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Let {{\bf F}_q} be a finite field of order {q = p^n}, and let {C} be an absolutely irreducible smooth projective curve defined over {{\bf F}_q} (and hence over the algebraic closure {k := \overline{{\bf F}_q}} of that field). For instance, {C} could be the projective elliptic curve

\displaystyle C = \{ [x,y,z]: y^2 z = x^3 + ax z^2 + b z^3 \}

in the projective plane {{\bf P}^2 = \{ [x,y,z]: (x,y,z) \neq (0,0,0) \}}, where {a,b \in {\bf F}_q} are coefficients whose discriminant {-16(4a^3+27b^2)} is non-vanishing, which is the projective version of the affine elliptic curve

\displaystyle \{ (x,y): y^2 = x^3 + ax + b \}.

To each such curve {C} one can associate a genus {g}, which we will define later; for instance, elliptic curves have genus {1}. We can also count the cardinality {|C({\bf F}_q)|} of the set {C({\bf F}_q)} of {{\bf F}_q}-points of {C}. The Hasse-Weil bound relates the two:

Theorem 1 (Hasse-Weil bound) {||C({\bf F}_q)| - q - 1| \leq 2g\sqrt{q}}.

The usual proofs of this bound proceed by first establishing a trace formula of the form

\displaystyle |C({\bf F}_{p^n})| = p^n - \sum_{i=1}^{2g} \alpha_i^n + 1 \ \ \ \ \ (1)

 

for some complex numbers {\alpha_1,\dots,\alpha_{2g}} independent of {n}; this is in fact a special case of the Lefschetz-Grothendieck trace formula, and can be interpreted as an assertion that the zeta function associated to the curve {C} is rational. The task is then to establish a bound {|\alpha_i| \leq \sqrt{p}} for all {i=1,\dots,2g}; this (or more precisely, the slightly stronger assertion {|\alpha_i| = \sqrt{p}}) is the Riemann hypothesis for such curves. This can be done either by passing to the Jacobian variety of {C} and using a certain duality available on the cohomology of such varieties, known as Rosati involution; alternatively, one can pass to the product surface {C \times C} and apply the Riemann-Roch theorem for that surface.

In 1969, Stepanov introduced an elementary method (a version of what is now known as the polynomial method) to count (or at least to upper bound) the quantity {|C({\bf F}_q)|}. The method was initially restricted to hyperelliptic curves, but was soon extended to general curves. In particular, Bombieri used this method to give a short proof of the following weaker version of the Hasse-Weil bound:

Theorem 2 (Weak Hasse-Weil bound) If {q} is a perfect square, and {q \geq (g+1)^4}, then {|C({\bf F}_q)| \leq q + (2g+1) \sqrt{q} + 1}.

In fact, the bound on {|C({\bf F}_q)|} can be sharpened a little bit further, as we will soon see.

Theorem 2 is only an upper bound on {|C({\bf F}_q)|}, but there is a Galois-theoretic trick to convert (a slight generalisation of) this upper bound to a matching lower bound, and if one then uses the trace formula (1) (and the “tensor power trick” of sending {n} to infinity to control the weights {\alpha_i}) one can then recover the full Hasse-Weil bound. We discuss these steps below the fold.

I’ve discussed Bombieri’s proof of Theorem 2 in this previous post (in the special case of hyperelliptic curves), but now wish to present the full proof, with some minor simplifications from Bombieri’s original presentation; it is mostly elementary, with the deepest fact from algebraic geometry needed being Riemann’s inequality (a weak form of the Riemann-Roch theorem).

The first step is to reinterpret {|C({\bf F}_q)|} as the number of points of intersection between two curves {C_1,C_2} in the surface {C \times C}. Indeed, if we define the Frobenius endomorphism {\hbox{Frob}_q} on any projective space by

\displaystyle \hbox{Frob}_q( [x_0,\dots,x_n] ) := [x_0^q, \dots, x_n^q]

then this map preserves the curve {C}, and the fixed points of this map are precisely the {{\bf F}_q} points of {C}:

\displaystyle C({\bf F}_q) = \{ z \in C: \hbox{Frob}_q(z) = z \}.

Thus one can interpret {|C({\bf F}_q)|} as the number of points of intersection between the diagonal curve

\displaystyle \{ (z,z): z \in C \}

and the Frobenius graph

\displaystyle \{ (z, \hbox{Frob}_q(z)): z \in C \}

which are copies of {C} inside {C \times C}. But we can use the additional hypothesis that {q} is a perfect square to write this more symmetrically, by taking advantage of the fact that the Frobenius map has a square root

\displaystyle \hbox{Frob}_q = \hbox{Frob}_{\sqrt{q}}^2

with {\hbox{Frob}_{\sqrt{q}}} also preserving {C}. One can then also interpret {|C({\bf F}_q)|} as the number of points of intersection between the curve

\displaystyle C_1 := \{ (z, \hbox{Frob}_{\sqrt{q}}(z)): z \in C \} \ \ \ \ \ (2)

 

and its transpose

\displaystyle C_2 := \{ (\hbox{Frob}_{\sqrt{q}}(w), w): w \in C \}.

Let {k(C \times C)} be the field of rational functions on {C \times C} (with coefficients in {k}), and define {k(C_1)}, {k(C_2)}, and {k(C_1 \cap C_2)} analogously )(although {C_1 \cap C_2} is likely to be disconnected, so {k(C_1 \cap C_2)} will just be a ring rather than a field. We then (morally) have the commuting square

\displaystyle \begin{array}{ccccc} && k(C \times C) && \\ & \swarrow & & \searrow & \\ k(C_1) & & & & k(C_2) \\ & \searrow & & \swarrow & \\ && k(C_1 \cap C_2) && \end{array},

if we ignore the issue that a rational function on, say, {C \times C}, might blow up on all of {C_1} and thus not have a well-defined restriction to {C_1}. We use {\pi_1: k(C \times C) \rightarrow k(C_1)} and {\pi_2: k(C \times C) \rightarrow k(C_2)} to denote the restriction maps. Furthermore, we have obvious isomorphisms {\iota_1: k(C_1) \rightarrow k(C)}, {\iota_2: k(C_2) \rightarrow k(C)} coming from composing with the graphing maps {z \mapsto (z, \hbox{Frob}_{\sqrt{q}}(z))} and {w \mapsto (\hbox{Frob}_{\sqrt{q}}(w), w)}.

The idea now is to find a rational function {f \in k(C \times C)} on the surface {C \times C} of controlled degree which vanishes when restricted to {C_1}, but is non-vanishing (and not blowing up) when restricted to {C_2}. On {C_2}, we thus get a non-zero rational function {f \downharpoonright_{C_2}} of controlled degree which vanishes on {C_1 \cap C_2} – which then lets us bound the cardinality of {C_1 \cap C_2} in terms of the degree of {f \downharpoonright_{C_2}}. (In Bombieri’s original argument, one required vanishing to high order on the {C_1} side, but in our presentation, we have factored out a {\hbox{Frob}_{\sqrt{q}}} term which removes this high order vanishing condition.)

