A truncated elementary Selberg sieve of Pintz

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

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

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

For any fixed natural number , define an admissible -tuple to be a fixed tuple of distinct integers which avoids at least one residue class modulo for each prime . Our objective is to obtain the following conjecture for as small a value of the parameter as possible:

Conjecture 1 () Let be a fixed admissible -tuple. Then there exist infinitely many translates of that contain at least two primes.

The twin prime conjecture asserts that holds for as small as , but currently we are only able to establish this result for (see this comment). However, with the new truncated sieve of Pintz described in this post, we expect to be able to lower this threshold somewhat.

In previous posts, we deduced from a technical variant of the Elliot-Halberstam conjecture for certain choices of parameters , . We will use the following formulation of :

Conjecture 2 () Let be a fixed -tuple (not necessarily admissible) for some fixed , and let be a primitive residue class. Then

for any fixed , where , are the square-free integers whose prime factors lie in , and is the quantity

and is the set of congruence classes

and is the polynomial

The conjecture is currently known to hold whenever (see this comment and this confirmation). Actually, we can prove a stronger result than in this regime in a couple ways. Firstly, the congruence classes can be replaced by a more general systetm of congruence classes obeying a certain controlled multiplicity axiom; see this post. Secondly, and more importantly for this post, the requirement that the modulus lies in can be relaxed; see below.

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

Theorem 3 Let , and be such that we have the inequality

where is the first positive zero of the Bessel function , and are the quantities

and

Then implies .

Actually there have been some slight improvements to the quantities ; see the comments to this previous post. However, the main error remains roughly of the order , which limits one from taking too small.

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

Definition 4 Let . A positive integer is said to be -densely divisible if for every , one can find a factor of in the interval . We let denote the set of positive integers that are -densely divisible.

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

We now define to be the same statement as , but with the condition replaced by the weaker condition . The arguments in previous posts then also establish whenever .

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

Theorem 5 Let , , , and be such that

where

and

and

Then implies .

This theorem has rather messy constants, but we can isolate some special cases which are a bit easier to compute with. Setting , we see that vanishes (and the argument below will show that we only need rather than ), and we obtain the following slight improvement of Theorem 3:

Theorem 6 Let , and be such that we have the inequality

where

Then implies .

This is a little better than Theorem 3, because the error has size about , which compares favorably with the error in Theorem 3 which is about . This should already give a “cheap” improvement to our current threshold , though it will fall short of what one would get if one fully optimised over all parameters in the above theorem.

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

We can crudely bound

and then optimise in to obtain

Because of the factor in the integrand for and , we expect the ratio to be of the order of , although one will need some theoretical or numerical estimates on Bessel functions to make this heuristic more precise. Setting to be something like , we get a good bound here as long as , which at current values of is a mild condition.

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

Estimation of the Type III sums

This is the final continuation of the online reading seminar of Zhang’s paper for the polymath8 project. (There are two other continuations; this previous post, which deals with the combinatorial aspects of the second part of Zhang’s paper, and this previous post, that covers the Type I and Type II sums.) The main purpose of this post is to present (and hopefully, to improve upon) the treatment of the final and most innovative of the key estimates in Zhang’s paper, namely the Type III estimate.

The main estimate was already stated as Theorem 17 in the previous post, but we quickly recall the relevant definitions here. As in other posts, we always take to be a parameter going off to infinity, with the usual asymptotic notation associated to this parameter.

Definition 1 (Coefficient sequences) A coefficient sequence is a finitely supported sequence that obeys the bounds

for all , where is the divisor function.

• (i) If is a coefficient sequence and is a primitive residue class, the (signed) discrepancy of in the sequence is defined to be the quantity

  

• (ii) A coefficient sequence is said to be at scale for some if it is supported on an interval of the form .
• (iii) A coefficient sequence at scale is said to be smooth if it takes the form for some smooth function supported on obeying the derivative bounds

  

  for all fixed (note that the implied constant in the notation may depend on ).

For any , let denote the square-free numbers whose prime factors lie in . The main result of this post is then the following result of Zhang:

Theorem 2 (Type III estimate) Let be fixed quantities, and let be quantities such that

and

and

for some fixed . Let be coefficient sequences at scale respectively with smooth. Then for any we have

In fact we have the stronger “pointwise” estimate

for all with and all , and some fixed .

(This is very slightly stronger than previously claimed, in that the condition has been dropped.)

It turns out that Zhang does not exploit any averaging of the factor, and matters reduce to the following:

Theorem 3 (Type III estimate without ) Let be fixed, and let be quantities such that

and

and

for some fixed . Let be smooth coefficient sequences at scales respectively. Then we have

for all and some fixed .

Let us quickly see how Theorem 3 implies Theorem 2. To show (4), it suffices to establish the bound

for all , where denotes a quantity that is independent of (but can depend on other quantities such as ). The left-hand side can be rewritten as

From Theorem 3 we have

where the quantity does not depend on or . Inserting this asymptotic and using crude bounds on (see Lemma 8 of this previous post) we conclude (4) as required (after modifying slightly).

