and Yildirim, as well as the subsequent paper of Zhang, the choice

$\displaystyle a_d := \log(\frac{R}{d})_+^{k_0+\ell_0}$

is used for some additional parameter ${\ell_0 > 0}$ to be optimised over. More generally, one can take

$\displaystyle a_d := g( \frac{\log d}{\log R} )$

for some suitable (in particular, sufficiently smooth) cutoff function ${g: {\bf R} \rightarrow {\bf R}}$. We will refer to this choice of sieve weights as the “analytic Selberg sieve”; this is the choice used in the analysis in the previous post.

However, there is a slight variant choice of sieve weights that one can use, which I will call the “elementary Selberg sieve”, and it takes the form

$\displaystyle a_d := \frac{1}{\Phi(d) \Delta(d)} \sum_{q \in {\mathcal S}_I: (q,d)=1} \frac{1}{\Phi(q)} f'( \frac{\log dq}{\log R}) \ \ \ \ \ (11)$

for a sufficiently smooth function ${f: {\bf R} \rightarrow {\bf R}}$, where

$\displaystyle \Phi(d) := \prod_{p|d} \frac{p-k_0}{k_0}$

for ${d \in {\mathcal S}_I}$ is a ${k_0}$-variant of the Euler totient function, and

$\displaystyle \Delta(d) := \prod_{p|d} \frac{k_0}{p} = \frac{k_0^{\Omega(d)}}{d}$

for ${d \in {\mathcal S}_I}$ is a ${k_0}$-variant of the function ${1/d}$. (The derivative on the ${f}$ cutoff is convenient for computations, as will be made clearer later in this post.) This choice of weights ${a_d}$ may seem somewhat arbitrary, but it arises naturally when considering how to optimise the quadratic form

$\displaystyle \sum_{d_1,d_2 \in {\mathcal S}_I} \mu(d_1) a_{d_1} \mu(d_2) a_{d_2} \Delta([d_1,d_2])$

(which arises naturally in the estimation of ${\alpha}$ in (6)) subject to a fixed value of ${a_1}$ (which morally is associated to the estimation of ${\beta}$ in (7)); this is discussed in any sieve theory text as part of the general theory of the Selberg sieve, e.g. Friedlander-Iwaniec.

The use of the elementary Selberg sieve for the bounded prime gaps problem was studied by Motohashi and Pintz. Their arguments give an alternate derivation of ${DHL[k_0,2]}$ from ${MPZ[\varpi,\theta]}$ for ${k_0}$ sufficiently large, although unfortunately we were not able to confirm some of their calculations regarding the precise dependence of ${k_0}$ on ${\varpi,\theta}$, and in particular we have not yet been able to improve upon the specific criterion in Theorem 6 using the elementary sieve. However it is quite plausible that such improvements could become available with additional arguments.

Below the fold we describe how the elementary Selberg sieve can be used to reprove Theorem 3, and discuss how they could potentially be used to improve upon Theorem 6. (But the elementary Selberg sieve and the analytic Selberg sieve are in any event closely related; see the appendix of this paper of mine with Ben Green for some further discussion.) For the purposes of polymath8, either developing the elementary Selberg sieve or continuing the analysis of the analytic Selberg sieve from the previous post would be a relevant topic of conversation in the comments to this post.

Suppose one is given a ${k_0}$-tuple ${{\mathcal H} = (h_1,\ldots,h_{k_0})}$ of ${k_0}$ distinct integers for some ${k_0 \geq 1}$, arranged in increasing order. When is it possible to find infinitely many translates ${n + {\mathcal H} =(n+h_1,\ldots,n+h_{k_0})}$ of ${{\mathcal H}}$ which consists entirely of primes? The case ${k_0=1}$ is just Euclid’s theorem on the infinitude of primes, but the case ${k_0=2}$ is already open in general, with the ${{\mathcal H} = (0,2)}$ case being the notorious twin prime conjecture.

On the other hand, there are some tuples ${{\mathcal H}}$ for which one can easily answer the above question in the negative. For instance, the only translate of ${(0,1)}$ that consists entirely of primes is ${(2,3)}$, basically because each translate of ${(0,1)}$ must contain an even number, and the only even prime is ${2}$. More generally, if there is a prime ${p}$ such that ${{\mathcal H}}$ meets each of the ${p}$ residue classes ${0 \hbox{ mod } p, 1 \hbox{ mod } p, \ldots, p-1 \hbox{ mod } p}$, then every translate of ${{\mathcal H}}$ contains at least one multiple of ${p}$; since ${p}$ is the only multiple of ${p}$ that is prime, this shows that there are only finitely many translates of ${{\mathcal H}}$ that consist entirely of primes.