To find this {f}, we will use linear algebra. Namely, we will locate a finite-dimensional subspace {V} of {k(C \times C)} (consisting of certain “controlled degree” rational functions) which projects injectively to {k(C_2)}, but whose projection to {k(C_1)} has strictly smaller dimension than {V} itself. The rank-nullity theorem then forces the existence of a non-zero element {P} of {V} whose projection to {k(C_1)} vanishes, but whose projection to {k(C_2)} is non-zero.

Now we build {V}. Pick a {{\bf F}_q} point {P_\infty} of {C}, which we will think of as being a point at infinity. (For the purposes of proving Theorem 2, we may clearly assume that {C({\bf F}_q)} is non-empty.) Thus {P_\infty} is fixed by {\hbox{Frob}_q}. To simplify the exposition, we will also assume that {P_\infty} is fixed by the square root {\hbox{Frob}_{\sqrt{q}}} of {\hbox{Frob}_q}; in the opposite case when {\hbox{Frob}_{\sqrt{q}}} has order two when acting on {P_\infty}, the argument is essentially the same, but all references to {P_\infty} in the second factor of {C \times C} need to be replaced by {\hbox{Frob}_{\sqrt{q}} P_\infty} (we leave the details to the interested reader).

For any natural number {n}, define {R_n} to be the set of rational functions {f \in k(C)} which are allowed to have a pole of order up to {n} at {P_\infty}, but have no other poles on {C}; note that as we are assuming {C} to be smooth, it is unambiguous what a pole is (and what order it will have). (In the fancier language of divisors and Cech cohomology, we have {R_n = H^0( C, {\mathcal O}_C(-n P_\infty) )}.) The space {R_n} is clearly a vector space over {k}; one can view intuitively as the space of “polynomials” on {C} of “degree” at most {n}. When {n=0}, {R_0} consists just of the constant functions. Indeed, if {f \in R_0}, then the image {f(C)} of {f} avoids {\infty} and so lies in the affine line {k = {\mathbf P}^1 \backslash \{\infty\}}; but as {C} is projective, the image {f(C)} needs to be compact (hence closed) in {{\mathbf P}^1}, and must therefore be a point, giving the claim.

For higher {n \geq 1}, we have the easy relations

\displaystyle \hbox{dim}(R_{n-1}) \leq \hbox{dim}(R_n) \leq \hbox{dim}(R_{n-1})+1. \ \ \ \ \ (3)

 

The former inequality just comes from the trivial inclusion {R_{n-1} \subset R_n}. For the latter, observe that if two functions {f, g} lie in {R_n}, so that they each have a pole of order at most {n} at {P_\infty}, then some linear combination of these functions must have a pole of order at most {n-1} at {P_\infty}; thus {R_{n-1}} has codimension at most one in {R_n}, giving the claim.

From (3) and induction we see that each of the {R_n} are finite dimensional, with the trivial upper bound

\displaystyle \hbox{dim}(R_n) \leq n+1. \ \ \ \ \ (4)

 

Riemann’s inequality complements this with the lower bound

\displaystyle \hbox{dim}(R_n) \geq n+1-g, \ \ \ \ \ (5)

 

thus one has {\hbox{dim}(R_n) = \hbox{dim}(R_{n-1})+1} for all but at most {g} exceptions (in fact, exactly {g} exceptions as it turns out). This is a consequence of the Riemann-Roch theorem; it can be proven from abstract nonsense (the snake lemma) if one defines the genus {g} in a non-standard fashion (as the dimension of the first Cech cohomology {H^1(C)} of the structure sheaf {{\mathcal O}_C} of {C}), but to obtain this inequality with a standard definition of {g} (e.g. as the dimension of the zeroth Cech cohomolgy {H^0(C, \Omega_C^1)} of the line bundle of differentials) requires the more non-trivial tool of Serre duality.

At any rate, now that we have these vector spaces {R_n}, we will define {V \subset k(C \times C)} to be a tensor product space

\displaystyle V = R_\ell \otimes R_m

for some natural numbers {\ell, m \geq 0} which we will optimise in later. That is to say, {V} is spanned by functions of the form {(z,w) \mapsto f(z) g(w)} with {f \in R_\ell} and {g \in R_m}. This is clearly a linear subspace of {k(C \times C)} of dimension {\hbox{dim}(R_\ell) \hbox{dim}(R_m)}, and hence by Rieman’s inequality we have

\displaystyle \hbox{dim}(V) \geq (\ell+1-g) (m+1-g) \ \ \ \ \ (6)

 

if

\displaystyle \ell,m \geq g-1. \ \ \ \ \ (7)

 

Observe that {\iota_1 \circ \pi_1} maps a tensor product {(z,w) \mapsto f(z) g(w)} to a function {z \mapsto f(z) g(\hbox{Frob}_{\sqrt{q}} z)}. If {f \in R_\ell} and {g \in R_m}, then we see that the function {z \mapsto f(z) g(\hbox{Frob}_{\sqrt{q}} z)} has a pole of order at most {\ell+m\sqrt{q}} at {P_\infty}. We conclude that

\displaystyle \iota_1 \circ \pi_1( V ) \subset R_{\ell + m\sqrt{q}} \ \ \ \ \ (8)

 

and in particular by (4)

\displaystyle \hbox{dim}(\pi_1(V)) \leq \ell + m \sqrt{q} + 1 \ \ \ \ \ (9)

 

and similarly

\displaystyle \hbox{dim}(\pi_2(V)) \leq \ell \sqrt{q} + m + 1. \ \ \ \ \ (10)

 

We will choose {m} to be a bit bigger than {\ell}, to make the {\pi_2} image of {V} smaller than that of {\pi_1}. From (6), (10) we see that if we have the inequality

\displaystyle (\ell+1-g) (m+1-g) > \ell \sqrt{q}+m + 1 \ \ \ \ \ (11)

 

(together with (7)) then {\pi_2} cannot be injective.

On the other hand, we have the following basic fact:

Lemma 3 (Injectivity) If

\displaystyle \ell < \sqrt{q}, \ \ \ \ \ (12)

 

then {\pi_1: V \rightarrow \pi_1(V)} is injective.