It remains to establish Theorem 3. This is done by a set of tools similar to that used to control the Type I and Type II sums:

• (i) completion of sums;
• (ii) the Weil conjectures and bounds on Ramanujan sums;
• (iii) factorisation of smooth moduli ;
• (iv) the Cauchy-Schwarz and triangle inequalities (Weyl differencing).

The specifics are slightly different though. For the Type I and Type II sums, it was the classical Weil bound on Kloosterman sums that were the key source of power saving; Ramanujan sums only played a minor role, controlling a secondary error term. For the Type III sums, one needs a significantly deeper consequence of the Weil conjectures, namely the estimate of Bombieri and Birch on a three-dimensional variant of a Kloosterman sum. Furthermore, the Ramanujan sums – which are a rare example of sums that actually exhibit better than square root cancellation, thus going beyond even what the Weil conjectures can offer – make a crucial appearance, when combined with the factorisation of the smooth modulus (this new argument is arguably the most original and interesting contribution of Zhang).

A multi-dimensional Szemer\’edi theorem for the primes via a correspondence principle

Tamar Ziegler and I have just uploaded to the arXiv our joint paper “A multi-dimensional Szemerédi theorem for the primes via a correspondence principle“. This paper is related to an earlier result of Ben Green and mine in which we established that the primes contain arbitrarily long arithmetic progressions. Actually, in that paper we proved a more general result:

Theorem 1 (Szemerédi’s theorem in the primes) Let be a subset of the primes of positive relative density, thus . Then contains arbitrarily long arithmetic progressions.

This result was based in part on an earlier paper of Green that handled the case of progressions of length three. With the primes replaced by the integers, this is of course the famous theorem of Szemerédi.

Szemerédi’s theorem has now been generalised in many different directions. One of these is the multidimensional Szemerédi theorem of Furstenberg and Katznelson, who used ergodic-theoretic techniques to show that any dense subset of necessarily contained infinitely many constellations of any prescribed shape. Our main result is to relativise that theorem to the primes as well:

Theorem 2 (Multidimensional Szemerédi theorem in the primes) Let , and let be a subset of the Cartesian power of the primes of positive relative density, thus

Then for any , contains infinitely many “constellations” of the form with and a positive integer.

In the case when is itself a Cartesian product of one-dimensional sets (in particular, if is all of ), this result already follows from Theorem 1, but there does not seem to be a similarly easy argument to deduce the general case of Theorem 2 from previous results. Simultaneously with this paper, an independent proof of Theorem 2 using a somewhat different method has been established by Cook, Maygar, and Titichetrakun.

The result is reminiscent of an earlier result of mine on finding constellations in the Gaussian primes (or dense subsets thereof). That paper followed closely the arguments of my original paper with Ben Green, namely it first enclosed (a W-tricked version of) the primes or Gaussian primes (in a sieve theoretic-sense) by a slightly larger set (or more precisely, a weight function ) of almost primes or almost Gaussian primes, which one could then verify (using methods closely related to the sieve-theoretic methods in the ongoing Polymath8 project) to obey certain pseudorandomness conditions, known as the linear forms condition and the correlation condition. Very roughly speaking, these conditions assert statements of the following form: if is a randomly selected integer, then the events of simultaneously being an almost prime (or almost Gaussian prime) are approximately independent for most choices of . Once these conditions are satisfied, one can then run a transference argument (initially based on ergodic-theory methods, but nowadays there are simpler transference results based on the Hahn-Banach theorem, due to Gowers and Reingold-Trevisan-Tulsiani-Vadhan) to obtain relative Szemerédi-type theorems from their absolute counterparts.

However, when one tries to adapt these arguments to sets such as , a new difficulty occurs: the natural analogue of the almost primes would be the Cartesian square of the almost primes – pairs whose entries are both almost primes. (Actually, for technical reasons, one does not work directly with a set of almost primes, but would instead work with a weight function such as that is concentrated on a set such as , but let me ignore this distinction for now.) However, this set does not enjoy as many pseudorandomness conditions as one would need for a direct application of the transference strategy to work. More specifically, given any fixed , and random , the four events

do not behave independently (as they would if were replaced for instance by the Gaussian almost primes), because any three of these events imply the fourth. This blocks the transference strategy for constellations which contain some right-angles to them (e.g. constellations of the form ) as such constellations soon turn into rectangles such as the one above after applying Cauchy-Schwarz a few times. (But a few years ago, Cook and Magyar showed that if one restricted attention to constellations which were in general position in the sense that any coordinate hyperplane contained at most one element in the constellation, then this obstruction does not occur and one can establish Theorem 2 in this case through the transference argument.) It’s worth noting that very recently, Conlon, Fox, and Zhao have succeeded in removing of the pseudorandomness conditions (namely the correlation condition) from the transference principle, leaving only the linear forms condition as the remaining pseudorandomness condition to be verified, but unfortunately this does not completely solve the above problem because the linear forms condition also fails for (or for weights concentrated on ) when applied to rectangular patterns.