To avoid this obstruction, let us call a ${k_0}$-tuple ${{\mathcal H}}$ admissible if it avoids at least one residue class ${\hbox{ mod } p}$ for each prime ${p}$. It is easy to check for admissibility in practice, since a ${k_0}$-tuple is automatically admissible in every prime ${p}$ larger than ${k_0}$, so one only needs to check a finite number of primes in order to decide on the admissibility of a given tuple. For instance, ${(0,2)}$ or ${(0,2,6)}$ are admissible, but ${(0,2,4)}$ is not (because it covers all the residue classes modulo ${3}$). We then have the famous Hardy-Littlewood prime tuples conjecture:

Conjecture 1 (Prime tuples conjecture, qualitative form) If ${{\mathcal H}}$ is an admissible ${k_0}$-tuple, then there exists infinitely many translates of ${{\mathcal H}}$ that consist entirely of primes.

This conjecture is extremely difficult (containing the twin prime conjecture, for instance, as a special case), and in fact there is no explicitly known example of an admissible ${k_0}$-tuple with ${k_0 \geq 2}$ for which we can verify this conjecture (although, thanks to the recent work of Zhang, we know that ${(0,d)}$ satisfies the conclusion of the prime tuples conjecture for some ${0 < d < 70,000,000}$, even if we can’t yet say what the precise value of ${d}$ is).

Actually, Hardy and Littlewood conjectured a more precise version of Conjecture 1. Given an admissible ${k_0}$-tuple ${{\mathcal H} = (h_1,\ldots,h_{k_0})}$, and for each prime ${p}$, let ${\nu_p = \nu_p({\mathcal H}) := |{\mathcal H} \hbox{ mod } p|}$ denote the number of residue classes modulo ${p}$ that ${{\mathcal H}}$ meets; thus we have ${1 \leq \nu_p \leq p-1}$ for all ${p}$ by admissibility, and also ${\nu_p = k_0}$ for all ${p>h_{k_0}-h_1}$. We then define the singular series ${{\mathfrak G} = {\mathfrak G}({\mathcal H})}$ associated to ${{\mathcal H}}$ by the formula

$\displaystyle {\mathfrak G} := \prod_{p \in {\mathcal P}} \frac{1-\frac{\nu_p}{p}}{(1-\frac{1}{p})^{k_0}}$

where ${{\mathcal P} = \{2,3,5,\ldots\}}$ is the set of primes; by the previous discussion we see that the infinite product in ${{\mathfrak G}}$ converges to a finite non-zero number.

We will also need some asymptotic notation (in the spirit of “cheap nonstandard analysis“). We will need a parameter ${x}$ that one should think of going to infinity. Some mathematical objects (such as ${{\mathcal H}}$ and ${k_0}$) will be independent of ${x}$ and referred to as fixed; but unless otherwise specified we allow all mathematical objects under consideration to depend on ${x}$. If ${X}$ and ${Y}$ are two such quantities, we say that ${X = O(Y)}$ if one has ${|X| \leq CY}$ for some fixed ${C}$, and ${X = o(Y)}$ if one has ${|X| \leq c(x) Y}$ for some function ${c(x)}$ of ${x}$ (and of any fixed parameters present) that goes to zero as ${x \rightarrow \infty}$ (for each choice of fixed parameters).

Conjecture 2 (Prime tuples conjecture, quantitative form) Let ${k_0 \geq 1}$ be a fixed natural number, and let ${{\mathcal H}}$ be a fixed admissible ${k_0}$-tuple. Then the number of natural numbers ${n < x}$ such that ${n+{\mathcal H}}$ consists entirely of primes is ${({\mathfrak G} + o(1)) \frac{x}{\log^{k_0} x}}$.