Proof: From (3), we can find a linear basis {f_1,\dots,f_a} of {R_\ell} such that each of the {f_i} has a distinct order {d_i} of pole at {P_\infty} (somewhere between {0} and {\ell} inclusive). Similarly, we may find a linear basis {g_1,\dots,g_b} of {R_m} such that each of the {g_j} has a distinct order {e_j} of pole at {P_\infty} (somewhere between {0} and {m} inclusive). The functions {z \mapsto f_i(z) g_j(\hbox{Frob}_{\sqrt{q}} z)} then span {\iota_1(\pi_1(V))}, and the order of pole at {P_\infty} is {d_i + \sqrt{q} e_j}. But since {\ell < \sqrt{q}}, these orders are all distinct, and so these functions must be linearly independent. The claim follows. \Box

This gives us the following bound:

Proposition 4 Let {\ell,m} be natural numbers such that (7), (11), (12) hold. Then {|C({\bf F}_q)| \leq \ell + m \sqrt{q}}.

Proof: As {\pi_2} is not injective, we can find {f \in V} with {\pi_2(f)} vanishing. By the above lemma, the function {\iota_1(\pi_1(f))} is then non-zero, but it must also vanish on {\iota_1(C_1 \cap C_2)}, which has cardinality {|C({\bf F}_q)|}. On the other hand, by (8), {\iota_1(\pi_1(f))} has a pole of order at most {\ell+m\sqrt{q}} at {P_\infty} and no other poles. Since the number of poles and zeroes of a rational function on a projective curve must add up to zero, the claim follows. \Box

If {q \geq (g+1)^4}, we may make the explicit choice

\displaystyle m := \sqrt{q}+2g; \quad \ell := \lfloor \frac{g}{g+1} \sqrt{q} \rfloor + g + 1

and a brief calculation then gives Theorem 2. In some cases one can optimise things a bit further. For instance, in the genus zero case {g=0} (e.g. if {C} is just the projective line {{\mathbf P}^1}) one may take {\ell=1, m = \sqrt{q}} and conclude the absolutely sharp bound {|C({\bf F}_q)| \leq q+1} in this case; in the case of the projective line {{\mathbf P}^1}, the function {f} is in fact the very concrete function {f(z,w) := z - w^{\sqrt{q}}}.

Remark 1 When {q = p^{2n+1}} is not a perfect square, one can try to run the above argument using the factorisation {\hbox{Frob}_q = \hbox{Frob}_{p^n} \hbox{Frob}_{p^{n+1}}} instead of {\hbox{Frob}_q = \hbox{Frob}_{\sqrt{q}} \hbox{Frob}_{\sqrt{q}}}. This gives a weaker version of the above bound, of the shape {|C({\bf F}_q)| \leq q + O( \sqrt{p} \sqrt{q} )}. In the hyperelliptic case at least, one can erase this loss by working with a variant of the argument in which one requires {f} to vanish to high order at {C_1}, rather than just to first order; see this survey article of mine for details.

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This is the eighth thread for the Polymath8b project to obtain new bounds for the quantity

\displaystyle  H_m := \liminf_{n \rightarrow\infty} (p_{n+m} - p_n),

either for small values of {m} (in particular {m=1,2}) or asymptotically as {m \rightarrow \infty}. The previous thread may be found here. The currently best known bounds on {H_m} can be found at the wiki page.

The big news since the last thread is that we have managed to obtain the (sieve-theoretically) optimal bound of {H_1 \leq 6} assuming the generalised Elliott-Halberstam conjecture (GEH), which pretty much closes off that part of the story. Unconditionally, our bound on {H_1} is still {H_1 \leq 270}. This bound was obtained using the “vanilla” Maynard sieve, in which the cutoff {F} was supported in the original simplex {\{ t_1+\dots+t_k \leq 1\}}, and only Bombieri-Vinogradov was used. In principle, we can enlarge the sieve support a little bit further now; for instance, we can enlarge to {\{ t_1+\dots+t_k \leq \frac{k}{k-1} \}}, but then have to shrink the J integrals to {\{t_1+\dots+t_{k-1} \leq 1-\epsilon\}}, provided that the marginals vanish for {\{ t_1+\dots+t_{k-1} \geq 1+\epsilon \}}. However, we do not yet know how to numerically work with these expanded problems.

Given the substantial progress made so far, it looks like we are close to the point where we should declare victory and write up the results (though we should take one last look to see if there is any room to improve the {H_1 \leq 270} bounds). There is actually a fair bit to write up:

  • Improvements to the Maynard sieve (pushing beyond the simplex, the epsilon trick, and pushing beyond the cube);
  • Asymptotic bounds for {M_k} and hence {H_m};
  • Explicit bounds for {H_m, m \geq 2} (using the Polymath8a results)
  • {H_1 \leq 270};
  • {H_1 \leq 6} on GEH (and parity obstructions to any further improvement).

I will try to create a skeleton outline of such a paper in the Polymath8 Dropbox folder soon. It shouldn’t be nearly as big as the Polymath8a paper, but it will still be quite sizeable.

There are multiple purposes to this blog post.

The first purpose is to announce the uploading of the paper “New equidistribution estimates of Zhang type, and bounded gaps between primes” by D.H.J. Polymath, which is the main output of the Polymath8a project on bounded gaps between primes, to the arXiv, and to describe the main results of this paper below the fold.

The second purpose is to roll over the previous thread on all remaining Polymath8a-related matters (e.g. updates on the submission status of the paper) to a fresh thread. (Discussion of the ongoing Polymath8b project is however being kept on a separate thread, to try to reduce confusion.)

The final purpose of this post is to coordinate the writing of a retrospective article on the Polymath8 experience, which has been solicited for the Newsletter of the European Mathematical Society. I suppose that this could encompass both the Polymath8a and Polymath8b projects, even though the second one is still ongoing (but I think we will soon be entering the endgame there). I think there would be two main purposes of such a retrospective article. The first one would be to tell a story about the process of conducting mathematical research, rather than just describe the outcome of such research; this is an important aspect of the subject which is given almost no attention in most mathematical writing, and it would be good to be able to capture some sense of this process while memories are still relatively fresh. The other would be to draw some tentative conclusions with regards to what the strengths and weaknesses of a Polymath project are, and how appropriate such a format would be for other mathematical problems than bounded gaps between primes. In my opinion, the bounded gaps problem had some fairly unique features that made it particularly amenable to a Polymath project, such as (a) a high level of interest amongst the mathematical community in the problem; (b) a very focused objective (“improve {H}!”), which naturally provided an obvious metric to measure progress; (c) the modular nature of the project, which allowed for people to focus on one aspect of the problem only, and still make contributions to the final goal; and (d) a very reasonable level of ambition (for instance, we did not attempt to prove the twin prime conjecture, which in my opinion would make a terrible Polymath project at our current level of mathematical technology). This is not an exhaustive list of helpful features of the problem; I would welcome other diagnoses of the project by other participants.