There are now two ways known to get around this problem and establish Theorem 2 in full generality. The approach of Cook, Magyar, and Titichetrakun proceeds by starting with one of the known proofs of the multidimensional Szemerédi theorem – namely, the proof that proceeds through hypergraph regularity and hypergraph removal – and attach pseudorandom weights directly within the proof itself, rather than trying to add the weights to the result of that proof through a transference argument. (A key technical issue is that weights have to be added to all the levels of the hypergraph – not just the vertices and top-order edges – in order to circumvent the failure of naive pseudorandomness.) As one has to modify the entire proof of the multidimensional Szemerédi theorem, rather than use that theorem as a black box, the Cook-Magyar-Titichetrakun argument is lengthier than ours; on the other hand, it is more general and does not rely on some difficult theorems about primes that are used in our paper.

In our approach, we continue to use the multidimensional Szemerédi theorem (or more precisely, the equivalent theorem of Furstenberg and Katznelson concerning multiple recurrence for commuting shifts) as a black box. The difference is that instead of using a transference principle to connect the relative multidimensional Szemerédi theorem we need to the multiple recurrence theorem, we instead proceed by a version of the Furstenberg correspondence principle, similar to the one that connects the absolute multidimensional Szemerédi theorem to the multiple recurrence theorem. I had discovered this approach many years ago in an unpublished note, but had abandoned it because it required an infinite number of linear forms conditions (in contrast to the transference technique, which only needed a finite number of linear forms conditions and (until the recent work of Conlon-Fox-Zhao) a correlation condition). The reason for this infinite number of conditions is that the correspondence principle has to build a probability measure on an entire -algebra; for this, it is not enough to specify the measure of a single set such as , but one also has to specify the measure of “cylinder sets” such as where could be arbitrarily large. The larger gets, the more linear forms conditions one needs to keep the correspondence under control.

With the sieve weights we were using at the time, standard sieve theory methods could indeed provide a finite number of linear forms conditions, but not an infinite number, so my idea was abandoned. However, with my later work with Green and Ziegler on linear equations in primes (and related work on the Mobius-nilsequences conjecture and the inverse conjecture on the Gowers norm), Tamar and I realised that the primes themselves obey an infinite number of linear forms conditions, so one can basically use the primes (or a proxy for the primes, such as the von Mangoldt function ) as the enveloping sieve weight, rather than a classical sieve. Thus my old idea of using the Furstenberg correspondence principle to transfer Szemerédi-type theorems to the primes could actually be realised. In the one-dimensional case, this simply produces a much more complicated proof of Theorem 1 than the existing one; but it turns out that the argument works as well in higher dimensions and yields Theorem 2 relatively painlessly, except for the fact that it needs the results on linear equations in primes, the known proofs of which are extremely lengthy (and also require some of the transference machinery mentioned earlier). The problem of correlations in rectangles is avoided in the correspondence principle approach because one can compensate for such correlations by performing a suitable weighted limit to compute the measure of cylinder sets, with each requiring a different weighted correction. (This may be related to the Cook-Magyar-Titichetrakun strategy of weighting all of the facets of the hypergraph in order to recover pseudorandomness, although our contexts are rather different.)

Estimation of the Type I and Type II sums

This is one of the continuations of the online reading seminar of Zhang’s paper for the polymath8 project. (There are two other continuations; this previous post, which deals with the combinatorial aspects of the second part of Zhang’s paper, and a post to come that covers the Type III sums.) The main purpose of this post is to present (and hopefully, to improve upon) the treatment of two of the three key estimates in Zhang’s paper, namely the Type I and Type II estimates.

The main estimate was already stated as Theorem 16 in the previous post, but we quickly recall the relevant definitions here. As in other posts, we always take to be a parameter going off to infinity, with the usual asymptotic notation associated to this parameter.

Definition 1 (Coefficient sequences) A coefficient sequence is a finitely supported sequence that obeys the bounds

for all , where is the divisor function.

• (i) If is a coefficient sequence and is a primitive residue class, the (signed) discrepancy of in the sequence is defined to be the quantity

  

• (ii) A coefficient sequence is said to be at scale for some if it is supported on an interval of the form .
• (iii) A coefficient sequence at scale is said to obey the Siegel-Walfisz theorem if one has

  

  for any , any fixed , and any primitive residue class .

• (iv) A coefficient sequence at scale is said to be smooth if it takes the form for some smooth function supported on obeying the derivative bounds

  

  for all fixed (note that the implied constant in the notation may depend on .

In Lemma 8 of this previous post we established a collection of “crude estimates” which assert, roughly speaking, that for the purposes of averaged estimates one may ignore the factor in (1) and pretend that was in fact . We shall rely frequently on these “crude estimates” without further citation to that precise lemma.

For any , let denote the square-free numbers whose prime factors lie in .

Definition 2 (Singleton congruence class system) Let . A singleton congruence class system on is a collection of primitive residue classes for each , obeying the Chinese remainder theorem property

whenever are coprime. We say that such a system has controlled multiplicity if the

obeys the estimate

for any fixed and any congruence class with .

