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Many problems and results in analytic prime number theory can be formulated in the following general form: given a collection of (affine-)linear forms ${L_1(n),\dots,L_k(n)}$, none of which is a multiple of any other, find a number ${n}$ such that a certain property ${P( L_1(n),\dots,L_k(n) )}$ of the linear forms ${L_1(n),\dots,L_k(n)}$ are true. For instance:

• For the twin prime conjecture, one can use the linear forms ${L_1(n) := n}$, ${L_2(n) := n+2}$, and the property ${P( L_1(n), L_2(n) )}$ in question is the assertion that ${L_1(n)}$ and ${L_2(n)}$ are both prime.
• For the even Goldbach conjecture, the claim is similar but one uses the linear forms ${L_1(n) := n}$, ${L_2(n) := N-n}$ for some even integer ${N}$.
• For Chen’s theorem, we use the same linear forms ${L_1(n),L_2(n)}$ as in the previous two cases, but now ${P(L_1(n), L_2(n))}$ is the assertion that ${L_1(n)}$ is prime and ${L_2(n)}$ is an almost prime (in the sense that there are at most two prime factors).
• In the recent results establishing bounded gaps between primes, we use the linear forms ${L_i(n) = n + h_i}$ for some admissible tuple ${h_1,\dots,h_k}$, and take ${P(L_1(n),\dots,L_k(n))}$ to be the assertion that at least two of ${L_1(n),\dots,L_k(n)}$ are prime.

For these sorts of results, one can try a sieve-theoretic approach, which can broadly be formulated as follows:

1. First, one chooses a carefully selected sieve weight ${\nu: {\bf N} \rightarrow {\bf R}^+}$, which could for instance be a non-negative function having a divisor sum form

$\displaystyle \nu(n) := \sum_{d_1|L_1(n), \dots, d_k|L_k(n); d_1 \dots d_k \leq x^{1-\varepsilon}} \lambda_{d_1,\dots,d_k}$

for some coefficients ${\lambda_{d_1,\dots,d_k}}$, where ${x}$ is a natural scale parameter. The precise choice of sieve weight is often quite a delicate matter, but will not be discussed here. (In some cases, one may work with multiple sieve weights ${\nu_1, \nu_2, \dots}$.)

2. Next, one uses tools from analytic number theory (such as the Bombieri-Vinogradov theorem) to obtain upper and lower bounds for sums such as

$\displaystyle \sum_n \nu(n) \ \ \ \ \ (1)$

or

$\displaystyle \sum_n \nu(n) 1_{L_i(n) \hbox{ prime}} \ \ \ \ \ (2)$

or more generally of the form

$\displaystyle \sum_n \nu(n) f(L_i(n)) \ \ \ \ \ (3)$

where ${f(L_i(n))}$ is some “arithmetic” function involving the prime factorisation of ${L_i(n)}$ (we will be a bit vague about what this means precisely, but a typical choice of ${f}$ might be a Dirichlet convolution ${\alpha*\beta(L_i(n))}$ of two other arithmetic functions ${\alpha,\beta}$).

3. Using some combinatorial arguments, one manipulates these upper and lower bounds, together with the non-negative nature of ${\nu}$, to conclude the existence of an ${n}$ in the support of ${\nu}$ (or of at least one of the sieve weights ${\nu_1, \nu_2, \dots}$ being considered) for which ${P( L_1(n), \dots, L_k(n) )}$ holds

For instance, in the recent results on bounded gaps between primes, one selects a sieve weight ${\nu}$ for which one has upper bounds on

$\displaystyle \sum_n \nu(n)$

and lower bounds on

$\displaystyle \sum_n \nu(n) 1_{n+h_i \hbox{ prime}}$

so that one can show that the expression

$\displaystyle \sum_n \nu(n) (\sum_{i=1}^k 1_{n+h_i \hbox{ prime}} - 1)$

is strictly positive, which implies the existence of an ${n}$ in the support of ${\nu}$ such that at least two of ${n+h_1,\dots,n+h_k}$ are prime. As another example, to prove Chen’s theorem to find ${n}$ such that ${L_1(n)}$ is prime and ${L_2(n)}$ is almost prime, one uses a variety of sieve weights to produce a lower bound for

$\displaystyle S_1 := \sum_{n \leq x} 1_{L_1(n) \hbox{ prime}} 1_{L_2(n) \hbox{ rough}}$

and an upper bound for

$\displaystyle S_2 := \sum_{z \leq p < x^{1/3}} \sum_{n \leq x} 1_{L_1(n) \hbox{ prime}} 1_{p|L_2(n)} 1_{L_2(n) \hbox{ rough}}$

and

$\displaystyle S_3 := \sum_{n \leq x} 1_{L_1(n) \hbox{ prime}} 1_{L_2(n)=pqr \hbox{ for some } z \leq p \leq x^{1/3} < q \leq r},$

where ${z}$ is some parameter between ${1}$ and ${x^{1/3}}$, and “rough” means that all prime factors are at least ${z}$. One can observe that if ${S_1 - \frac{1}{2} S_2 - \frac{1}{2} S_3 > 0}$, then there must be at least one ${n}$ for which ${L_1(n)}$ is prime and ${L_2(n)}$ is almost prime, since for any rough number ${m}$, the quantity

$\displaystyle 1 - \frac{1}{2} \sum_{z \leq p < x^{1/3}} 1_{p|m} - \frac{1}{2} \sum_{z \leq p \leq x^{1/3} < q \leq r} 1_{m = pqr}$

is only positive when ${m}$ is an almost prime (if ${m}$ has three or more factors, then either it has at least two factors less than ${x^{1/3}}$, or it is of the form ${pqr}$ for some ${p \leq x^{1/3} < q \leq r}$). The upper and lower bounds on ${S_1,S_2,S_3}$ are ultimately produced via asymptotics for expressions of the form (1), (2), (3) for various divisor sums ${\nu}$ and various arithmetic functions ${f}$.

Unfortunately, there is an obstruction to sieve-theoretic techniques working for certain types of properties ${P(L_1(n),\dots,L_k(n))}$, which Zeb Brady and I recently formalised at an AIM workshop this week. To state the result, we recall the Liouville function ${\lambda(n)}$, defined by setting ${\lambda(n) = (-1)^j}$ whenever ${n}$ is the product of exactly ${j}$ primes (counting multiplicity). Define a sign pattern to be an element ${(\epsilon_1,\dots,\epsilon_k)}$ of the discrete cube ${\{-1,+1\}^k}$. Given a property ${P(l_1,\dots,l_k)}$ of ${k}$ natural numbers ${l_1,\dots,l_k}$, we say that a sign pattern ${(\epsilon_1,\dots,\epsilon_k)}$ is forbidden by ${P}$ if there does not exist any natural numbers ${l_1,\dots,l_k}$ obeying ${P(l_1,\dots,l_k)}$ for which

$\displaystyle (\lambda(l_1),\dots,\lambda(l_k)) = (\epsilon_1,\dots,\epsilon_k).$

Example 1 Let ${P(l_1,l_2,l_3)}$ be the property that at least two of ${l_1,l_2,l_3}$ are prime. Then the sign patterns ${(+1,+1,+1)}$, ${(+1,+1,-1)}$, ${(+1,-1,+1)}$, ${(-1,+1,+1)}$ are forbidden, because prime numbers have a Liouville function of ${-1}$, so that ${P(l_1,l_2,l_3)}$ can only occur when at least two of ${\lambda(l_1),\lambda(l_2), \lambda(l_3)}$ are equal to ${-1}$.

Example 2 Let ${P(l_1,l_2)}$ be the property that ${l_1}$ is prime and ${l_2}$ is almost prime. Then the only forbidden sign patterns are ${(+1,+1)}$ and ${(+1,-1)}$.

Example 3 Let ${P(l_1,l_2)}$ be the property that ${l_1}$ and ${l_2}$ are both prime. Then ${(+1,+1), (+1,-1), (-1,+1)}$ are all forbidden sign patterns.

We then have a parity obstruction as soon as ${P}$ has “too many” forbidden sign patterns, in the following (slightly informal) sense:

Claim 1 (Parity obstruction) Suppose ${P(l_1,\dots,l_k)}$ is such that that the convex hull of the forbidden sign patterns of ${P}$ contains the origin. Then one cannot use the above sieve-theoretic approach to establish the existence of an ${n}$ such that ${P(L_1(n),\dots,L_k(n))}$ holds.

Thus for instance, the property in Example 3 is subject to the parity obstruction since ${0}$ is a convex combination of ${(+1,-1)}$ and ${(-1,+1)}$, whereas the properties in Examples 1, 2 are not. One can also check that the property “at least ${j}$ of the ${k}$ numbers ${l_1,\dots,l_k}$ is prime” is subject to the parity obstruction as soon as ${j \geq \frac{k}{2}+1}$. Thus, the largest number of elements of a ${k}$-tuple that one can force to be prime by purely sieve-theoretic methods is ${k/2}$, rounded up.