With these two objectives in mind, I propose a format for the retrospective article consisting of a brief introduction to the polymath concept in general and the polymath8 project in particular, followed by a collection of essentially independent contributions by different participants on their own experiences and thoughts. Finally we could have a conclusion section in which we make some general remarks on the polymath project (such as the remarks above). I’ve started a dropbox subfolder for this article (currently in a very skeletal outline form only), and will begin writing a section on my own experiences; other participants are of course encouraged to add their own sections (it is probably best to create separate files for these, and then input them into the main file retrospective.tex, to reduce edit conflicts. If there are participants who wish to contribute but do not currently have access to the Dropbox folder, please email me and I will try to have you added (or else you can supply your thoughts by email, or in the comments to this post; we may have a section for shorter miscellaneous comments from more casual participants, for people who don’t wish to write a lengthy essay on the subject).

As for deadlines, the EMS Newsletter would like a submitted article by mid-April in order to make the June issue, but in the worst case, it will just be held over until the issue after that.

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This is the seventh thread for the Polymath8b project to obtain new bounds for the quantity

\displaystyle  H_m := \liminf_{n \rightarrow\infty} (p_{n+m} - p_n),

either for small values of {m} (in particular {m=1,2}) or asymptotically as {m \rightarrow \infty}. The previous thread may be found here. The currently best known bounds on {H_m} can be found at the wiki page.

The current focus is on improving the upper bound on {H_1} under the assumption of the generalised Elliott-Halberstam conjecture (GEH) from {H_1 \leq 8} to {H_1 \leq 6}. Very recently, we have been able to exploit GEH more fully, leading to a promising new expansion of the sieve support region. The problem now reduces to the following:

Problem 1 Does there exist a (not necessarily convex) polytope {R \subset [0,2]^3} with quantities {0 \leq \varepsilon_1,\varepsilon_2,\varepsilon_3 \leq 1}, and a non-trivial square-integrable function {F: {\bf R}^3 \rightarrow {\bf R}} supported on {R} such that

  • {R + R \subset \{ (x,y,z) \in [0,4]^3: \min(x+y,y+z,z+x) \leq 2 \},}
  • {\int_0^\infty F(x,y,z)\ dx = 0} when {y+z \geq 1+\varepsilon_1};
  • {\int_0^\infty F(x,y,z)\ dy = 0} when {x+z \geq 1+\varepsilon_2};
  • {\int_0^\infty F(x,y,z)\ dz = 0} when {x+y \geq 1+\varepsilon_3};

and such that we have the inequality

\displaystyle  \int_{y+z \leq 1-\varepsilon_1} (\int_{\bf R} F(x,y,z)\ dx)^2\ dy dz

\displaystyle + \int_{z+x \leq 1-\varepsilon_2} (\int_{\bf R} F(x,y,z)\ dy)^2\ dz dx

\displaystyle + \int_{x+y \leq 1-\varepsilon_3} (\int_{\bf R} F(x,y,z)\ dz)^2\ dx dy

\displaystyle  > 2 \int_R F(x,y,z)^2\ dx dy dz?

An affirmative answer to this question will imply {H_1 \leq 6} on GEH. We are “within two percent” of this claim; we cannot quite reach {2} yet, but have got as far as {1.962998}. However, we have not yet fully optimised {F} in the above problem. In particular, the simplex

\displaystyle  R = \{ (x,y,z) \in [0,2]^3: x+y+z \leq 3/2 \}

is now available, and should lead to some noticeable improvement in the numerology.

There is also a very slim chance that the twin prime conjecture is now provable on GEH. It would require an affirmative solution to the following problem:

Problem 2 Does there exist a (not necessarily convex) polytope {R \subset [0,2]^2} with quantities {0 \leq \varepsilon_1,\varepsilon_2 \leq 1}, and a non-trivial square-integrable function {F: {\bf R}^2 \rightarrow {\bf R}} supported on {R} such that

  • {R + R \subset \{ (x,y) \in [0,4]^2: \min(x,y) \leq 2 \}}

    \displaystyle  = [0,2] \times [0,4] \cup [0,4] \times [0,2],

  • {\int_0^\infty F(x,y)\ dx = 0} when {y \geq 1+\varepsilon_1};
  • {\int_0^\infty F(x,y)\ dy = 0} when {x \geq 1+\varepsilon_2};

and such that we have the inequality

\displaystyle  \int_{y \leq 1-\varepsilon_1} (\int_{\bf R} F(x,y)\ dx)^2\ dy

\displaystyle + \int_{x \leq 1-\varepsilon_2} (\int_{\bf R} F(x,y)\ dy)^2\ dx

\displaystyle  > 2 \int_R F(x,y)^2\ dx dy?

We suspect that the answer to this question is negative, but have not formally ruled it out yet.

For the rest of this post, I will justify why positive answers to these sorts of variational problems are sufficient to get bounds on {H_1} (or more generally {H_m}).

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This is the fourth thread for the Polymath8b project to obtain new bounds for the quantity

\displaystyle  H_m := \liminf_{n \rightarrow\infty} (p_{n+m} - p_n),

either for small values of {m} (in particular {m=1,2}) or asymptotically as {m \rightarrow \infty}. The previous thread may be found here. The currently best known bounds on {H_m} are:

  • (Maynard) Assuming the Elliott-Halberstam conjecture, {H_1 \leq 12}.
  • (Polymath8b, tentative) {H_1 \leq 272}. Assuming Elliott-Halberstam, {H_2 \leq 272}.
  • (Polymath8b, tentative) {H_2 \leq 429{,}822}. Assuming Elliott-Halberstam, {H_4 \leq 493{,}408}.
  • (Polymath8b, tentative) {H_3 \leq 26{,}682{,}014}. (Presumably a comparable bound also holds for {H_6} on Elliott-Halberstam, but this has not been computed.)
  • (Polymath8b) {H_m \leq \exp( 3.817 m )} for sufficiently large {m}. Assuming Elliott-Halberstam, {H_m \ll m e^{2m}} for sufficiently large {m}.

While the {H_1} bound on the Elliott-Halberstam conjecture has not improved since the start of the Polymath8b project, there is reason to hope that it will soon fall, hopefully to {8}. This is because we have begun to exploit more fully the fact that when using “multidimensional Selberg-GPY” sieves of the form

\displaystyle  \nu(n) := \sigma_{f,k}(n)^2

with

\displaystyle  \sigma_{f,k}(n) := \sum_{d_1|n+h_1,\dots,d_k|n+h_k} \mu(d_1) \dots \mu(d_k) f( \frac{\log d_1}{\log R},\dots,\frac{\log d_k}{\log R}),

where {R := x^{\theta/2}}, it is not necessary for the smooth function {f: [0,+\infty)^k \rightarrow {\bf R}} to be supported on the simplex

\displaystyle {\cal R}_k := \{ (t_1,\dots,t_k)\in [0,1]^k: t_1+\dots+t_k \leq 1\},

but can in fact be allowed to range on larger sets. First of all, {f} may instead be supported on the slightly larger polytope

\displaystyle {\cal R}'_k := \{ (t_1,\dots,t_k)\in [0,1]^k: t_1+\dots+t_{j-1}+t_{j+1}+\dots+t_k \leq 1

\displaystyle  \hbox{ for all } j=1,\dots,k\}.