The main result of this post is then the following:

Theorem 3 (Type I/II estimate) Let be fixed quantities such that

and

and let be coefficient sequences at scales respectively with

and

with obeying a Siegel-Walfisz theorem. Then for any and any singleton congruence class system with controlled multiplicity we have

The proof of this theorem relies on five basic tools:

• (i) the Bombieri-Vinogradov theorem;
• (ii) completion of sums;
• (iii) and the Weil conjectures;
• (iv) factorisation of smooth moduli ; and
• (v) the Cauchy-Schwarz and triangle inequalities (the dispersion method).

For the purposes of numerics, it is the interplay between (ii), (iii), and (v) that drives the final conditions (7), (8). The Weil conjectures are the primary source of power savings ( for some fixed ) in the argument, but they need to overcome power losses coming from completion of sums, and also each use of Cauchy-Schwarz tends to halve any power savings present in one’s estimates. Naively, one could thus expect to get better estimates by relying more on the Weil conjectures, and less on completion of sums and on Cauchy-Schwarz.

Further analysis of the truncated GPY sieve

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 be a large asymptotic parameter, and a sufficiently slowly growing function of . Let and be such that holds (see this previous post for a definition of this assertion). We let be a fixed admissible -tuple, let , let be the square-free numbers with prime divisors in , and consider the truncated GPY sieve

where

where , is the polynomial

and is a fixed smooth function supported on . As discussed in the previous post, we are interested in obtaining an upper bound of the form

as well as a lower bound of the form

for all (where when is prime and otherwise), since this will give the conjecture (i.e. infinitely many prime gaps of size at most ) whenever

It turns out we in fact have precise asymptotics

and

although the exact formulae for are a little complicated. (The fact that 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

and (3) is similarly equivalent to

Here is the number of prime factors of .

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

We would now like to replace the truncated interval with the untruncated interval , where . Unfortunately this replacement was not quite done correctly in the previous post, and this will now be corrected here. We first observe that if is any finitely supported function, then by Möbius inversion we have

Note that if and only if we have a factorisation , with and coprime to , and that this factorisation is unique. From this, we see that we may rearrange the previous expression as

Applying this to (4), and relabeling as , we conclude that the left-hand side of (4) is equal to

which may be rearranged as

This is almost the same formula that we had in the previous post, except that the Möbius function of the greatest common divisor of was missing, and also the coprimality condition was not handled properly in the previous post.

We may now eliminate the condition as follows. Suppose that there is a prime that divides both and . The expression

can then be bounded by

which may be factorised as

which by Mertens’ theorem (or the simple pole of at ) is

Summing over all gives a negligible contribution to (6) for the purposes of (4). Thus we may effectively replace (6) by

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

where is the function

Note that vanishes if or . In practice, we will work with functions in which has a definite sign (in our normalisations, will be non-positive), making non-negative.

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

We may factor for some with ; in particular, . The previous expression now becomes

Using Mertens’ theorem, we thus conclude an exact formula for , and similarly for :

Proposition 1 (Exact formula) We have

where

Similarly we have

where and are defined similarly to and by replacing all occurrences of with .

These formulae are unwieldy. However if we make some monotonicity hypotheses, namely that is positive, is negative, and is positive on , then we can get some good estimates on the (which are now non-negative functions) and hence on . Namely, if is negative but increasing then we have

for and , which implies that

for any . A similar argument in fact gives

for any . Iterating this we conclude that

and similarly

From Cauchy-Schwarz we thus have

Observe from the binomial formula that of the pairs with , of them have even, and of them have odd. We thus have

We have thus established the upper bound

where

By symmetry we may factorise

The expression is explicitly computable for any given . Following the recent preprint of Pintz, one can get a slightly looser, but cleaner, bound by using the upper bound

and so

Note that

and hence

where

In practice we expect the term to dominate, thus we have the heuristic approximation

Now we turn to the estimation of . We have an analogue of (8), namely

But we have an improvment in the lower bound in the case, because in this case we have

From the positive decreasing nature of we see that and so is non-negative and can thus be ignored for the purposes of lower bounds. (There are similar improvements available for higher but this seems to only give negligible improvements and will not be pursued here.) Thus we obtain

where

Estimating similarly to we conclude that

where

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

By Theorem 14 and Lemma 15 of this previous post, we may take the ratio to be arbitrarily close to . We conclude the following theorem.

Theorem 2 Let and be such that holds. Let be an integer, define

and

and suppose that

Then holds.

As noted earlier, we heuristically have

and is negligible. This constraint is a bit better than the previous condition, in which was not present and was replaced by a quantity roughly of the form .

A combinatorial subset sum problem associated with bounded prime gaps

The purpose of this post is to isolate a combinatorial optimisation problem regarding subset sums; any improvement upon the current known bounds for this problem would lead to numerical improvements for the quantities pursued in the Polymath8 project. (UPDATE: Unfortunately no purely combinatorial improvement is possible, see comments.) We will also record the number-theoretic details of how this combinatorial problem is used in Zhang’s argument establishing bounded prime gaps.

First, some (rough) motivational background, omitting all the number-theoretic details and focusing on the combinatorics. (But readers who just want to see the combinatorial problem can skip the motivation and jump ahead to Lemma 5.) As part of the Polymath8 project we are trying to establish a certain estimate called for as wide a range of as possible. Currently the best result we have is:

Theorem 1 (Zhang’s theorem, numerically optimised) holds whenever .