This claim is not precisely a theorem, because it presumes a certain “Liouville pseudorandomness conjecture” (a very close cousin of the more well known “Möbius pseudorandomness conjecture”) which is a bit difficult to formalise precisely. However, this conjecture is widely believed by analytic number theorists, see e.g. this blog post for a discussion. (Note though that there are scenarios, most notably the “Siegel zero” scenario, in which there is a severe breakdown of this pseudorandomness conjecture, and the parity obstruction then disappears. A typical instance of this is Heath-Brown’s proof of the twin prime conjecture (which would ordinarily be subject to the parity obstruction) under the hypothesis of a Siegel zero.) The obstruction also does not prevent the establishment of an ${n}$ such that ${P(L_1(n),\dots,L_k(n))}$ holds by introducing additional sieve axioms beyond upper and lower bounds on quantities such as (1), (2), (3). The proof of the Friedlander-Iwaniec theorem is a good example of this latter scenario.

Now we give a (slightly nonrigorous) proof of the claim.

Proof: (Nonrigorous) Suppose that the convex hull of the forbidden sign patterns contain the origin. Then we can find non-negative numbers ${p_{\epsilon_1,\dots,\epsilon_k}}$ for sign patterns ${(\epsilon_1,\dots,\epsilon_k)}$, which sum to ${1}$, are non-zero only for forbidden sign patterns, and which have mean zero in the sense that

$\displaystyle \sum_{(\epsilon_1,\dots,\epsilon_k)} p_{\epsilon_1,\dots,\epsilon_k} \epsilon_i = 0$

for all ${i=1,\dots,k}$. By Fourier expansion (or Lagrange interpolation), one can then write ${p_{\epsilon_1,\dots,\epsilon_k}}$ as a polynomial

$\displaystyle p_{\epsilon_1,\dots,\epsilon_k} = 1 + Q( \epsilon_1,\dots,\epsilon_k)$

where ${Q(t_1,\dots,t_k)}$ is a polynomial in ${k}$ variables that is a linear combination of monomials ${t_{i_1} \dots t_{i_r}}$ with ${i_1 < \dots < i_r}$ and ${r \geq 2}$ (thus ${Q}$ has no constant or linear terms, and no monomials with repeated terms). If we now consider the weight function

$\displaystyle w(n) := 1 + Q( \lambda(L_1(n)), \dots, \lambda(L_k(n)) )$

then ${w}$ is non-negative, is supported solely on ${n}$ for which ${(\lambda(L_1(n)),\dots,\lambda(L_k(n)))}$ is a forbidden pattern, and is equal to ${1}$ plus a linear combination of monomials ${\lambda(L_{i_1}(n)) \dots \lambda(L_{i_r}(n))}$ with ${r>2}$.

The Liouville pseudorandomness principle then predicts that sums of the form

$\displaystyle \sum_n \nu(n) Q( \lambda(L_1(n)), \dots, \lambda(L_k(n)) )$

and

$\displaystyle \sum_n \nu(n) Q( \lambda(L_1(n)), \dots, \lambda(L_k(n)) ) 1_{L_i(n) \hbox{ prime}}$

or more generally

$\displaystyle \sum_n \nu(n) Q( \lambda(L_1(n)), \dots, \lambda(L_k(n)) ) f(L_i(n))$

should be asymptotically negligible; intuitively, the point here is that the prime factorisation of ${L_i(n)}$ should not influence the Liouville function of ${L_j(n)}$, even on the short arithmetic progressions that the divisor sum ${\nu}$ is built out of, and so any monomial ${\lambda(L_{i_1}(n)) \dots \lambda(L_{i_r}(n))}$ occurring in ${Q( \lambda(L_1(n)), \dots, \lambda(L_k(n)) )}$ should exhibit strong cancellation for any of the above sums. If one accepts this principle, then all the expressions (1), (2), (3) should be essentially unchanged when ${\nu(n)}$ is replaced by ${\nu(n) w(n)}$.

Suppose now for sake of contradiction that one could use sieve-theoretic methods to locate an ${n}$ in the support of some sieve weight ${\nu(n)}$ obeying ${P( L_1(n),\dots,L_k(n))}$. Then, by reweighting all sieve weights by the additional multiplicative factor of ${w(n)}$, the same arguments should also be able to locate ${n}$ in the support of ${\nu(n) w(n)}$ for which ${P( L_1(n),\dots,L_k(n))}$ holds. But ${w}$ is only supported on those ${n}$ whose Liouville sign pattern is forbidden, a contradiction. $\Box$

Claim 1 is sharp in the following sense: if the convex hull of the forbidden sign patterns of ${P}$ do not contain the origin, then by the Hahn-Banach theorem (in the hyperplane separation form), there exist real coefficients ${c_1,\dots,c_k}$ such that

$\displaystyle c_1 \epsilon_1 + \dots + c_k \epsilon_k < -c$

for all forbidden sign patterns ${(\epsilon_1,\dots,\epsilon_k)}$ and some ${c>0}$. On the other hand, from Liouville pseudorandomness one expects that

$\displaystyle \sum_n \nu(n) (c_1 \lambda(L_1(n)) + \dots + c_k \lambda(L_k(n)))$

is negligible (as compared against ${\sum_n \nu(n)}$ for any reasonable sieve weight ${\nu}$. We conclude that for some ${n}$ in the support of ${\nu}$, that

$\displaystyle c_1 \lambda(L_1(n)) + \dots + c_k \lambda(L_k(n)) > -c \ \ \ \ \ (4)$

and hence ${(\lambda(L_1(n)),\dots,\lambda(L_k(n)))}$ is not a forbidden sign pattern. This does not actually imply that ${P(L_1(n),\dots,L_k(n))}$ holds, but it does not prevent ${P(L_1(n),\dots,L_k(n))}$ from holding purely from parity considerations. Thus, we do not expect a parity obstruction of the type in Claim 1 to hold when the convex hull of forbidden sign patterns does not contain the origin.

Example 4 Let ${G}$ be a graph on ${k}$ vertices ${\{1,\dots,k\}}$, and let ${P(l_1,\dots,l_k)}$ be the property that one can find an edge ${\{i,j\}}$ of ${G}$ with ${l_i,l_j}$ both prime. We claim that this property is subject to the parity problem precisely when ${G}$ is two-colourable. Indeed, if ${G}$ is two-colourable, then we can colour ${\{1,\dots,k\}}$ into two colours (say, red and green) such that all edges in ${G}$ connect a red vertex to a green vertex. If we then consider the two sign patterns in which all the red vertices have one sign and the green vertices have the opposite sign, these are two forbidden sign patterns which contain the origin in the convex hull, and so the parity problem applies. Conversely, suppose that ${G}$ is not two-colourable, then it contains an odd cycle. Any forbidden sign pattern then must contain more ${+1}$s on this odd cycle than ${-1}$s (since otherwise two of the ${-1}$s are adjacent on this cycle by the pigeonhole principle, and this is not forbidden), and so by convexity any tuple in the convex hull of this sign pattern has a positive sum on this odd cycle. Hence the origin is not in the convex hull, and the parity obstruction does not apply. (See also this previous post for a similar obstruction ultimately coming from two-colourability).

Example 5 An example of a parity-obstructed property (supplied by Zeb Brady) that does not come from two-colourability: we let ${P( l_{\{1,2\}}, l_{\{1,3\}}, l_{\{1,4\}}, l_{\{2,3\}}, l_{\{2,4\}}, l_{\{3,4\}} )}$ be the property that ${l_{A_1},\dots,l_{A_r}}$ are prime for some collection ${A_1,\dots,A_r}$ of pair sets that cover ${\{1,\dots,4\}}$. For instance, this property holds if ${l_{\{1,2\}}, l_{\{3,4\}}}$ are both prime, or if ${l_{\{1,2\}}, l_{\{1,3\}}, l_{\{1,4\}}}$ are all prime, but not if ${l_{\{1,2\}}, l_{\{1,3\}}, l_{\{2,3\}}}$ are the only primes. An example of a forbidden sign pattern is the pattern where ${\{1,2\}, \{2,3\}, \{1,3\}}$ are given the sign ${-1}$, and the other three pairs are given ${+1}$. Averaging over permutations of ${1,2,3,4}$ we see that zero lies in the convex hull, and so this example is blocked by parity. However, there is no sign pattern such that it and its negation are both forbidden, which is another formulation of two-colourability.

Of course, the absence of a parity obstruction does not automatically mean that the desired claim is true. For instance, given an admissible ${5}$-tuple ${h_1,\dots,h_5}$, parity obstructions do not prevent one from establishing the existence of infinitely many ${n}$ such that at least three of ${n+h_1,\dots,n+h_5}$ are prime, however we are not yet able to actually establish this, even assuming strong sieve-theoretic hypotheses such as the generalised Elliott-Halberstam hypothesis. (However, the argument giving (4) does easily give the far weaker claim that there exist infinitely many ${n}$ such that at least three of ${n+h_1,\dots,n+h_5}$ have a Liouville function of ${-1}$.)

Remark 1 Another way to get past the parity problem in some cases is to take advantage of linear forms that are constant multiples of each other (which correlates the Liouville functions to each other). For instance, on GEH we can find two ${E_3}$ numbers (products of exactly three primes) that differ by exactly ${60}$; a direct sieve approach using the linear forms ${n,n+60}$ fails due to the parity obstruction, but instead one can first find ${n}$ such that two of ${n,n+4,n+10}$ are prime, and then among the pairs of linear forms ${(15n,15n+60)}$, ${(6n,6n+60)}$, ${(10n+40,10n+100)}$ one can find a pair of ${E_3}$ numbers that differ by exactly ${60}$. See this paper of Goldston, Graham, Pintz, and Yildirim for more examples of this type.

I thank John Friedlander and Sid Graham for helpful discussions and encouragement.