However, it turns out that more is true: given a sufficiently general version of the Elliott-Halberstam conjecture {EH[\theta]} at the given value of {\theta}, one may work with functions {f} supported on more general domains {R}, so long as the sumset {R+R := \{ t+t': t,t'\in R\}} is contained in the non-convex region

\displaystyle  \bigcup_{j=1}^k \{ (t_1,\dots,t_k)\in [0,\frac{2}{\theta}]^k: t_1+\dots+t_{j-1}+t_{j+1}+\dots+t_k \leq 2 \} \cup \frac{2}{\theta} \cdot {\cal R}_k, \ \ \ \ \ (1)

and also provided that the restriction

\displaystyle  (t_1,\dots,t_{j-1},t_{j+1},\dots,t_k) \mapsto f(t_1,\dots,t_{j-1},0,t_{j+1},\dots,t_k) \ \ \ \ \ (2)

is supported on the simplex

\displaystyle {\cal R}_{k-1} := \{ (t_1,\dots,t_{j-1},t_{j+1},\dots,t_k)\in [0,1]^{k-1}:

\displaystyle t_1+\dots+t_{j-1}+t_{j+1}+\dots t_k \leq 1\}.

More precisely, if {f} is a smooth function, not identically zero, with the above properties for some {R}, and the ratio

\displaystyle  \sum_{j=1}^k \int_{{\cal R}_{k-1}} f_{1,\dots,j-1,j+1,\dots,k}(t_1,\dots,t_{j-1},0,t_{j+1},\dots,t_k)^2 \ \ \ \ \ (3)

\displaystyle dt_1 \dots dt_{j-1} dt_{j+1} \dots dt_k

\displaystyle  / \int_R f_{1,\dots,k}^2(t_1,\dots,t_k)\ dt_1 \dots dt_k

is larger than {\frac{2m}{\theta}}, then the claim {DHL[k,m+1]} holds (assuming {EH[\theta]}), and in particular {H_m \leq H(k)}.

I’ll explain why one can do this below the fold. Taking this for granted, we can rewrite this criterion in terms of the mixed derivative {F := f_{1,\dots,k}}, the upshot being that if one can find a smooth function {F} supported on {R} that obeys the vanishing marginal conditions

\displaystyle  \int F( t_1,\dots,t_k )\ dt_j = 0

whenever {1 \leq j \leq k} and {t_1+\dots+t_{j-1}+t_{j+1}+\dots+t_k > 1}, and the ratio

\displaystyle  \frac{\sum_{j=1}^k J_k^{(j)}(F)}{I_k(F)} \ \ \ \ \ (4)

is larger than {\frac{2m}{\theta}}, where

\displaystyle  I_k(F) := \int_R F(t_1,\dots,t_k)^2\ dt_1 \dots dt_k

and

\displaystyle  J_k^{(j)}(F) := \int_{{\cal R}_{k-1}} (\int_0^{1/\theta} F(t_1,\dots,t_k)\ dt_j)^2 dt_1 \dots dt_{j-1} dt_{j+1} \dots dt_k

then {DHL[k,m+1]} holds. (To equate these two formulations, it is convenient to assume that {R} is a downset, in the sense that whenever {(t_1,\dots,t_k) \in R}, the entire box {[0,t_1] \times \dots \times [0,t_k]} lie in {R}, but one can easily enlarge {R} to be a downset without destroying the containment of {R+R} in the non-convex region (1).) One initially requires {F} to be smooth, but a limiting argument allows one to relax to bounded measurable {F}. (To approximate a rough {F} by a smooth {F} while retaining the required moment conditions, one can first apply a slight dilation and translation so that the marginals of {F} are supported on a slightly smaller version of the simplex {{\cal R}_{k-1}}, and then convolve by a smooth approximation to the identity to make {F} smooth, while keeping the marginals supported on {{\cal R}_{k-1}}.)

We are now exploring various choices of {R} to work with, including the prism

\displaystyle  \{ (t_1,\dots,t_k) \in [0,1/\theta]^k: t_1+\dots+t_{k-1} \leq 1 \}

and the symmetric region

\displaystyle  \{ (t_1,\dots,t_k) \in [0,1/\theta]^k: t_1+\dots+t_k \leq \frac{k}{k-1} \}.

By suitably subdividing these regions into polytopes, and working with piecewise polynomial functions {F} that are polynomial of a specified degree on each subpolytope, one can phrase the problem of optimising (4) as a quadratic program, which we have managed to work with for {k=3}. Extending this program to {k=4}, there is a decent chance that we will be able to obtain {DHL[4,2]} on EH.

We have also been able to numerically optimise {M_k} quite accurately for medium values of {k} (e.g. {k \sim 50}), which has led to improved values of {H_1} without EH. For large {k}, we now also have the asymptotic {M_k=\log k - O(1)} with explicit error terms (details here) which have allowed us to slightly improve the {m=2} numerology, and also to get explicit {m=3} numerology for the first time.

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Mertens’ theorems are a set of classical estimates concerning the asymptotic distribution of the prime numbers:

Theorem 1 (Mertens’ theorems) In the asymptotic limit {x \rightarrow \infty}, we have

\displaystyle  \sum_{p\leq x} \frac{\log p}{p} = \log x + O(1), \ \ \ \ \ (1)

\displaystyle  \sum_{p\leq x} \frac{1}{p} = \log \log x + O(1), \ \ \ \ \ (2)

and

\displaystyle  \sum_{p\leq x} \log(1-\frac{1}{p}) = -\log \log x - \gamma + o(1) \ \ \ \ \ (3)

where {\gamma} is the Euler-Mascheroni constant, defined by requiring that

\displaystyle  1 + \frac{1}{2} + \ldots + \frac{1}{n} = \log n + \gamma + o(1) \ \ \ \ \ (4)

in the limit {n \rightarrow \infty}.

The third theorem (3) is usually stated in exponentiated form

\displaystyle  \prod_{p \leq x} (1-\frac{1}{p}) = \frac{e^{-\gamma}+o(1)}{\log x},

but in the logarithmic form (3) we see that it is strictly stronger than (2), in view of the asymptotic {\log(1-\frac{1}{p}) = -\frac{1}{p} + O(\frac{1}{p^2})}.