Enlarging this region would lead to a better value of certain parameters , which in turn control the best bound on asymptotic gaps between consecutive primes. See this previous post for more discussion of this. At present, the best value of is applied by taking sufficiently close to , so improving Theorem 1 in the neighbourhood of this value is particularly desirable.

I’ll state exactly what is below the fold. For now, suffice to say that it involves a certain number-theoretic function, the von Mangoldt function . To prove the theorem, the first step is to use a certain identity (the Heath-Brown identity) to decompose into a lot of pieces, which take the form

for some bounded (in Zhang’s paper never exceeds ) and various weights supported at various scales that multiply up to approximately :

We can write , thus ignoring negligible errors, are non-negative real numbers that add up to :

A key technical feature of the Heath-Brown identity is that the weight associated to sufficiently large values of (e.g. ) are “smooth” in a certain sense; this will be detailed below the fold.

The operation is Dirichlet convolution, which is commutative and associative. We can thus regroup the convolution (1) in a number of ways. For instance, given any partition into disjoint sets , we can rewrite (1) as

where is the convolution of those with , and similarly for .

Zhang’s argument splits into two major pieces, in which certain classes of (1) are established. Cheating a little bit, the following three results are established:

Theorem 2 (Type 0 estimate, informal version) The term (1) gives an acceptable contribution to whenever

for some .

Theorem 3 (Type I/II estimate, informal version) The term (1) gives an acceptable contribution to whenever one can find a partition such that

where is a quantity such that

Theorem 4 (Type III estimate, informal version) The term (1) gives an acceptable contribution to whenever one can find with distinct with

and

The above assertions are oversimplifications; there are some additional minor smallness hypotheses on that are needed but at the current (small) values of under consideration they are not relevant and so will be omitted.

The deduction of Theorem 1 from Theorems 2, 3, 4 is then accomplished from the following, purely combinatorial, lemma:

Lemma 5 (Subset sum lemma) Let be such that

Let be non-negative reals such that

Then at least one of the following statements hold:

• (Type 0) There is such that .
• (Type I/II) There is a partition such that

  

  where is a quantity such that

  

• (Type III) One can find distinct with

  

  and

  

The purely combinatorial question is whether the hypothesis (2) can be relaxed here to a weaker condition. This would allow us to improve the ranges for Theorem 1 (and hence for the values of and alluded to earlier) without needing further improvement on Theorems 2, 3, 4 (although such improvement is also going to be a focus of Polymath8 investigations in the future).

Let us review how this lemma is currently proven. The key sublemma is the following:

Lemma 6 Let , and let be non-negative numbers summing to . Then one of the following three statements hold:

• (Type 0) There is a with .
• (Type I/II) There is a partition such that

  

• (Type III) There exist distinct with and .

Proof: Suppose Type I/II never occurs, then every partial sum is either “small” in the sense that it is less than or equal to , or “large” in the sense that it is greater than or equal to , since otherwise we would be in the Type I/II case either with as is and the complement of , or vice versa.

Call a summand “powerless” if it cannot be used to turn a small partial sum into a large partial sum, thus there are no such that is small and is large. We then split where are the powerless elements and are the powerful elements.

By induction we see that if and is small, then is also small. Thus every sum of powerful summand is either less than or larger than . Since a powerful element must be able to convert a small sum to a large sum (in fact it must be able to convert a small sum of powerful summands to a large sum, by stripping out the powerless summands), we conclude that every powerful element has size greater than . We may assume we are not in Type 0, then every powerful summand is at least and at most . In particular, there have to be at least three powerful summands, otherwise cannot be as large as . As , we have , and we conclude that the sum of any two powerful summands is large (which, incidentally, shows that there are exactly three powerful summands). Taking to be three powerful summands in increasing order we land in Type III.

Now we see how Lemma 6 implies Lemma 5. Let be as in Lemma 5. We take almost as large as we can for the Type I/II case, thus we set

for some sufficiently small . We observe from (2) that we certainly have

and

with plenty of room to spare. We then apply Lemma 6. The Type 0 case of that lemma then implies the Type 0 case of Lemma 5, while the Type I/II case of Lemma 6 also implies the Type I/II case of Lemma 5. Finally, suppose that we are in the Type III case of Lemma 6. Since

we thus have

and so we will be done if

Inserting (3) and taking small enough, it suffices to verify that

but after some computation this is equivalent to (2).

It seems that there is some slack in this computation; some of the conclusions of the Type III case of Lemma 5, in particular, ended up being “wasted”, and it is possible that one did not fully exploit all the partial sums that could be used to create a Type I/II situation. So there may be a way to make improvements through purely combinatorial arguments. (UPDATE: As it turns out, this is sadly not the case: consderation of the case when , , and shows that one cannot obtain any further improvement without actually improving the Type I/II and Type III analysis.)