Two of the most famous open problems in additive prime number theory are the twin prime conjecture and the binary Goldbach conjecture. They have quite similar forms:

• Twin prime conjecture The equation ${p_1 - p_2 = 2}$ has infinitely many solutions with ${p_1,p_2}$ prime.
• Binary Goldbach conjecture The equation ${p_1 + p_2 = N}$ has at least one solution with ${p_1,p_2}$ prime for any given even ${N \geq 4}$.

In view of this similarity, it is not surprising that the partial progress on these two conjectures have tracked each other fairly closely; the twin prime conjecture is generally considered slightly easier than the binary Goldbach conjecture, but broadly speaking any progress made on one of the conjectures has also led to a comparable amount of progress on the other. (For instance, Chen’s theorem has a version for the twin prime conjecture, and a version for the binary Goldbach conjecture.) Also, the notorious parity obstruction is present in both problems, preventing a solution to either conjecture by almost all known methods (see this previous blog post for more discussion).

In this post, I would like to note a divergence from this general principle, with regards to bounded error versions of these two conjectures:

• Twin prime with bounded error The inequalities ${0 < p_1 - p_2 < H}$ has infinitely many solutions with ${p_1,p_2}$ prime for some absolute constant ${H}$.
• Binary Goldbach with bounded error The inequalities ${N \leq p_1+p_2 \leq N+H}$ has at least one solution with ${p_1,p_2}$ prime for any sufficiently large ${N}$ and some absolute constant ${H}$.

The first of these statements is now a well-known theorem of Zhang, and the Polymath8b project hosted on this blog has managed to lower ${H}$ to ${H=246}$ unconditionally, and to ${H=6}$ assuming the generalised Elliott-Halberstam conjecture. However, the second statement remains open; the best result that the Polymath8b project could manage in this direction is that (assuming GEH) at least one of the binary Goldbach conjecture with bounded error, or the twin prime conjecture with no error, had to be true.

All the known proofs of Zhang’s theorem proceed through sieve-theoretic means. Basically, they take as input equidistribution results that control the size of discrepancies such as

$\displaystyle \Delta(f; a\ (q)) := \sum_{x \leq n \leq 2x; n=a\ (q)} f(n) - \frac{1}{\phi(q)} \sum_{x \leq n \leq 2x} f(n) \ \ \ \ \ (1)$

for various congruence classes ${a\ (q)}$ and various arithmetic functions ${f}$, e.g. ${f(n) = \Lambda(n+h_i)}$ (or more generaly ${f(n) = \alpha * \beta(n+h_i)}$ for various ${\alpha,\beta}$). After taking some carefully chosen linear combinations of these discrepancies, and using the trivial positivity lower bound

$\displaystyle a_n \geq 0 \hbox{ for all } n \implies \sum_n a_n \geq 0 \ \ \ \ \ (2)$

one eventually obtains (for suitable ${H}$) a non-trivial lower bound of the form

$\displaystyle \sum_{x \leq n \leq 2x} \nu(n) 1_A(n) > 0$

where ${\nu}$ is some weight function, and ${A}$ is the set of ${n}$ such that there are at least two primes in the interval ${[n,n+H]}$. This implies at least one solution to the inequalities ${0 < p_1 - p_2 < H}$ with ${p_1,p_2 \sim x}$, and Zhang’s theorem follows.

In a similar vein, one could hope to use bounds on discrepancies such as (1) (for ${x}$ comparable to ${N}$), together with the trivial lower bound (2), to obtain (for sufficiently large ${N}$, and suitable ${H}$) a non-trivial lower bound of the form

$\displaystyle \sum_{n \leq N} \nu(n) 1_B(n) > 0 \ \ \ \ \ (3)$

for some weight function ${\nu}$, where ${B}$ is the set of ${n}$ such that there is at least one prime in each of the intervals ${[n,n+H]}$ and ${[N-n-H,n]}$. This would imply the binary Goldbach conjecture with bounded error.

However, the parity obstruction blocks such a strategy from working (for much the same reason that it blocks any bound of the form ${H \leq 4}$ in Zhang’s theorem, as discussed in the Polymath8b paper.) The reason is as follows. The sieve-theoretic arguments are linear with respect to the ${n}$ summation, and as such, any such sieve-theoretic argument would automatically also work in a weighted setting in which the ${n}$ summation is weighted by some non-negative weight ${\omega(n) \geq 0}$. More precisely, if one could control the weighted discrepancies

$\displaystyle \Delta(f\omega; a\ (q)) = \sum_{x \leq n \leq 2x; n=a\ (q)} f(n) \omega(n) - \frac{1}{\phi(q)} \sum_{x \leq n \leq 2x} f(n) \omega(n)$

to essentially the same accuracy as the unweighted discrepancies (1), then thanks to the trivial weighted version

$\displaystyle a_n \geq 0 \hbox{ for all } n \implies \sum_n a_n \omega(n) \geq 0$

of (2), any sieve-theoretic argument that was capable of proving (3) would also be capable of proving the weighted estimate

$\displaystyle \sum_{n \leq N} \nu(n) 1_B(n) \omega(n) > 0. \ \ \ \ \ (4)$

However, (4) may be defeated by a suitable choice of weight ${\omega}$, namely

$\displaystyle \omega(n) := \prod_{i=1}^H (1 + \lambda(n) \lambda(n+i)) \times \prod_{j=0}^H (1 - \lambda(n) \lambda(N-n-j))$

where ${n \mapsto \lambda(n)}$ is the Liouville function, which counts the parity of the number of prime factors of a given number ${n}$. Since ${\lambda(n)^2 = 1}$, one can expand out ${\omega(n)}$ as the sum of ${1}$ and a finite number of other terms, each of which consists of the product of two or more translates (or reflections) of ${\lambda}$. But from the Möbius randomness principle (or its analogue for the Liouville function), such products of ${\lambda}$ are widely expected to be essentially orthogonal to any arithmetic function ${f(n)}$ that is arising from a single multiplicative function such as ${\Lambda}$, even on very short arithmetic progressions. As such, replacing ${1}$ by ${\omega(n)}$ in (1) should have a negligible effect on the discrepancy. On the other hand, in order for ${\omega(n)}$ to be non-zero, ${\lambda(n+i)}$ has to have the same sign as ${\lambda(n)}$ and hence the opposite sign to ${\lambda(N-n-j)}$ cannot simultaneously be prime for any ${0 \leq i,j \leq H}$, and so ${1_B(n) \omega(n)}$ vanishes identically, contradicting (4). This indirectly rules out any modification of the Goldston-Pintz-Yildirim/Zhang method for establishing the binary Goldbach conjecture with bounded error.

The above argument is not watertight, and one could envisage some ways around this problem. One of them is that the Möbius randomness principle could simply be false, in which case the parity obstruction vanishes. A good example of this is the result of Heath-Brown that shows that if there are infinitely many Siegel zeroes (which is a strong violation of the Möbius randomness principle), then the twin prime conjecture holds. Another way around the obstruction is to start controlling the discrepancy (1) for functions ${f}$ that are combinations of more than one multiplicative function, e.g. ${f(n) = \Lambda(n) \Lambda(n+2)}$. However, controlling such functions looks to be at least as difficult as the twin prime conjecture (which is morally equivalent to obtaining non-trivial lower-bounds for ${\sum_{x \leq n \leq 2x} \Lambda(n) \Lambda(n+2)}$). A third option is not to use a sieve-theoretic argument, but to try a different method (e.g. the circle method). However, most other known methods also exhibit linearity in the “${n}$” variable and I would suspect they would be vulnerable to a similar obstruction. (In any case, the circle method specifically has some other difficulties in tackling binary problems, as discussed 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. As the previous post was getting somewhat full, we are rolling the thread over to the current post. We also take the opportunity to correct some errors in the treatment of the truncated GPY sieve from this previous post.

As usual, we let ${x}$ be a large asymptotic parameter, and ${w}$ a sufficiently slowly growing function of ${x}$. Let ${0 < \varpi < 1/4}$ and ${0 < \delta < 1/4+\varpi}$ be such that ${MPZ[\varpi,\delta]}$ holds (see this previous post for a definition of this assertion). We let ${{\mathcal H}}$ be a fixed admissible ${k_0}$-tuple, let ${I := [w,x^\delta]}$, let ${{\mathcal S}_I}$ be the square-free numbers with prime divisors in ${I}$, and consider the truncated GPY sieve

$\displaystyle \nu(n) := \lambda(n)^2$

where

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

where ${R := x^{1/4+\varpi}}$, ${P}$ is the polynomial

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

and ${g: {\bf R} \rightarrow {\bf R}}$ is a fixed smooth function supported on ${[-1,1]}$. As discussed in the previous post, we are interested in obtaining an upper bound of the form

$\displaystyle \sum_{x \leq n \leq 2x} \nu(n) \leq (\alpha+o(1)) (\frac{W}{\phi(W)})^{k_0} \frac{x}{W \log^{k_0} R}$

as well as a lower bound of the form

$\displaystyle \sum_{x \leq n \leq 2x} \nu(n) \theta(n+h) \geq (\beta+o(1)) (\frac{W}{\phi(W)})^{k_0} \frac{x}{W \log^{k_0-1} R}$

for all ${h \in {\mathcal H}}$ (where ${\theta(n) = \log n}$ when ${n}$ is prime and ${\theta(n)=0}$ otherwise), since this will give the conjecture ${DHL[k_0,2]}$ (i.e. infinitely many prime gaps of size at most ${k_0}$) whenever