Thus, for instance, if Conjecture 2 holds, then the number of twin primes less than ${x}$ should equal ${(2 \Pi_2 + o(1)) \frac{x}{\log^2 x}}$, where ${\Pi_2}$ is the twin prime constant

$\displaystyle \Pi_2 := \prod_{p \in {\mathcal P}: p>2} (1 - \frac{1}{(p-1)^2}) = 0.6601618\ldots.$

As this conjecture is stronger than Conjecture 1, it is of course open. However there are a number of partial results on this conjecture. For instance, this conjecture is known to be true if one introduces some additional averaging in ${{\mathcal H}}$; see for instance this previous post. From the methods of sieve theory, one can obtain an upper bound of ${(C_{k_0} {\mathfrak G} + o(1)) \frac{x}{\log^{k_0} x}}$ for the number of ${n < x}$ with ${n + {\mathcal H}}$ all prime, where ${C_{k_0}}$ depends only on ${k_0}$. Sieve theory can also give analogues of Conjecture 2 if the primes are replaced by a suitable notion of almost prime (or more precisely, by a weight function concentrated on almost primes).

Another type of partial result towards Conjectures 1, 2 come from the results of Goldston-Pintz-Yildirim, Motohashi-Pintz, and of Zhang. Following the notation of this recent paper of Pintz, for each ${k_0>2}$, let ${DHL[k_0,2]}$ denote the following assertion (DHL stands for “Dickson-Hardy-Littlewood”):

Conjecture 3 (${DHL[k_0,2]}$) Let ${{\mathcal H}}$ be a fixed admissible ${k_0}$-tuple. Then there are infinitely many translates ${n+{\mathcal H}}$ of ${{\mathcal H}}$ which contain at least two primes.

This conjecture gets harder as ${k_0}$ gets smaller. Note for instance that ${DHL[2,2]}$ would imply all the ${k_0=2}$ cases of Conjecture 1, including the twin prime conjecture. More generally, if one knew ${DHL[k_0,2]}$ for some ${k_0}$, then one would immediately conclude that there are an infinite number of pairs of consecutive primes of separation at most ${H(k_0)}$, where ${H(k_0)}$ is the minimal diameter ${h_{k_0}-h_1}$ amongst all admissible ${k_0}$-tuples ${{\mathcal H}}$. Values of ${H(k_0)}$ for small ${k_0}$ can be found at this link (with ${H(k_0)}$ denoted ${w}$ in that page). For large ${k_0}$, the best upper bounds on ${H(k_0)}$ have been found by using admissible ${k_0}$-tuples ${{\mathcal H}}$ of the form

$\displaystyle {\mathcal H} = ( - p_{m+\lfloor k_0/2\rfloor - 1}, \ldots, - p_{m+1}, -1, +1, p_{m+1}, \ldots, p_{m+\lfloor (k_0+1)/2\rfloor - 1} )$

where ${p_n}$ denotes the ${n^{th}}$ prime and ${m}$ is a parameter to be optimised over (in practice it is an order of magnitude or two smaller than ${k_0}$); see this blog post for details. The upshot is that one can bound ${H(k_0)}$ for large ${k_0}$ by a quantity slightly smaller than ${k_0 \log k_0}$ (and the large sieve inequality shows that this is sharp up to a factor of two, see e.g. this previous post for more discussion).

In a key breakthrough, Goldston, Pintz, and Yildirim were able to establish the following conditional result a few years ago:

Theorem 4 (Goldston-Pintz-Yildirim) Suppose that the Elliott-Halberstam conjecture ${EH[\theta]}$ is true for some ${1/2 < \theta < 1}$. Then ${DHL[k_0,2]}$ is true for some finite ${k_0}$. In particular, this establishes an infinite number of pairs of consecutive primes of separation ${O(1)}$.

The dependence of constants between ${k_0}$ and ${\theta}$ given by the Goldston-Pintz-Yildirim argument is basically of the form ${k_0 \sim (\theta-1/2)^{-2}}$. (UPDATE: as recently observed by Farkas, Pintz, and Revesz, this relationship can be improved to ${k_0 \sim (\theta-1/2)^{-3/2}}$.)

Unfortunately, the Elliott-Halberstam conjecture (which we will state properly below) is only known for ${\theta<1/2}$, an important result known as the Bombieri-Vinogradov theorem. If one uses the Bombieri-Vinogradov theorem instead of the Elliott-Halberstam conjecture, Goldston, Pintz, and Yildirim were still able to show the highly non-trivial result that there were infinitely many pairs ${p_{n+1},p_n}$ of consecutive primes with ${(p_{n+1}-p_n) / \log p_n \rightarrow 0}$ (actually they showed more than this; see e.g. this survey of Soundararajan for details).