A technical remark: for the application to Theorem 1, it is possible to enforce a bound on the number of summands in Lemma 5. More precisely, we may assume that is an even number of size at most for any natural number we please, at the cost of adding the additioal constraint to the Type III conclusion. Since is already at least , which is at least , one can safely take , so can be taken to be an even number of size at most , which in principle makes the problem of optimising Lemma 5 a fixed linear programming problem. (Zhang takes , but this appears to be overkill. On the other hand, does not appear to be a parameter that overly influences the final numerical bounds.)

Below the fold I give the number-theoretic details of the combinatorial aspects of Zhang’s argument that correspond to the combinatorial problem described above.

The elementary Selberg sieve and bounded prime gaps

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 going to infinity, and a quantity depending on that goes to infinity sufficiently slowly with , and (the -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 and . Let us first state , which has an integer parameter :

Conjecture 1 () Let be a fixed admissible -tuple. Then there are infinitely many translates of which contain at least two primes.

Zhang was the first to prove a result of this type with . Since then the value of has been lowered substantially; at this time of writing, the current record is .

There are two basic ways known currently to attain this conjecture. The first is to use the Elliott-Halberstam conjecture for some :

Conjecture 2 () One has

for all fixed . Here we use the abbreviation for .

Here of course is the von Mangoldt function and the Euler totient function. It is conjectured that holds for all , but this is currently only known for , an important result known as the Bombieri-Vinogradov theorem.

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

for any , where depends on . 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 be a real number, and let be an integer obeying the inequality

where is the first positive zero of the Bessel function . Then implies .

Note that the right-hand side of (2) is larger than , but tends asymptotically to as . 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 by an easier conjecture for some and , at the cost of degrading the sufficient condition (2) slightly. In our notation, this conjecture takes the following form for each choice of parameters :

Conjecture 4 () Let be a fixed -tuple (not necessarily admissible) for some fixed , and let be a primitive residue class. Then

for any fixed , where , are the square-free integers whose prime factors lie in , and is the quantity

and is the set of congruence classes

and is the polynomial

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

Proposition 5 (EH implies MPZ) Let and . Then implies for any . (In abbreviated form: implies .)

In particular, since is conjecturally true for all , we conjecture to be true for all and .

Proof: Define

then the hypothesis (applied to and and then subtracting) tells us that

for any fixed . From the Chinese remainder theorem and the Siegel-Walfisz theorem we have

for any coprime to (and in particular for ). Since , where is the number of prime divisors of , we can thus bound the left-hand side of (3) by

The contribution of the second term is by standard estimates (see Proposition 8 below). Using the very crude bound

and standard estimates we also have

and the claim now follows from the Cauchy-Schwarz inequality.

In practice, the conjecture is easier to prove than due to the restriction of the residue classes to , and also the restriction of the modulus to -smooth numbers. Zhang proved for any . 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 whenever are such that

The work of Motohashi and Pintz, and later Zhang, implicitly describe arguments that allow one to deduce from provided that is sufficiently large depending on . 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 , , and let be an integer obeying the constraint

where is the quantity

Then implies .

This complicated version of is roughly of size . It is unlikely to be optimal; the work of Motohashi-Pintz and Pintz suggests that it can essentially be improved to , 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 term in results such as Theorem 6.

We remark that as (5) is an open condition, it is unaffected by infinitesimal modifications to , and so we do not ascribe much importance to such modifications (e.g. replacing by for some arbitrarily small ).

The known deductions of from claims such as or rely on the following elementary observation of Goldston, Pintz, and Yildirim (essentially a weighted pigeonhole principle), which we have placed in “-tricked form”:

Lemma 7 (Criterion for DHL) Let . Suppose that for each fixed admissible -tuple and each congruence class such that is coprime to for all , one can find a non-negative weight function , fixed quantities , a quantity , and a fixed positive power of such that one has the upper bound

the lower bound

for all , and the key inequality

holds. Then holds. Here is defined to equal when is prime and otherwise.

Proof: Consider the quantity

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

By (8), this expression is positive for all sufficiently large . On the other hand, (9) can only be positive if at least one summand is positive, which only can happen when contains at least two primes for some with . Letting we obtain as claimed.

In practice, the quantity (referred to as the sieve level) is a power of such as or , and reflects the strength of the distribution hypothesis or that is available; the quantity will also be a key parameter in the definition of the sieve weight . The factor reflects the order of magnitude of the expected density of in the residue class ; it could be absorbed into the sieve weight by dividing that weight by , but it is convenient to not enforce such a normalisation so as not to clutter up the formulae. In practice, will some combination of and .

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

where takes the form

for some weights vanishing for that are to be chosen, where is an interval and is the polynomial . If the distribution hypothesis is , one takes and ; if the distribution hypothesis is instead , one takes and .

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

is used for some additional parameter to be optimised over. More generally, one can take

for some suitable (in particular, sufficiently smooth) cutoff function . 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

for a sufficiently smooth function , where

for is a -variant of the Euler totient function, and

for is a -variant of the function . (The derivative on the cutoff is convenient for computations, as will be made clearer later in this post.) This choice of weights may seem somewhat arbitrary, but it arises naturally when considering how to optimise the quadratic form

(which arises naturally in the estimation of in (6)) subject to a fixed value of (which morally is associated to the estimation of 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 from for sufficiently large, although unfortunately we were not able to confirm some of their calculations regarding the precise dependence of on , 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.