$\displaystyle 1+4\varpi > \frac{4\alpha}{k_0 \beta}. \ \ \ \ \ (1)$

It turns out we in fact have precise asymptotics

$\displaystyle \sum_{x \leq n \leq 2x} \nu(n) = (\alpha+o(1)) (\frac{W}{\phi(W)})^{k_0} \frac{x}{W \log^{k_0} R} \ \ \ \ \ (2)$

and

$\displaystyle \sum_{x \leq n \leq 2x} \nu(n) \theta(n+h) = (\beta+o(1)) (\frac{W}{\phi(W)})^{k_0} \frac{x}{W \log^{k_0} R} \ \ \ \ \ (3)$

although the exact formulae for ${\alpha,\beta}$ are a little complicated. (The fact that ${\alpha,\beta}$ could be computed exactly was already anticipated in Zhang’s paper; see the remark on page 24.) We proceed as in the previous post. Indeed, from the arguments in that post, (2) is equivalent to

$\displaystyle \sum_{d_1,d_2 \in {\mathcal S}_I} \mu(d_1) g(\frac{\log d_1}{\log R}) \mu(d_2) g(\frac{\log d_2}{\log R}) \frac{k_0^{\Omega([d_1,d_2])}}{[d_1,d_2]} \ \ \ \ \ (4)$

$\displaystyle = (\alpha + o(1)) (\frac{W}{\phi(W)})^{k_0} \log^{-k_0} R$

and (3) is similarly equivalent to

$\displaystyle \sum_{d_1,d_2 \in {\mathcal S}_I} \mu(d_1) g(\frac{\log d_1}{\log R}) \mu(d_2) g(\frac{\log d_2}{\log R}) \frac{(k_0-1)^{\Omega([d_1,d_2])}}{[d_1,d_2]} \ \ \ \ \ (5)$

$\displaystyle = (\beta + o(1)) (\frac{W}{\phi(W)})^{k_0-1} \log^{-k_0+1} R.$

Here ${\Omega(d)}$ is the number of prime factors of ${d}$.

We will work for now with (4), as the treatment of (5) is almost identical.

We would now like to replace the truncated interval ${I = [w,x^\delta]}$ with the untruncated interval ${I \cup J = [w,\infty)}$, where ${J = (x^\delta,\infty)}$. Unfortunately this replacement was not quite done correctly in the previous post, and this will now be corrected here. We first observe that if ${F(d_1,d_2)}$ is any finitely supported function, then by Möbius inversion we have

$\displaystyle \sum_{d_1,d_2 \in {\mathcal S}_I} F(d_1,d_2) = \sum_{d_1,d_2 \in {\mathcal S}_{I \cup J}} F(d_1,d_2) \sum_{a \in {\mathcal S}_J} \mu(a) 1_{a|[d_1,d_2]}.$

Note that ${a|[d_1,d_2]}$ if and only if we have a factorisation ${d_1 = a_1 d'_1}$, ${d_2 = a_2 d'_2}$ with ${[a_1,a_2] = a}$ and ${d'_1 d'_2}$ coprime to ${a_1 a_2}$, and that this factorisation is unique. From this, we see that we may rearrange the previous expression as

$\displaystyle \sum_{a_1,a_2 \in {\mathcal S}_J} \mu( [a_1,a_2] ) \sum_{d'_1,d'_2 \in {\mathcal S}_{I \cup J}: (d'_1 d'_2, a_1 a_2) = 1} F( a_1 d'_1, a_2 d'_2 ).$

Applying this to (4), and relabeling ${d'_1,d'_2}$ as ${d_1,d_2}$, we conclude that the left-hand side of (4) is equal to

$\displaystyle \sum_{a_1,a_2 \in {\mathcal S}_J} \mu( [a_1,a_2] ) \sum_{d_1,d_2 \in {\mathcal S}_{I \cup J}: (d_1d_2,a_1a_2)=1}$

$\displaystyle \mu(a_1d_1) g(\frac{\log a_1d_1}{\log R}) \mu(a_2d_2) g(\frac{\log a_2d_2}{\log R}) \frac{k_0^{\Omega([a_1 d_1,a_2 d_2])}}{[a_1 d_1,a_2 d_2]}$

which may be rearranged as

$\displaystyle \sum_{a_1,a_2 \in {\mathcal S}_J} \frac{\mu( (a_1,a_2) ) k_0^{\Omega([a_1,a_2])}}{[a_1,a_2]} \sum_{d_1,d_2\in {\mathcal S}_{I \cup J}: (d_1d_2,a_1a_2)=1} \ \ \ \ \ (6)$

$\displaystyle \mu(d_1) g(\frac{\log a_1d_1}{\log R}) \mu(d_2) g(\frac{\log a_1 d_2}{\log R}) \frac{k_0^{\Omega([d_1,d_2])}}{[d_1, d_2]}.$

This is almost the same formula that we had in the previous post, except that the Möbius function ${\mu((a_1,a_2))}$ of the greatest common divisor ${(a_1,a_2)}$ of ${a_1,a_2}$ was missing, and also the coprimality condition ${(d_1d_2,a_1a_2)=1}$ was not handled properly in the previous post.

We may now eliminate the condition ${(d_1d_2,a_1a_2)=1}$ as follows. Suppose that there is a prime ${p_* \in J}$ that divides both ${d_1d_2}$ and ${a_1a_2}$. The expression

$\displaystyle \sum_{a_1,a_2 \in {\mathcal S}_J} \frac{k_0^{\Omega([a_1,a_2])}}{[a_1,a_2]} \sum_{d_1,d_2 \in {\mathcal S}_{I \cup J}: p_* | (d_1d_2,a_1a_2)}$

$\displaystyle |g(\frac{\log a_1d_1}{\log R})| |g(\frac{\log a_1 d_2}{\log R})| \frac{k_0^{\Omega([d_1,d_2])}}{[d_1, d_2]}$

can then be bounded by

$\displaystyle \ll \sum_{a_1,a_2} \sum_{d_1,d_2: p_* | (d_1d_2,a_1a_2)} \frac{k_0^{\Omega([a_1,a_2])}}{[a_1,a_2]} \frac{k_0^{\Omega([d_1,d_2])}}{[d_1, d_2]} (a_1 a_2 d_1 d_2)^{-1/\log R}$

which may be factorised as

$\displaystyle \ll \frac{1}{p_*^2} \prod_p (1 + \frac{O(1)}{p^{1+1/\log R}})$

which by Mertens’ theorem (or the simple pole of ${\zeta(s)}$ at ${s=1}$) is

$\displaystyle \ll \frac{\log^{O(1)} R}{p_*^2}.$

Summing over all ${p_* > x^\varpi}$ gives a negligible contribution to (6) for the purposes of (4). Thus we may effectively replace (6) by

$\displaystyle \sum_{a_1,a_2 \in {\mathcal S}_J} \frac{\mu( (a_1,a_2) ) k_0^{\Omega([a_1,a_2])}}{[a_1,a_2]} \sum_{d_1,d_2\in {\mathcal S}_{I \cup J}}$

$\displaystyle \mu(d_1) g(\frac{\log a_1d_1}{\log R}) \mu(d_2) g(\frac{\log a_1 d_2}{\log R}) \frac{k_0^{\Omega([d_1,d_2])}}{[d_1, d_2]}.$

The inner summation can be treated using Proposition 10 of the previous post. We can then reduce (4) to

$\displaystyle \sum_{a_1,a_2 \in {\mathcal S}_J} \frac{\mu( (a_1,a_2) ) k_0^{\Omega([a_1,a_2])}}{[a_1,a_2]} G_{k_0}( \frac{\log a_1}{\log R}, \frac{\log a_2}{\log R} ) = \alpha+o(1) \ \ \ \ \ (7)$

where ${G_{k_0}}$ is the function

$\displaystyle G_{k_0}(t_1,t_2) := \int_0^1 g^{(k_0)}(t+t_1) g^{(k_0)}(t+t_2) \frac{t^{k_0-1}}{(k_0-1)!}\ dt.$

Note that ${G}$ vanishes if ${t_1 \geq 1}$ or ${t_2 \geq 1}$. In practice, we will work with functions ${g}$ in which ${g^{(k_0)}}$ has a definite sign (in our normalisations, ${g^{(k_0)}}$ will be non-positive), making ${G_{k_0}}$ non-negative.