Remarkably, these theorems can be proven without the assistance of the prime number theorem

\displaystyle  \sum_{p \leq x} 1 = \frac{x}{\log x} + o( \frac{x}{\log x} ),

which was proven about two decades after Mertens’ work. (But one can certainly use versions of the prime number theorem with good error term, together with summation by parts, to obtain good estimates on the various errors in Mertens’ theorems.) Roughly speaking, the reason for this is that Mertens’ theorems only require control on the Riemann zeta function {\zeta(s) = \sum_{n=1}^\infty \frac{1}{n^s}} in the neighbourhood of the pole at {s=1}, whereas (as discussed in this previous post) the prime number theorem requires control on the zeta function on (a neighbourhood of) the line {\{ 1+it: t \in {\bf R} \}}. Specifically, Mertens’ theorem is ultimately deduced from the Euler product formula

\displaystyle  \zeta(s) = \prod_p (1-\frac{1}{p^s})^{-1}, \ \ \ \ \ (5)

valid in the region {\hbox{Re}(s) > 1} (which is ultimately a Fourier-Dirichlet transform of the fundamental theorem of arithmetic), and following crude asymptotics:

Proposition 2 (Simple pole) For {s} sufficiently close to {1} with {\hbox{Re}(s) > 1}, we have

\displaystyle  \zeta(s) = \frac{1}{s-1} + O(1) \ \ \ \ \ (6)

and

\displaystyle  \zeta'(s) = \frac{-1}{(s-1)^2} + O(1).

Proof: For {s} as in the proposition, we have {\frac{1}{n^s} = \frac{1}{t^s} + O(\frac{1}{n^2})} for any natural number {n} and {n \leq t \leq n+1}, and hence

\displaystyle  \frac{1}{n^s} = \int_n^{n+1} \frac{1}{t^s}\ dt + O( \frac{1}{n^2} ).

Summing in {n} and using the identity {\int_1^\infty \frac{1}{t^s}\ dt = \frac{1}{s-1}}, we obtain the first claim. Similarly, we have

\displaystyle  \frac{-\log n}{n^s} = \int_n^{n+1} \frac{-\log t}{t^s}\ dt + O( \frac{\log n}{n^2} ),

and by summing in {n} and using the identity {\int_1^\infty \frac{-\log t}{t^s}\ dt = \frac{-1}{(s-1)^2}} (the derivative of the previous identity) we obtain the claim. \Box

The first two of Mertens’ theorems (1), (2) are relatively easy to prove, and imply the third theorem (3) except with {\gamma} replaced by an unspecified absolute constant. To get the specific constant {\gamma} requires a little bit of additional effort. From (4), one might expect that the appearance of {\gamma} arises from the refinement

\displaystyle  \zeta(s) = \frac{1}{s-1} + \gamma + O(|s-1|) \ \ \ \ \ (7)

that one can obtain to (6). However, it turns out that the connection is not so much with the zeta function, but with the Gamma function, and specifically with the identity {\Gamma'(1) = - \gamma} (which is of course related to (7) through the functional equation for zeta, but can be proven without any reference to zeta functions). More specifically, we have the following asymptotic for the exponential integral:

Proposition 3 (Exponential integral asymptotics) For sufficiently small {\epsilon}, one has

\displaystyle  \int_\epsilon^\infty \frac{e^{-t}}{t}\ dt = \log \frac{1}{\epsilon} - \gamma + O(\epsilon).

A routine integration by parts shows that this asymptotic is equivalent to the identity

\displaystyle  \int_0^\infty e^{-t} \log t\ dt = -\gamma

which is the identity {\Gamma'(1)=-\gamma} mentioned previously.

Proof: We start by using the identity {\frac{1}{i} = \int_0^1 x^{i-1}\ dx} to express the harmonic series {H_n := 1+\frac{1}{2}+\ldots+\frac{1}{n}} as

\displaystyle  H_n = \int_0^1 1 + x + \ldots + x^{n-1}\ dx

or on summing the geometric series

\displaystyle  H_n = \int_0^1 \frac{1-x^n}{1-x}\ dx.

Since {\int_0^{1-1/n} \frac{1}{1-x} = \log n}, we thus have

\displaystyle  H_n - \log n = \int_0^1 \frac{1_{[1-1/n,1]}(x) - x^n}{1-x}\ dx;

making the change of variables {x = 1-\frac{t}{n}}, this becomes

\displaystyle  H_n - \log n = \int_0^n \frac{1_{[0,1]}(t) - (1-\frac{t}{n})^n}{t}\ dt.

As {n \rightarrow \infty}, {\frac{1_{[0,1]}(t) - (1-\frac{t}{n})^n}{t}} converges pointwise to {\frac{1_{[0,1]}(t) - e^{-t}}{t}} and is pointwise dominated by {O( e^{-t} )}. Taking limits as {n \rightarrow \infty} using dominated convergence, we conclude that

\displaystyle  \gamma = \int_0^\infty \frac{1_{[0,1]}(t) - e^{-t}}{t}\ dt.

or equivalently

\displaystyle  \int_0^\infty \frac{e^{-t} - 1_{[0,\epsilon]}(t)}{t}\ dt = \log \frac{1}{\epsilon} - \gamma.

The claim then follows by bounding the {\int_0^\epsilon} portion of the integral on the left-hand side. \Box

Below the fold I would like to record how Proposition 2 and Proposition 3 imply Theorem 1; the computations are utterly standard, and can be found in most analytic number theory texts, but I wanted to write them down for my own benefit (I always keep forgetting, in particular, how the third of Mertens’ theorems is proven).

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This is the third thread for the Polymath8b project to obtain new bounds for the quantity

\displaystyle H_m := \liminf_{n \rightarrow\infty} (p_{n+m} - p_n),

either for small values of {m} (in particular {m=1,2}) or asymptotically as {m \rightarrow \infty}. The previous thread may be found here. The currently best known bounds on {H_m} are:

  • (Maynard) Assuming the Elliott-Halberstam conjecture, {H_1 \leq 12}.
  • (Polymath8b, tentative) {H_1 \leq 330}. Assuming Elliott-Halberstam, {H_2 \leq 330}.
  • (Polymath8b, tentative) {H_2 \leq 484{,}126}. Assuming Elliott-Halberstam, {H_4 \leq 493{,}408}.
  • (Polymath8b) {H_m \leq \exp( 3.817 m )} for sufficiently large {m}. Assuming Elliott-Halberstam, {H_m \ll e^{2m} m \log m} for sufficiently large {m}.