Online reading seminar for Zhang’s “bounded gaps between primes”

In a recent paper, Yitang Zhang has proven the following theorem:

Theorem 1 (Bounded gaps between primes) There exists a natural number such that there are infinitely many pairs of distinct primes with .

Zhang obtained the explicit value of for . A polymath project has been proposed to lower this value and also to improve the understanding of Zhang’s results; as of this time of writing, the current “world record” is (and the link given should stay updated with the most recent progress.

Zhang’s argument naturally divides into three steps, which we describe in reverse order. The last step, which is the most elementary, is to deduce the above theorem from the following weak version of the Dickson-Hardy-Littlewood (DHL) conjecture for some :

Theorem 2 () Let be an admissible -tuple, that is to say a tuple of distinct integers which avoids at least one residue class mod for every prime . Then there are infinitely many translates of that contain at least two primes.

Zhang obtained for . The current best value of is , as discussed in this previous blog post. To get from to Theorem 1, one has to exhibit an admissible -tuple of diameter at most . For instance, with , the narrowest admissible -tuple that we can construct has diameter , which explains the current world record. There is an active discussion on trying to improve the constructions of admissible tuples at this blog post; it is conceivable that some combination of computer search and clever combinatorial constructions could obtain slightly better values of for a given value of . The relationship between and is approximately of the form (and a classical estimate of Montgomery and Vaughan tells us that we cannot make much narrower than , see this previous post for some related discussion).

The second step in Zhang’s argument, which is somewhat less elementary (relying primarily on the sieve theory of Goldston, Yildirim, Pintz, and Motohashi), is to deduce from a certain conjecture for some . Here is one formulation of the conjecture, more or less as (implicitly) stated in Zhang’s paper:

Conjecture 3 () Let be an admissible tuple, let be an element of , let be a large parameter, and define

for any natural number , and

for any function . Let equal when is a prime , and otherwise. Then one has

for any fixed .

Note that this is slightly different from the formulation of in the previous post; I have reverted to Zhang’s formulation here as the primary purpose of this post is to read through Zhang’s paper. However, I have distinguished two separate parameters here instead of one, as it appears that there is some room to optimise by making these two parameters different.

In the previous post, I described how one can deduce from . Ignoring an exponentially small error , it turns out that one can deduce from whenever one can find a smooth function vanishing to order at least at such that

By selecting for a real parameter to optimise over, and ignoring the technical term alluded to previously (which is the only quantity here that depends on ), this gives from whenever

It may be possible to do better than this by choosing smarter choices for , or performing some sort of numerical calculus of variations or spectral theory; people interested in this topic are invited to discuss it in the previous post.

The final, and deepest, part of Zhang’s work is the following theorem (Theorem 2 from Zhang’s paper, whose proof occupies Sections 6-13 of that paper, and is about 32 pages long):

Theorem 4 (Zhang) is true for all .

The significance of the fraction is that Zhang’s argument proceeds for a general choice of , but ultimately the argument only closes if one has

(see page 53 of Zhang) which is equivalent to . Plugging in this choice of into (1) then gives with as stated previously.

Improving the value of in Theorem 4 would lead to improvements in and then as discussed above. The purpose of this reading seminar is then twofold:

1. Going through Zhang’s argument in order to improve the value of (perhaps by decreasing ); and
2. Gaining a more holistic understanding of Zhang’s argument (and perhaps to find some more “global” improvements to that argument), as well as related arguments such as the prior work of Bombieri, Fouvry, Friedlander, and Iwaniec that Zhang’s work is based on.

In addition to reading through Zhang’s paper, the following material is likely to be relevant:

• A recent blog post of Emmanuel Kowalski on the technical details of Zhang’s argument.
• Scanned notes from a talk related to the above blog post.
• A recent expository note by Fouvry, Kowalski, and Michel on a Friedlander-Iwaniec character sum relevant to this argument.
• This 1981 paper of Fouvry and Iwaniec which is the first result in the literature which is roughly of the type . (This paper seems to give a related result for and , if I read it correctly; I don’t yet understand what prevents this result or modifications thereof from being used in place of Theorem 4.)

I envisage a loose, unstructured format for the reading seminar. In the comments below, I am going to post my own impressions, questions, and remarks as I start going through the material, and I encourage other participants to do the same. The most obvious thing to do is to go through Zhang’s Sections 6-13 in linear order, but it may make sense for some participants to follow a different path. One obvious near-term goal is to carefully go through Zhang’s arguments for instead of , and record exactly how various exponents depend on , and what inequalities these parameters need to obey for the arguments to go through. It may be that this task can be done at a fairly superficial level without the need to carefully go through the analytic number theory estimates in that paper, though of course we should also be doing that as well. This may lead to some “cheap” optimisations of which can then propagate to improved bounds on and thanks to the other parts of the Polymath project.