We rewrite the left-hand side of (7) as

$\displaystyle \sum_{a \in {\mathcal S}_J} \frac{k_0^{\Omega(a)}}{a} \sum_{a_1,a_2: [a_1,a_2] = a} \mu((a_1,a_2)) G_{k_0}( \frac{\log a_1}{\log R}, \frac{\log a_2}{\log R} ).$

We may factor ${a = p_1 \ldots p_n}$ for some ${x^\delta < p_1 < \ldots < p_n}$ with ${p_1 \ldots p_n \leq R}$; in particular, ${n < \frac{1 + 4\varpi}{4\delta}}$. The previous expression now becomes

$\displaystyle \sum_{0 \leq n < \frac{1+4\varpi}{4\delta}} k_0^n \sum_{x^\delta < p_1 < \ldots < p_n} \frac{1}{p_1 \ldots p_n}$

$\displaystyle \sum_{\{1,\ldots,n\} = S \cup T} (-1)^{|S \cap T|} G_{k_0}( \sum_{i \in S} \frac{\log p_i}{\log R}, \sum_{j \in T} \frac{\log p_j}{\log R} ).$

Using Mertens’ theorem, we thus conclude an exact formula for ${\alpha}$, and similarly for ${\beta}$:

Proposition 1 (Exact formula) We have

$\displaystyle \alpha = \sum_{0 \leq n < \frac{1+4\varpi}{4\delta}} k_0^n \int_{\frac{4\delta}{1+4\varpi} < t_1 < \ldots < t_n} G_{k_0,n}(t_1,\ldots,t_n) \frac{dt_1 \ldots dt_n}{t_1 \ldots t_n}$

where

$\displaystyle G_{k_0,n}(t_1,\ldots,t_n) := \sum_{\{1,\ldots,n\} = S \cup T} (-1)^{|S \cap T|} G_{k_0}( \sum_{i \in S} t_i, \sum_{j \in T} t_j ).$

Similarly we have

$\displaystyle \beta = \sum_{0 \leq n < \frac{1+4\varpi}{4\delta}} (k_0-1)^n \int_{\frac{4\delta}{1+4\varpi} < t_1 < \ldots < t_n} G_{k_0-1,n}(t_1,\ldots,t_n) \frac{dt_1 \ldots dt_n}{t_1 \ldots t_n}$

where ${G_{k_0-1}}$ and ${G_{k_0-1,n}}$ are defined similarly to ${G_{k_0}}$ and ${G_{k_0,n}}$ by replacing all occurrences of ${k_0}$ with ${k_0-1}$.

These formulae are unwieldy. However if we make some monotonicity hypotheses, namely that ${g^{(k_0-1)}}$ is positive, ${g^{(k_0)}}$ is negative, and ${g^{(k_0+1)}}$ is positive on ${[0,1)}$, then we can get some good estimates on the ${G_{k_0}, G_{k_0-1}}$ (which are now non-negative functions) and hence on ${\alpha,\beta}$. Namely, if ${g^{(k_0)}}$ is negative but increasing then we have

$\displaystyle -g^{(k_0)}(t+t_1) \leq -g^{(k_0)}(\frac{t}{1-t_1})$

for ${0 \leq t_1 < 1}$ and ${t \in [0,1]}$, which implies that

$\displaystyle G_{k_0}(t_1,t_1) \leq (1-t_1)_+^{k_0} G_{k_0}(0,0)$

for any ${t_1 \geq 0}$. A similar argument in fact gives

$\displaystyle G_{k_0}(t_1+t_2,t_1+t_2) \leq (1-t_1)_+^{k_0} G_{k_0}(t_2,t_2)$

for any ${t_1,t_2 \geq 0}$. Iterating this we conclude that

$\displaystyle G_{k_0}(\sum_{i \in S} t_i, \sum_{i \in S} t_i) \leq (\prod_{i \in S} (1-t_i)_+^{k_0}) G_{k_0}(0,0)$

and similarly

$\displaystyle G_{k_0}(\sum_{i \in T} t_i, \sum_{i \in T} t_i) \leq (\prod_{i \in T} (1-t_i)_+^{k_0}) G_{k_0}(0,0).$

From Cauchy-Schwarz we thus have

$\displaystyle G_{k_0}( \sum_{i \in S} t_i, \sum_{i \in T} t_i ) \leq (\prod_{i=1}^n (1 - t_i)_+^{k_0/2}) G_{k_0}(0,0).$

Observe from the binomial formula that of the ${3^n}$ pairs ${(S,T)}$ with ${S \cup T = \{1,\ldots,n\}}$, ${\frac{3^n+1}{2}}$ of them have ${|S \cap T|}$ even, and ${\frac{3^n-1}{2}}$ of them have ${|S \cap T|}$ odd. We thus have

$\displaystyle -\frac{3^n-1}{2} (\prod_{i=1}^n (1 - t_i)_+^{k_0/2}) G_{k_0}(0,0) \leq G_{k_0,n}(t_1,\ldots,t_n) \ \ \ \ \ (8)$

$\displaystyle \leq \frac{3^n+1}{2} (\prod_{i=1}^n (1 - t_i)_+^{k_0/2}) G_{k_0}(0,0).$

We have thus established the upper bound

$\displaystyle \alpha \leq G_{k_0}(0,0) (1 + \kappa) \ \ \ \ \ (9)$

where

$\displaystyle \kappa := \sum_{1 \leq n < \frac{1+4\varpi}{4\delta}} \frac{3^n+1}{2} k_0^n \int_{\frac{4\delta}{1+4\varpi} < t_1 < \ldots < t_n} (\prod_{i=1}^n (1 - t_i)_+^{k_0/2}) \frac{dt_1 \ldots dt_n}{t_1 \ldots t_n}.$

By symmetry we may factorise

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

The expression ${\kappa}$ is explicitly computable for any given ${\varpi,\delta,k_0}$. Following the recent preprint of Pintz, one can get a slightly looser, but cleaner, bound by using the upper bound

$\displaystyle 1-t \leq \exp(-t)$

and so

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

Note that

$\displaystyle \int_{4\delta/(1+4\varpi)}^\infty \exp( - \frac{k_0}{2} t )\ \frac{dt}{t} = \int_1^\infty \exp( - \frac{2k_0 \delta}{1+4\varpi} t)\ \frac{dt}{t}$

$\displaystyle < \int_1^\infty \exp( - \frac{2k_0 \delta}{1+4\varpi} t)\ dt$

$\displaystyle = \frac{1+4\varpi}{2k_0\delta} \exp( - \frac{2k_0 \delta}{1+4\varpi} )$

and hence

$\displaystyle \kappa \leq \tilde \kappa$

where

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

In practice we expect the ${n=1}$ term to dominate, thus we have the heuristic approximation

$\displaystyle \kappa \lessapprox \frac{1+4\varpi}{\delta} \exp( - \frac{2k_0 \delta}{1+4\varpi} ).$

Now we turn to the estimation of ${\beta}$. We have an analogue of (8), namely

$\displaystyle -\frac{3^n-1}{2} (\prod_{i=1}^n (1-t_i)^{(k_0-1)/2}) G_{k_0-1}(0,0) \leq G_{k_0-1,n}(t_1,\ldots,t_n)$

$\displaystyle \leq \frac{3^n+1}{2} (\prod_{i=1}^n (1-t_i)^{(k_0-1)/2}) G_{k_0-1}(0,0).$

But we have an improvment in the lower bound in the ${n=1}$ case, because in this case we have

$\displaystyle G_{k_0-1,n}(t) = G_{k_0-1}(t,0) + G_{k_0-1}(0,t) - G_{k_0-1}(t,t).$

From the positive decreasing nature of ${g^{(k_0-1)}}$ we see that ${G_{k_0-1}(t,t) \leq G_{k_0-1}(t,0)}$ and so ${G_{k_0-1,n}(t)}$ is non-negative and can thus be ignored for the purposes of lower bounds. (There are similar improvements available for higher ${n}$ but this seems to only give negligible improvements and will not be pursued here.) Thus we obtain

$\displaystyle \beta \geq G_{k_0-1}(0,0) (1-\kappa') \ \ \ \ \ (10)$

where

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

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

Estimating ${\kappa'}$ similarly to ${\kappa}$ we conclude that

$\displaystyle \kappa' \leq \tilde \kappa'$

where

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

By (9), (10), we see that the condition (1) is implied by

$\displaystyle (1+4\varpi) (1-\kappa') > \frac{4G_{k_0}(0,0)}{k_0 G_{k_0-1}(0,0)} (1+\kappa).$

By Theorem 14 and Lemma 15 of this previous post, we may take the ratio ${\frac{4G_{k_0}(0,0)}{k_0 G_{k_0-1}(0,0)}}$ to be arbitrarily close to ${\frac{j_{k_0-2}^2}{k_0(k_0-1)}}$. We conclude the following theorem.

Theorem 2 Let ${0 < \varpi < 1/4}$ and ${0 < \delta < 1/4 + \varpi}$ be such that ${MPZ[\varpi,\delta]}$ holds. Let ${k_0 \geq 2}$ be an integer, define

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

and

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

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

and suppose that

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

Then ${DHL[k_0,2]}$ holds.

As noted earlier, we heuristically have

$\displaystyle \tilde \kappa \approx \frac{1+4\varpi}{\delta} \exp( - \frac{2k_0 \delta}{1+4\varpi} )$

and ${\tilde \kappa'}$ is negligible. This constraint is a bit better than the previous condition, in which ${\tilde \kappa'}$ was not present and ${\tilde \kappa}$ was replaced by a quantity roughly of the form ${2 \log(2) k_0 \exp( - \frac{2k_0 \delta}{1+4\varpi})}$.

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 Bombieri-Friedlander-Iwaniec 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 this final set of course notes, we discuss how (a generalisation of) the expansion results obtained in the preceding notes can be used for some nnumber-theoretic applications, and in particular to locate almost primes inside orbits of thin groups, following the work of Bourgain, Gamburd, and Sarnak. We will not attempt here to obtain the sharpest or most general results in this direction, but instead focus on the simplest instances of these results which are still illustrative of the ideas involved.