Much of the current focus of the Polymath8b project is on the quantity

\displaystyle M_k = M_k({\cal R}_k) := \sup_F \frac{\sum_{m=1}^k J_k^{(m)}(F)}{I_k(F)}

where {F} ranges over square-integrable functions on the simplex

\displaystyle {\cal R}_k := \{ (t_1,\ldots,t_k) \in [0,+\infty)^k: t_1+\ldots+t_k \leq 1 \}

with {I_k, J_k^{(m)}} being the quadratic forms

\displaystyle I_k(F) := \int_{{\cal R}_k} F(t_1,\ldots,t_k)^2\ dt_1 \ldots dt_k

and

\displaystyle J_k^{(m)}(F) := \int_{{\cal R}_{k-1}} (\int_0^{1-\sum_{i \neq m} t_i} F(t_1,\ldots,t_k)\ dt_m)^2

\displaystyle dt_1 \ldots dt_{m-1} dt_{m+1} \ldots dt_k.

It was shown by Maynard that one has {H_m \leq H(k)} whenever {M_k > 4m}, where {H(k)} is the narrowest diameter of an admissible {k}-tuple. As discussed in the previous post, we have slight improvements to this implication, but they are currently difficult to implement, due to the need to perform high-dimensional integration. The quantity {M_k} does seem however to be close to the theoretical limit of what the Selberg sieve method can achieve for implications of this type (at the Bombieri-Vinogradov level of distribution, at least); it seems of interest to explore more general sieves, although we have not yet made much progress in this direction.

The best asymptotic bounds for {M_k} we have are

\displaystyle \log k - \log\log\log k + O(1) \leq M_k \leq \frac{k}{k-1} \log k \ \ \ \ \ (1)

 

which we prove below the fold. The upper bound holds for all {k > 1}; the lower bound is only valid for sufficiently large {k}, and gives the upper bound {H_m \ll e^{2m} \log m} on Elliott-Halberstam.

For small {k}, the upper bound is quite competitive, for instance it provides the upper bound in the best values

\displaystyle 1.845 \leq M_4 \leq 1.848

and

\displaystyle 2.001162 \leq M_5 \leq 2.011797

we have for {M_4} and {M_5}. The situation is a little less clear for medium values of {k}, for instance we have

\displaystyle 3.95608 \leq M_{59} \leq 4.148

and so it is not yet clear whether {M_{59} > 4} (which would imply {H_1 \leq 300}). See this wiki page for some further upper and lower bounds on {M_k}.

The best lower bounds are not obtained through the asymptotic analysis, but rather through quadratic programming (extending the original method of Maynard). This has given significant numerical improvements to our best bounds (in particular lowering the {H_1} bound from {600} to {330}), but we have not yet been able to combine this method with the other potential improvements (enlarging the simplex, using MPZ distributional estimates, and exploiting upper bounds on two-point correlations) due to the computational difficulty involved.

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This is the second thread for the Polymath8b project to obtain new bounds for the quantity

\displaystyle  H_m := \liminf_{n \rightarrow\infty} (p_{n+m} - p_n),

either for small values of {m} (in particular {m=1,2}) or asymptotically as {m \rightarrow \infty}. The previous thread may be found here. The currently best known bounds on {H_m} are:

  • (Maynard) {H_1 \leq 600}.
  • (Polymath8b, tentative) {H_2 \leq 484,276}.
  • (Polymath8b, tentative) {H_m \leq \exp( 3.817 m )} for sufficiently large {m}.
  • (Maynard) Assuming the Elliott-Halberstam conjecture, {H_1 \leq 12}, {H_2 \leq 600}, and {H_m \ll m^3 e^{2m}}.

Following the strategy of Maynard, the bounds on {H_m} proceed by combining four ingredients:

  1. Distribution estimates {EH[\theta]} or {MPZ[\varpi,\delta]} for the primes (or related objects);
  2. Bounds for the minimal diameter {H(k)} of an admissible {k}-tuple;
  3. Lower bounds for the optimal value {M_k} to a certain variational problem;
  4. Sieve-theoretic arguments to convert the previous three ingredients into a bound on {H_m}.

Accordingly, the most natural routes to improve the bounds on {H_m} are to improve one or more of the above four ingredients.

Ingredient 1 was studied intensively in Polymath8a. The following results are known or conjectured (see the Polymath8a paper for notation and proofs):

  • (Bombieri-Vinogradov) {EH[\theta]} is true for all {0 < \theta < 1/2}.
  • (Polymath8a) {MPZ[\varpi,\delta]} is true for {\frac{600}{7} \varpi + \frac{180}{7}\delta < 1}.
  • (Polymath8a, tentative) {MPZ[\varpi,\delta]} is true for {\frac{1080}{13} \varpi + \frac{330}{13} \delta < 1}.
  • (Elliott-Halberstam conjecture) {EH[\theta]} is true for all {0 < \theta < 1}.

Ingredient 2 was also studied intensively in Polymath8a, and is more or less a solved problem for the values of {k} of interest (with exact values of {H(k)} for {k \leq 342}, and quite good upper bounds for {H(k)} for {k < 5000}, available at this page). So the main focus currently is on improving Ingredients 3 and 4.

For Ingredient 3, the basic variational problem is to understand the quantity

\displaystyle  M_k({\cal R}_k) := \sup_F \frac{\sum_{m=1}^k J_k^{(m)}(F)}{I_k(F)}

for {F: {\cal R}_k \rightarrow {\bf R}} bounded measurable functions, not identically zero, on the simplex

\displaystyle  {\cal R}_k := \{ (t_1,\ldots,t_k) \in [0,+\infty)^k: t_1+\ldots+t_k \leq 1 \}

with {I_k, J_k^{(m)}} being the quadratic forms

\displaystyle  I_k(F) := \int_{{\cal R}_k} F(t_1,\ldots,t_k)^2\ dt_1 \ldots dt_k

and

\displaystyle  J_k^{(m)}(F) := \int_{{\cal R}_{k-1}} (\int_0^{1-\sum_{i \neq m} t_i} F(t_1,\ldots,t_k)\ dt_i)^2 dt_1 \ldots dt_{m-1} dt_{m+1} \ldots dt_k.

Equivalently, one has

\displaystyle  M_k({\cal R}_k) := \sup_F \frac{\int_{{\cal R}_k} F {\cal L}_k F}{\int_{{\cal R}_k} F^2}

where {{\cal L}_k: L^2({\cal R}_k) \rightarrow L^2({\cal R}_k)} is the positive semi-definite bounded self-adjoint operator

\displaystyle  {\cal L}_k F(t_1,\ldots,t_k) = \sum_{m=1}^k \int_0^{1-\sum_{i \neq m} t_i} F(t_1,\ldots,t_{m-1},s,t_{m+1},\ldots,t_k)\ ds,

so {M_k} is the operator norm of {{\cal L}}. Another interpretation of {M_k({\cal R}_k)} is that the probability that a rook moving randomly in the unit cube {[0,1]^k} stays in simplex {{\cal R}_k} for {n} moves is asymptotically {(M_k({\cal R}_k)/k + o(1))^n}.