Actually, the full strength of the Elliott-Halberstam conjecture is not needed for these results. There is a technical specialisation of the Elliott-Halberstam conjecture which does not presently have a commonly accepted name; I will call it the Motohashi-Pintz-Zhang conjecture ${MPZ[\varpi]}$ in this post, where ${0 < \varpi < 1/4}$ is a parameter. We will define this conjecture more precisely later, but let us remark for now that ${MPZ[\varpi]}$ is a consequence of ${EH[\frac{1}{2}+2\varpi]}$.

We then have the following two theorems. Firstly, we have the following strengthening of Theorem 4:

Theorem 5 (Motohashi-Pintz-Zhang) Suppose that ${MPZ[\varpi]}$ is true for some ${0 < \varpi < 1/4}$. Then ${DHL[k_0,2]}$ is true for some ${k_0}$.

A version of this result (with a slightly different formulation of ${MPZ[\varpi]}$) appears in this paper of Motohashi and Pintz, and in the paper of Zhang, Theorem 5 is proven for the concrete values ${\varpi = 1/1168}$ and ${k_0 = 3,500,000}$. We will supply a self-contained proof of Theorem 5 below the fold, the constants upon those in Zhang’s paper (in particular, for ${\varpi = 1/1168}$, we can take ${k_0}$ as low as ${341,640}$, with further improvements on the way). As with Theorem 4, we have an inverse quadratic relationship ${k_0 \sim \varpi^{-2}}$.

In his paper, Zhang obtained for the first time an unconditional advance on ${MPZ[\varpi]}$:

Theorem 6 (Zhang) ${MPZ[\varpi]}$ is true for all ${0 < \varpi \leq 1/1168}$.

This is a deep result, building upon the work of Fouvry-Iwaniec, Friedlander-Iwaniec and BombieriFriedlanderIwaniec which established results of a similar nature to ${MPZ[\varpi]}$ but simpler in some key respects. We will not discuss this result further here, except to say that they rely on the (higher-dimensional case of the) Weil conjectures, which were famously proven by Deligne using methods from l-adic cohomology. Also, it was believed among at least some experts that the methods of Bombieri, Fouvry, Friedlander, and Iwaniec were not quite strong enough to obtain results of the form ${MPZ[\varpi]}$, making Theorem 6 a particularly impressive achievement.

Combining Theorem 6 with Theorem 5 we obtain ${DHL[k_0,2]}$ for some finite ${k_0}$; Zhang obtains this for ${k_0 = 3,500,000}$ but as detailed below, this can be lowered to ${k_0 = 341,640}$. This in turn gives infinitely many pairs of consecutive primes of separation at most ${H(k_0)}$. Zhang gives a simple argument that bounds ${H(3,500,000)}$ by ${70,000,000}$, giving his famous result that there are infinitely many pairs of primes of separation at most ${70,000,000}$; by being a bit more careful (as discussed in this post) one can lower the upper bound on ${H(3,500,000)}$ to ${57,554,086}$, and if one instead uses the newer value ${k_0 = 341,640}$ for ${k_0}$ one can instead use the bound ${H(341,640) \leq 4,982,086}$. (Many thanks to Scott Morrison for these numerics.) UPDATE: These values are now obsolete; see this web page for the latest bounds.

In this post we would like to give a self-contained proof of both Theorem 4 and Theorem 5, which are both sieve-theoretic results that are mainly elementary in nature. (But, as stated earlier, we will not discuss the deepest new result in Zhang’s paper, namely Theorem 6.) Our presentation will deviate a little bit from the traditional sieve-theoretic approach in a few places. Firstly, there is a portion of the argument that is traditionally handled using contour integration and properties of the Riemann zeta function; we will present a “cheaper” approach (which Ben Green and I used in our papers, e.g. in this one) using Fourier analysis, with the only property used about the zeta function ${\zeta(s)}$ being the elementary fact that blows up like ${\frac{1}{s-1}}$ as one approaches ${1}$ from the right. To deal with the contribution of small primes (which is the source of the singular series ${{\mathfrak G}}$), it will be convenient to use the “${W}$-trick” (introduced in this paper of mine with Ben), passing to a single residue class mod ${W}$ (where ${W}$ is the product of all the small primes) to end up in a situation in which all small primes have been “turned off” which leads to better pseudorandomness properties (for instance, once one eliminates all multiples of small primes, almost all pairs of remaining numbers will be coprime).

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.)