One of the basic general problems in analytic number theory is to locate tuples of primes of a certain form; for instance, the famous (and still unsolved) twin prime conjecture asserts that there are infinitely many pairs ${(n_1,n_2)}$ in the line ${\{ (n_1,n_2) \in {\bf Z}^2: n_2-n_1=2\}}$ in which both entries are prime. In a similar spirit, one of the Landau conjectures (also still unsolved) asserts that there are infinitely many primes in the set ${\{ n^2+1: n \in {\bf Z} \}}$. The Mersenne prime conjecture (also unsolved) asserts that there are infinitely many primes in the set ${\{ 2^n - 1: n \in {\bf Z} \}}$, and so forth.

More generally, given some explicit subset ${V}$ in ${{\bf R}^d}$ (or ${{\bf C}^d}$, if one wishes), such as an algebraic variety, one can ask the question of whether there are infinitely many integer lattice points ${(n_1,\ldots,n_d)}$ in ${V \cap {\bf Z}^d}$ in which all the coefficients ${n_1,\ldots,n_d}$ are simultaneously prime; let us refer to such points as prime points.

At this level of generality, this problem is impossibly difficult. Indeed, even the much simpler problem of deciding whether the set ${V \cap {\bf Z}^d}$ is non-empty (let alone containing prime points) when ${V}$ is a hypersurface ${\{ x \in {\bf R}^d: P(x) = 0 \}}$ cut out by a polynomial ${P}$ is essentially Hilbert’s tenth problem, which is known to be undecidable in general by Matiyasevich’s theorem. So one needs to restrict attention to a more special class of sets ${V}$, in which the question of finding integer points is not so difficult. One model case is to consider orbits ${V = \Gamma b}$, where ${b \in {\bf Z}^d}$ is a fixed lattice vector and ${\Gamma}$ is some discrete group that acts on ${{\bf Z}^d}$ somehow (e.g. ${\Gamma}$ might be embedded as a subgroup of the special linear group ${SL_d({\bf Z})}$, or on the affine group ${SL_d({\bf Z}) \ltimes {\bf Z}^d}$). In such a situation it is then quite easy to show that ${V = V \cap {\bf Z}^d}$ is large; for instance, ${V}$ will be infinite precisely when the stabiliser of ${b}$ in ${\Gamma}$ has infinite index in ${\Gamma}$.

Even in this simpler setting, the question of determining whether an orbit ${V = \Gamma b}$ contains infinitely prime points is still extremely difficult; indeed the three examples given above of the twin prime conjecture, Landau conjecture, and Mersenne prime conjecture are essentially of this form (possibly after some slight modification of the underlying ring ${{\bf Z}}$, see this paper of Bourgain-Gamburd-Sarnak for details), and are all unsolved (and generally considered well out of reach of current technology). Indeed, the list of non-trivial orbits ${V = \Gamma b}$ which are known to contain infinitely many prime points is quite slim; Euclid’s theorem on the infinitude of primes handles the case ${V = {\bf Z}}$, Dirichlet’s theorem handles infinite arithmetic progressions ${V = a{\bf Z} + r}$, and a somewhat complicated result of Green, Tao, and Ziegler handles “non-degenerate” affine lattices in ${{\bf Z}^d}$ of rank two or more (such as the lattice of length ${d}$ arithmetic progressions), but there are few other positive results known that are not based on the above cases (though we will note the remarkable theorem of Friedlander and Iwaniec that there are infinitely many primes of the form ${a^2+b^4}$, and the related result of Heath-Brown that there are infinitely many primes of the form ${a^3+2b^3}$, as being in a kindred spirit to the above results, though they are not explicitly associated to an orbit of a reasonable action as far as I know).

On the other hand, much more is known if one is willing to replace the primes by the larger set of almost primes – integers with a small number of prime factors (counting multiplicity). Specifically, for any ${r \geq 1}$, let us call an ${r}$-almost prime an integer which is the product of at most ${r}$ primes, and possibly by the unit ${-1}$ as well. Many of the above sorts of questions which are open for primes, are known for ${r}$-almost primes for ${r}$ sufficiently large. For instance, with regards to the twin prime conjecture, it is a result of Chen that there are infinitely many pairs ${p,p+2}$ where ${p}$ is a prime and ${p+2}$ is a ${2}$-almost prime; in a similar vein, it is a result of Iwaniec that there are infinitely many ${2}$-almost primes of the form ${n^2+1}$. On the other hand, it is still open for any fixed ${r}$ whether there are infinitely many Mersenne numbers ${2^n-1}$ which are ${r}$-almost primes. (For the superficially similar situation with the numbers ${2^n+1}$, it is in fact believed (but again unproven) that there are only finitely many ${r}$-almost primes for any fixed ${r}$ (the Fermat prime conjecture).)

The main tool that allows one to count almost primes in orbits is sieve theory. The reason for this lies in the simple observation that in order to ensure that an integer ${n}$ of magnitude at most ${x}$ is an ${r}$-almost prime, it suffices to guarantee that ${n}$ is not divisible by any prime less than ${x^{1/(r+1)}}$. Thus, to create ${r}$-almost primes, one can start with the integers up to some large threshold ${x}$ and remove (or “sieve out”) all the integers that are multiples of any prime ${p}$ less than ${x^{1/(r+1)}}$. The difficulty is then to ensure that a sufficiently non-trivial quantity of integers remain after this process, for the purposes of finding points in the given set ${V}$.

The most basic sieve of this form is the sieve of Eratosthenes, which when combined with the inclusion-exclusion principle gives the Legendre sieve (or exact sieve), which gives an exact formula for quantities such as the number ${\pi(x,z)}$ of natural numbers less than or equal to ${x}$ that are not divisible by any prime less than or equal to a given threshold ${z}$. Unfortunately, when one tries to evaluate this formula, one encounters error terms which grow exponentially in ${z}$, rendering this sieve useful only for very small thresholds ${z}$ (of logarithmic size in ${x}$). To improve the sieve level up to a small power of ${x}$ such as ${x^{1/(r+1)}}$, one has to replace the exact sieve by upper bound sieves and lower bound sieves which only seek to obtain upper or lower bounds on quantities such as ${\pi(x,z)}$, but contain a polynomial number of terms rather than an exponential number. There are a variety of such sieves, with the two most common such sieves being combinatorial sieves (such as the beta sieve), based on various combinatorial truncations of the inclusion-exclusion formula, and the Selberg upper bound sieve, based on upper bounds that are the square of a divisor sum. (There is also the large sieve, which is somewhat different in nature and based on ${L^2}$ almost orthogonality considerations, rather than on any actual sieving, to obtain upper bounds.) We will primarily work with a specific sieve in this notes, namely the beta sieve, and we will not attempt to optimise all the parameters of this sieve (which ultimately means that the almost primality parameter ${r}$ in our results will be somewhat large). For a more detailed study of sieve theory, see the classic text of Halberstam and Richert, or the more recent texts of Iwaniec-Kowalski or of Friedlander-Iwaniec.

Very roughly speaking, the end result of sieve theory is that excepting some degenerate and “exponentially thin” settings (such as those associated with the Mersenne primes), all the orbits which are expected to have a large number of primes, can be proven to at least have a large number of ${r}$-almost primes for some finite ${r}$. (Unfortunately, there is a major obstruction, known as the parity problem, which prevents sieve theory from lowering ${r}$ all the way to ${1}$; see this blog post for more discussion.) One formulation of this principle was established by Bourgain, Gamburd, and Sarnak:

Theorem 1 (Bourgain-Gamburd-Sarnak) Let ${\Gamma}$ be a subgroup of ${SL_2({\bf Z})}$ which is not virtually solvable. Let ${f: {\bf Z}^4 \rightarrow {\bf Z}}$ be a polynomial with integer coefficients obeying the following primitivity condition: for any positive integer ${q}$, there exists ${A \in \Gamma \subset {\bf Z}^4}$ such that ${f(A)}$ is coprime to ${q}$. Then there exists an ${r \geq 1}$ such that there are infinitely many ${A \in \Gamma}$ with ${f(A)}$ non-zero and ${r}$-almost prime.

This is not the strongest version of the Bourgain-Gamburd-Sarnak theorem, but it captures the general flavour of their results. Note that the theorem immediately implies an analogous result for orbits ${\Gamma b \subset {\bf Z}^2}$, in which ${f}$ is now a polynomial from ${{\bf Z}^2}$ to ${{\bf Z}}$, and one uses ${f(Ab)}$ instead of ${f(A)}$. It is in fact conjectured that one can set ${r=1}$ here, but this is well beyond current technology. For the purpose of reaching ${r=1}$, it is very natural to impose the primitivity condition, but as long as one is content with larger values of ${r}$, it is possible to relax the primitivity condition somewhat; see the paper of Bourgain, Gamburd, and Sarnak for more discussion.

By specialising to the polynomial ${f: \begin{pmatrix} a & b \\ c & d \end{pmatrix} \rightarrow abcd}$, we conclude as a corollary that as long as ${\Gamma}$ is primitive in the sense that it contains matrices with all coefficients coprime to ${q}$ for any given ${q}$, then ${\Gamma}$ contains infinitely many matrices whose elements are all ${r}$-almost primes for some ${r}$ depending only on ${\Gamma}$. For further applications of these sorts of results, for instance to Appolonian packings, see the paper of Bourgain, Gamburd, and Sarnak.

It turns out that to prove Theorem 1, the Cayley expansion results in ${SL_2(F_p)}$ from the previous set of notes are not quite enough; one needs a more general Cayley expansion result in ${SL_2({\bf Z}/q{\bf Z})}$ where ${q}$ is square-free but not necessarily prime. The proof of this expansion result uses the same basic methods as in the ${SL_2(F_p)}$ case, but is significantly more complicated technically, and we will only discuss it briefly here. As such, we do not give a complete proof of Theorem 1, but hopefully the portion of the argument presented here is still sufficient to give an impression of the ideas involved.