We now have a fairly good asymptotic understanding of {M_k({\cal R}_k)}, with the bounds

\displaystyle  \log k - 2 \log\log k -2 \leq M_k({\cal R}_k) \leq \log k + \log\log k + 2

holding for sufficiently large {k}. There is however still room to tighten the bounds on {M_k({\cal R}_k)} for small {k}; I’ll summarise some of the ideas discussed so far below the fold.

For Ingredient 4, the basic tool is this:

Theorem 1 (Maynard) If {EH[\theta]} is true and {M_k({\cal R}_k) > \frac{2m}{\theta}}, then {H_m \leq H(k)}.

Thus, for instance, it is known that {M_{105} > 4} and {H(105)=600}, and this together with the Bombieri-Vinogradov inequality gives {H_1\leq 600}. This result is proven in Maynard’s paper and an alternate proof is also given in the previous blog post.

We have a number of ways to relax the hypotheses of this result, which we also summarise below the fold.

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For each natural number {m}, let {H_m} denote the quantity

\displaystyle  H_m := \liminf_{n \rightarrow\infty} (p_{n+m} - p_n),

where {p_n} denotes the {n\textsuperscript{th}} prime. In other words, {H_m} is the least quantity such that there are infinitely many intervals of length {H_m} that contain {m+1} or more primes. Thus, for instance, the twin prime conjecture is equivalent to the assertion that {H_1 = 2}, and the prime tuples conjecture would imply that {H_m} is equal to the diameter of the narrowest admissible tuple of cardinality {m+1} (thus we conjecturally have {H_1 = 2}, {H_2 = 6}, {H_3 = 8}, {H_4 = 12}, {H_5 = 16}, and so forth; see this web page for further continuation of this sequence).

In 2004, Goldston, Pintz, and Yildirim established the bound {H_1 \leq 16} conditional on the Elliott-Halberstam conjecture, which remains unproven. However, no unconditional finiteness of {H_1} was obtained (although they famously obtained the non-trivial bound {p_{n+1}-p_n = o(\log p_n)}), and even on the Elliot-Halberstam conjecture no finiteness result on the higher {H_m} was obtained either (although they were able to show {p_{n+2}-p_n=o(\log p_n)} on this conjecture). In the recent breakthrough of Zhang, the unconditional bound {H_1 \leq 70,000,000} was obtained, by establishing a weak partial version of the Elliott-Halberstam conjecture; by refining these methods, the Polymath8 project (which I suppose we could retroactively call the Polymath8a project) then lowered this bound to {H_1 \leq 4,680}.

With the very recent preprint of James Maynard, we have the following further substantial improvements:

Theorem 1 (Maynard’s theorem) Unconditionally, we have the following bounds:

  • {H_1 \leq 600}.
  • {H_m \leq C m^3 e^{4m}} for an absolute constant {C} and any {m \geq 1}.

If one assumes the Elliott-Halberstam conjecture, we have the following improved bounds:

  • {H_1 \leq 12}.
  • {H_2 \leq 600}.
  • {H_m \leq C m^3 e^{2m}} for an absolute constant {C} and any {m \geq 1}.

The final conclusion {H_m \leq C m^3 e^{2m}} on Elliott-Halberstam is not explicitly stated in Maynard’s paper, but follows easily from his methods, as I will describe below the fold. (At around the same time as Maynard’s work, I had also begun a similar set of calculations concerning {H_m}, but was only able to obtain the slightly weaker bound {H_m \leq C \exp( C m )} unconditionally.) In the converse direction, the prime tuples conjecture implies that {H_m} should be comparable to {m \log m}. Granville has also obtained the slightly weaker explicit bound {H_m \leq e^{8m+5}} for any {m \geq 1} by a slight modification of Maynard’s argument.

The arguments of Maynard avoid using the difficult partial results on (weakened forms of) the Elliott-Halberstam conjecture that were established by Zhang and then refined by Polymath8; instead, the main input is the classical Bombieri-Vinogradov theorem, combined with a sieve that is closer in spirit to an older sieve of Goldston and Yildirim, than to the sieve used later by Goldston, Pintz, and Yildirim on which almost all subsequent work is based.

The aim of the Polymath8b project is to obtain improved bounds on {H_1, H_2}, and higher values of {H_m}, either conditional on the Elliott-Halberstam conjecture or unconditional. The likeliest routes for doing this are by optimising Maynard’s arguments and/or combining them with some of the results from the Polymath8a project. This post is intended to be the first research thread for that purpose. To start the ball rolling, I am going to give below a presentation of Maynard’s results, with some minor technical differences (most significantly, I am using the Goldston-Pintz-Yildirim variant of the Selberg sieve, rather than the traditional “elementary Selberg sieve” that is used by Maynard (and also in the Polymath8 project), although it seems that the numerology obtained by both sieves is essentially the same). An alternate exposition of Maynard’s work has just been completed also by Andrew Granville.

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I’ve just uploaded to the arXiv my article “Algebraic combinatorial geometry: the polynomial method in arithmetic combinatorics, incidence combinatorics, and number theory“, submitted to the new journal “EMS surveys in the mathematical sciences“.  This is the first draft of a survey article on the polynomial method – a technique in combinatorics and number theory for controlling a relevant set of points by comparing it with the zero set of a suitably chosen polynomial, and then using tools from algebraic geometry (e.g. Bezout’s theorem) on that zero set. As such, the method combines algebraic geometry with combinatorial geometry, and could be viewed as the philosophy of a combined field which I dub “algebraic combinatorial geometry”.   There is also an important extension of this method when one is working overthe reals, in which methods from algebraic topology (e.g. the ham sandwich theorem and its generalisation to polynomials), and not just algebraic geometry, come into play also.

The polynomial method has been used independently many times in mathematics; for instance, it plays a key role in the proof of Baker’s theorem in transcendence theory, or Stepanov’s method in giving an elementary proof of the Riemann hypothesis for finite fields over curves; in combinatorics, the nullstellenatz of Alon is also another relatively early use of the polynomial method.  More recently, it underlies Dvir’s proof of the Kakeya conjecture over finite fields and Guth and Katz’s near-complete solution to the Erdos distance problem in the plane, and can be used to give a short proof of the Szemeredi-Trotter theorem.  One of the aims of this survey is to try to present all of these disparate applications of the polynomial method in a somewhat unified context; my hope is that there will eventually be a systematic foundation for algebraic combinatorial geometry which naturally contains all of these different instances the polynomial method (and also suggests new instances to explore); but the field is unfortunately not at that stage of maturity yet.

This is something of a first draft, so comments and suggestions are even more welcome than usual.  (For instance, I have already had my attention drawn to some additional uses of the polynomial method in the literature that I was not previously aware of.)

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