One of the most fundamental principles in Fourier analysis is the uncertainty principle. It does not have a single canonical formulation, but one typical informal description of the principle is that if a function ${f}$ is restricted to a narrow region of physical space, then its Fourier transform ${\hat f}$ must be necessarily “smeared out” over a broad region of frequency space. Some versions of the uncertainty principle are discussed in this previous blog post.

In this post I would like to highlight a useful instance of the uncertainty principle, due to Hugh Montgomery, which is useful in analytic number theory contexts. Specifically, suppose we are given a complex-valued function ${f: {\bf Z} \rightarrow {\bf C}}$ on the integers. To avoid irrelevant issues at spatial infinity, we will assume that the support ${\hbox{supp}(f) := \{ n \in {\bf Z}: f(n) \neq 0 \}}$ of this function is finite (in practice, we will only work with functions that are supported in an interval ${[M+1,M+N]}$ for some natural numbers ${M,N}$). Then we can define the Fourier transform ${\hat f: {\bf R}/{\bf Z} \rightarrow {\bf C}}$ by the formula

$\displaystyle \hat f(\xi) := \sum_{n \in {\bf Z}} f(n) e(-n\xi)$

where ${e(x) := e^{2\pi i x}}$. (In some literature, the sign in the exponential phase is reversed, but this will make no substantial difference to the arguments below.)

The classical uncertainty principle, in this context, asserts that if ${f}$ is localised in an interval of length ${N}$, then ${\hat f}$ must be “smeared out” at a scale of at least ${1/N}$ (and essentially constant at scales less than ${1/N}$). For instance, if ${f}$ is supported in ${[M+1,M+N]}$, then we have the Plancherel identity

$\displaystyle \int_{{\bf R}/{\bf Z}} |\hat f(\xi)|^2\ d\xi = \| f \|_{\ell^2({\bf Z})}^2 \ \ \ \ \ (1)$

while from the Cauchy-Schwarz inequality we have

$\displaystyle |\hat f(\xi)| \leq N^{1/2} \| f \|_{\ell^2({\bf Z})}$

for each frequency ${\xi}$, and in particular that

$\displaystyle \int_I |\hat f(\xi)|^2\ d\xi \leq N |I| \| f \|_{\ell^2({\bf Z})}^2$

for any arc ${I}$ in the unit circle (with ${|I|}$ denoting the length of ${I}$). In particular, an interval of length significantly less than ${1/N}$ can only capture a fraction of the ${L^2}$ energy of the Fourier transform of ${\hat f}$, which is consistent with the above informal statement of the uncertainty principle.

Another manifestation of the classical uncertainty principle is the large sieve inequality. A particularly nice formulation of this inequality is due independently to Montgomery and Vaughan and Selberg: if ${f}$ is supported in ${[M+1,M+N]}$, and ${\xi_1,\ldots,\xi_J}$ are frequencies in ${{\bf R}/{\bf Z}}$ that are ${\delta}$-separated for some ${\delta>0}$, thus ${\| \xi_i-\xi_j\|_{{\bf R}/{\bf Z}} \geq \delta}$ for all ${1 \leq i (where ${\|\xi\|_{{\bf R}/{\bf Z}}}$ denotes the distance of ${\xi}$ to the origin in ${{\bf R}/{\bf Z}}$), then

$\displaystyle \sum_{j=1}^J |\hat f(\xi_j)|^2 \leq (N + \frac{1}{\delta}) \| f \|_{\ell^2({\bf Z})}^2. \ \ \ \ \ (2)$

The reader is encouraged to see how this inequality is consistent with the Plancherel identity (1) and the intuition that ${\hat f}$ is essentially constant at scales less than ${1/N}$. The factor ${N + \frac{1}{\delta}}$ can in fact be amplified a little bit to ${N + \frac{1}{\delta} - 1}$, which is essentially optimal, by using a neat dilation trick of Paul Cohen, in which one dilates ${[M+1,M+N]}$ to ${[MK+K, MK+NK]}$ (and replaces each frequency ${\xi_j}$ by their ${K^{th}}$ roots), and then sending ${K \rightarrow \infty}$ (cf. the tensor product trick); see this survey of Montgomery for details. But we will not need this refinement here.

In the above instances of the uncertainty principle, the concept of narrow support in physical space was formalised in the Archimedean sense, using the standard Archimedean metric ${d_\infty(n,m) := |n-m|}$ on the integers ${{\bf Z}}$ (in particular, the parameter ${N}$ is essentially the Archimedean diameter of the support of ${f}$). However, in number theory, the Archimedean metric is not the only metric of importance on the integers; the ${p}$-adic metrics play an equally important role; indeed, it is common to unify the Archimedean and ${p}$-adic perspectives together into a unified adelic perspective. In the ${p}$-adic world, the metric balls are no longer intervals, but are instead residue classes modulo some power of ${p}$. Intersecting these balls from different ${p}$-adic metrics together, we obtain residue classes with respect to various moduli ${q}$ (which may be either prime or composite). As such, another natural manifestation of the concept of “narrow support in physical space” is “vanishes on many residue classes modulo ${q}$“. This notion of narrowness is particularly common in sieve theory, when one deals with functions supported on thin sets such as the primes, which naturally tend to avoid many residue classes (particularly if one throws away the first few primes).

In this context, the uncertainty principle is this: the more residue classes modulo ${q}$ that ${f}$ avoids, the more the Fourier transform ${\hat f}$ must spread out along multiples of ${1/q}$. To illustrate a very simple example of this principle, let us take ${q=2}$, and suppose that ${f}$ is supported only on odd numbers (thus it completely avoids the residue class ${0 \mod 2}$). We write out the formulae for ${\hat f(\xi)}$ and ${\hat f(\xi+1/2)}$:

$\displaystyle \hat f(\xi) = \sum_n f(n) e(-n\xi)$

$\displaystyle \hat f(\xi+1/2) = \sum_n f(n) e(-n\xi) e(-n/2).$

If ${f}$ is supported on the odd numbers, then ${e(-n/2)}$ is always equal to ${-1}$ on the support of ${f}$, and so we have ${\hat f(\xi+1/2)=-\hat f(\xi)}$. Thus, whenever ${\hat f}$ has a significant presence at a frequency ${\xi}$, it also must have an equally significant presence at the frequency ${\xi+1/2}$; there is a spreading out across multiples of ${1/2}$. Note that one has a similar effect if ${f}$ was supported instead on the even integers instead of the odd integers.

A little more generally, suppose now that ${f}$ avoids a single residue class modulo a prime ${p}$; for sake of argument let us say that it avoids the zero residue class ${0 \mod p}$, although the situation for the other residue classes is similar. For any frequency ${\xi}$ and any ${j=0,\ldots,p-1}$, one has

$\displaystyle \hat f(\xi+j/p) = \sum_n f(n) e(-n\xi) e(-jn/p).$

From basic Fourier analysis, we know that the phases ${e(-jn/p)}$ sum to zero as ${j}$ ranges from ${0}$ to ${p-1}$ whenever ${n}$ is not a multiple of ${p}$. We thus have

$\displaystyle \sum_{j=0}^{p-1} \hat f(\xi+j/p) = 0.$

In particular, if ${\hat f(\xi)}$ is large, then one of the other ${\hat f(\xi+j/p)}$ has to be somewhat large as well; using the Cauchy-Schwarz inequality, we can quantify this assertion in an ${\ell^2}$ sense via the inequality

$\displaystyle \sum_{j=1}^{p-1} |\hat f(\xi+j/p)|^2 \geq \frac{1}{p-1} |\hat f(\xi)|^2. \ \ \ \ \ (3)$

Let us continue this analysis a bit further. Now suppose that ${f}$ avoids ${\omega(p)}$ residue classes modulo a prime ${p}$, for some ${0 \leq \omega(p) < p}$. (We exclude the case ${\omega(p)=p}$ as it is clearly degenerates by forcing ${f}$ to be identically zero.) Let ${\nu(n)}$ be the function that equals ${1}$ on these residue classes and zero away from these residue classes, then

$\displaystyle \sum_n f(n) e(-n\xi) \nu(n) = 0.$

Using the periodic Fourier transform, we can write

$\displaystyle \nu(n) = \sum_{j=0}^{p-1} c(j) e(-jn/p)$

for some coefficients ${c(j)}$, thus

$\displaystyle \sum_{j=0}^{p-1} \hat f(\xi+j/p) c(j) = 0.$

Some Fourier-analytic computations reveal that

$\displaystyle c(0) = \frac{\omega(p)}{p}$

and

$\displaystyle \sum_{j=0}^{p-1} |c(j)|^2 = \frac{\omega(p)}{p}$

and so after some routine algebra and the Cauchy-Schwarz inequality, we obtain a generalisation of (3):

$\displaystyle \sum_{j=1}^{p-1} |\hat f(\xi+j/p)|^2 \geq \frac{\omega(p)}{p-\omega(p)} |\hat f(\xi)|^2.$

Thus we see that the more residue classes mod ${p}$ we exclude, the more Fourier energy has to disperse along multiples of ${1/p}$. It is also instructive to consider the extreme case ${\omega(p)=p-1}$, in which ${f}$ is supported on just a single residue class ${a \mod p}$; in this case, one clearly has ${\hat f(\xi+j/p) = e(-aj/p) \hat f(\xi)}$, and so ${\hat f}$ spreads its energy completely evenly along multiples of ${1/p}$.

In 1968, Montgomery observed the following useful generalisation of the above calculation to arbitrary modulus:

Proposition 1 (Montgomery’s uncertainty principle) Let ${f: {\bf Z} \rightarrow{\bf C}}$ be a finitely supported function which, for each prime ${p}$, avoids ${\omega(p)}$ residue classes modulo ${p}$ for some ${0 \leq \omega(p) < p}$. Then for each natural number ${q}$, one has

$\displaystyle \sum_{1 \leq a \leq q: (a,q)=1} |\hat f(\xi+\frac{a}{q})|^2 \geq h(q) |\hat f(\xi)|^2$

where ${h(q)}$ is the quantity

$\displaystyle h(q) := \mu(q)^2 \prod_{p|q} \frac{\omega(p)}{p-\omega(p)} \ \ \ \ \ (4)$

where ${\mu}$ is the Möbius function.

We give a proof of this proposition below the fold.

Following the “adelic” philosophy, it is natural to combine this uncertainty principle with the large sieve inequality to take simultaneous advantage of localisation both in the Archimedean sense and in the ${p}$-adic senses. This leads to the following corollary:

Corollary 2 (Arithmetic large sieve inequality) Let ${f: {\bf Z} \rightarrow {\bf C}}$ be a function supported on an interval ${[M+1,M+N]}$ which, for each prime ${p}$, avoids ${\omega(p)}$ residue classes modulo ${p}$ for some ${0 \leq \omega(p) < p}$. Let ${\xi_1,\ldots,\xi_J \in {\bf R}/{\bf Z}}$, and let ${{\mathcal Q}}$ be a finite set of natural numbers. Suppose that the frequencies ${\xi_j + \frac{a}{q}}$ with ${1 \leq j \leq J}$, ${q \in {\mathcal Q}}$, and ${(a,q)=1}$ are ${\delta}$-separated. Then one has

$\displaystyle \sum_{j=1}^J |\hat f(\xi_j)|^2 \leq \frac{N + \frac{1}{\delta}}{\sum_{q \in {\mathcal Q}} h(q)} \| f \|_{\ell^2({\bf Z})}^2$

where ${h(q)}$ was defined in (4).

Indeed, from the large sieve inequality one has

$\displaystyle \sum_{j=1}^J \sum_{q \in {\mathcal Q}} \sum_{1 \leq a \leq q: (a,q)=1} |\hat f(\xi_j+\frac{a}{q})|^2 \leq (N + \frac{1}{\delta}) \| f \|_{\ell^2({\bf Z})}^2$

while from Proposition 1 one has

$\displaystyle \sum_{q \in {\mathcal Q}} \sum_{1 \leq a \leq q: (a,q)=1} |\hat f(\xi_j+\frac{a}{q})|^2 \geq (\sum_{q \in {\mathcal Q}} h(q)) |\hat f(\xi_j)|^2$

whence the claim.

There is a great deal of flexibility in the above inequality, due to the ability to select the set ${{\mathcal Q}}$, the frequencies ${\xi_1,\ldots,\xi_J}$, the omitted classes ${\omega(p)}$, and the separation parameter ${\delta}$. Here is a typical application concerning the original motivation for the large sieve inequality, namely in bounding the size of sets which avoid many residue classes:

Corollary 3 (Large sieve) Let ${A}$ be a set of integers contained in ${[M+1,M+N]}$ which avoids ${\omega(p)}$ residue classes modulo ${p}$ for each prime ${p}$, and let ${R>0}$. Then

$\displaystyle |A| \leq \frac{N+R^2}{G(R)}$

where

$\displaystyle G(R) := \sum_{q \leq R} h(q).$

Proof: We apply Corollary 2 with ${f = 1_A}$, ${J=1}$, ${\delta=1/R^2}$, ${\xi_1=0}$, and ${{\mathcal Q} := \{ q: q \leq R\}}$. The key point is that the Farey sequence of fractions ${a/q}$ with ${q \leq R}$ and ${(a,q)=1}$ is ${1/R^2}$-separated, since

$\displaystyle \| \frac{a}{q}-\frac{a'}{q'} \|_{{\bf R}/{\bf Z}} \geq \frac{1}{qq'} \geq \frac{1}{R^2}$

whenever ${a/q, a'/q'}$ are distinct fractions in this sequence. $\Box$

If, for instance, ${A}$ is the set of all primes in ${[M+1,M+N]}$ larger than ${R}$, then one can set ${\omega(p)=1}$ for all ${p \leq R}$, which makes ${h(q) = \frac{\mu^2(q)}{\phi(q)}}$, where ${\phi}$ is the Euler totient function. It is a classical estimate that

$\displaystyle G(R) = \log R + O(1).$

Using this fact and optimising in ${R}$, we obtain (a special case of) the Brun-Titchmarsh inequality

$\displaystyle \pi(M+N)-\pi(M) \leq (2+o_{N \rightarrow \infty}(1)) \frac{N}{\log N}$

where ${\pi(x)}$ is the prime counting function; a variant of the same argument gives the more general Brun-Titchmarsh inequality

$\displaystyle \pi(M+N;a,q)-\pi(M;a,q) \leq (2+o_{N \rightarrow \infty;q}(1)) \frac{q}{\phi(q)} \frac{N}{\log N}$

for any primitive residue class ${a \mod q}$, where ${\pi(M;a,q)}$ is the number of primes less than or equal to ${M}$ that are congruent to ${a \mod q}$. By performing a more careful optimisation using a slightly sharper version of the large sieve inequality (2) that exploits the irregular spacing of the Farey sequence, Montgomery and Vaughan were able to delete the error term in the Brun-Titchmarsh inequality, thus establishing the very nice inequality

$\displaystyle \pi(M+N;a,q)-\pi(M;a,q) \leq 2 \frac{q}{\phi(q)} \frac{N}{\log N}$

for any natural numbers ${M,N,a,q}$ with ${N>1}$. This is a particularly useful inequality in non-asymptotic analytic number theory (when one wishes to study number theory at explicit orders of magnitude, rather than the number theory of sufficiently large numbers), due to the absence of asymptotic notation.

I recently realised that Corollary 2 also establishes a stronger version of the “restriction theorem for the Selberg sieve” that Ben Green and I proved some years ago (indeed, one can view Corollary 2 as a “restriction theorem for the large sieve”). I’m placing the details below the fold.

This is my final Milliman lecture, in which I talk about the sum-product phenomenon in arithmetic combinatorics, and some selected recent applications of this phenomenon to uniform distribution of exponentials, expander graphs, randomness extractors, and detecting (sieving) almost primes in group orbits, particularly as developed by Bourgain and his co-authors.
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[This post is authored by Emmanuel Kowalski.]

This post may be seen as complementary to the post “The parity problem in sieve theory“. In addition to a survey of another important sieve technique, it might be interesting as a discussion of some of the foundational issues which were discussed in the comments to that post.

Many readers will certainly have heard already of one form or another of the “large sieve inequality”. The name itself is misleading however, and what is meant by this may be something having very little, if anything, to do with sieves. What I will discuss are genuine sieve situations.

The framework I will describe is explained in the preprint arXiv:math.NT/0610021, and in a forthcoming Cambridge Tract. I started looking at this first to have a common setting for the usual large sieve and a “sieve for Frobenius” I had devised earlier to study some arithmetic properties of families of zeta functions over finite fields. Another version of such a sieve was described by Zywina (“The large sieve and Galois representations”, preprint), and his approach was quite helpful in suggesting more general settings than I had considered at first. The latest generalizations more or less took life naturally when looking at new applications, such as discrete groups.

Unfortunately (maybe), there will be quite a bit of notation involved; hopefully, the illustrations related to the classical case of sieving integers to obtain the primes (or other subsets of integers with special multiplicative features) will clarify the general case, and the “new” examples will motivate readers to find yet more interesting applications of sieves.

The parity problem is a notorious problem in sieve theory: this theory was invented in order to count prime patterns of various types (e.g. twin primes), but despite superb success in obtaining upper bounds on the number of such patterns, it has proven to be somewhat disappointing in obtaining lower bounds. [Sieves can also be used to study many other things than primes, of course, but we shall focus only on primes in this post.] Even the task of reproving Euclid’s theorem – that there are infinitely many primes – seems to be extremely difficult to do by sieve theoretic means, unless one of course injects into the theory an estimate at least as strong as Euclid’s theorem (e.g. the prime number theorem). The main obstruction is the parity problem: even assuming such strong hypotheses as the Elliott-Halberstam conjecture (a sort of “super-generalised Riemann Hypothesis” for sieves), sieve theory is largely (but not completely) unable to distinguish numbers with an odd number of prime factors from numbers with an even number of prime factors. This “parity barrier” has been broken for some select patterns of primes by injecting some powerful non-sieve theory methods into the subject, but remains a formidable obstacle in general.

I’ll discuss the parity problem in more detail later in this post, but I want to first discuss how sieves work [drawing in part on some excellent unpublished lecture notes of Iwaniec]; the basic ideas are elementary and conceptually simple, but there are many details and technicalities involved in actually executing these ideas, and which I will try to suppress for sake of exposition.