Everyone is welcome to participate in this project (as per the usual polymath rules); however I would request that “meta” comments about the project that are not directly related to the task of reading Zhang’s paper and related works be placed instead on the polymath proposal page. (Similarly, comments regarding the optimisation of given and should be placed at this post, while comments on the optimisation of given should be given at this post. On the other hand, asking questions about Zhang’s paper, even (or especially!) “dumb” ones, would be very appropriate for this post and such questions are encouraged.

The prime tuples conjecture, sieve theory, and the work of Goldston-Pintz-Yildirim, Motohashi-Pintz, and Zhang

Suppose one is given a -tuple of distinct integers for some , arranged in increasing order. When is it possible to find infinitely many translates of which consists entirely of primes? The case is just Euclid’s theorem on the infinitude of primes, but the case is already open in general, with the case being the notorious twin prime conjecture.

On the other hand, there are some tuples for which one can easily answer the above question in the negative. For instance, the only translate of that consists entirely of primes is , basically because each translate of must contain an even number, and the only even prime is . More generally, if there is a prime such that meets each of the residue classes , then every translate of contains at least one multiple of ; since is the only multiple of that is prime, this shows that there are only finitely many translates of that consist entirely of primes.

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

Conjecture 1 (Prime tuples conjecture, qualitative form) If is an admissible -tuple, then there exists infinitely many translates of 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 -tuple with for which we can verify this conjecture (although, thanks to the recent work of Zhang, we know that satisfies the conclusion of the prime tuples conjecture for some , even if we can’t yet say what the precise value of is).

Actually, Hardy and Littlewood conjectured a more precise version of Conjecture 1. Given an admissible -tuple , and for each prime , let denote the number of residue classes modulo that meets; thus we have for all by admissibility, and also for all . We then define the singular series associated to by the formula

where is the set of primes; by the previous discussion we see that the infinite product in 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 that one should think of going to infinity. Some mathematical objects (such as and ) will be independent of and referred to as fixed; but unless otherwise specified we allow all mathematical objects under consideration to depend on . If and are two such quantities, we say that if one has for some fixed , and if one has for some function of (and of any fixed parameters present) that goes to zero as (for each choice of fixed parameters).

Conjecture 2 (Prime tuples conjecture, quantitative form) Let be a fixed natural number, and let be a fixed admissible -tuple. Then the number of natural numbers such that consists entirely of primes is .

Thus, for instance, if Conjecture 2 holds, then the number of twin primes less than should equal , where is the twin prime constant

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 ; see for instance this previous post. From the methods of sieve theory, one can obtain an upper bound of for the number of with all prime, where depends only on . 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 , let denote the following assertion (DHL stands for “Dickson-Hardy-Littlewood”):

Conjecture 3 () Let be a fixed admissible -tuple. Then there are infinitely many translates of which contain at least two primes.

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

where denotes the prime and is a parameter to be optimised over (in practice it is an order of magnitude or two smaller than ); see this blog post for details. The upshot is that one can bound for large by a quantity slightly smaller than (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 is true for some . Then is true for some finite . In particular, this establishes an infinite number of pairs of consecutive primes of separation .

The dependence of constants between and given by the Goldston-Pintz-Yildirim argument is basically of the form . (UPDATE: as recently observed by Farkas, Pintz, and Revesz, this relationship can be improved to .)

Unfortunately, the Elliott-Halberstam conjecture (which we will state properly below) is only known for , 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 of consecutive primes with (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 in this post, where is a parameter. We will define this conjecture more precisely later, but let us remark for now that is a consequence of .

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

Theorem 5 (Motohashi-Pintz-Zhang) Suppose that is true for some . Then is true for some .

A version of this result (with a slightly different formulation of ) appears in this paper of Motohashi and Pintz, and in the paper of Zhang, Theorem 5 is proven for the concrete values and . We will supply a self-contained proof of Theorem 5 below the fold, the constants upon those in Zhang’s paper (in particular, for , we can take as low as , with further improvements on the way). As with Theorem 4, we have an inverse quadratic relationship .

In his paper, Zhang obtained for the first time an unconditional advance on :

Theorem 6 (Zhang) is true for all .

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 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 , making Theorem 6 a particularly impressive achievement.

Combining Theorem 6 with Theorem 5 we obtain for some finite ; Zhang obtains this for but as detailed below, this can be lowered to . This in turn gives infinitely many pairs of consecutive primes of separation at most . Zhang gives a simple argument that bounds by , giving his famous result that there are infinitely many pairs of primes of separation at most ; by being a bit more careful (as discussed in this post) one can lower the upper bound on to , and if one instead uses the newer value for one can instead use the bound . (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 being the elementary fact that blows up like as one approaches from the right. To deal with the contribution of small primes (which is the source of the singular series ), it will be convenient to use the “-trick” (introduced in this paper of mine with Ben), passing to a single residue class mod (where 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).

A Fourier-analytic proof of Frobenius’ theorem

A finite group is said to be a Frobenius group if there is a non-trivial subgroup of (known as the Frobenius complement of ) such that the conjugates of are “disjoint as possible” in the sense that whenever . This gives a decomposition

where the Frobenius kernel of is defined as the identity element together with all the non-identity elements that are not conjugate to any element of . Taking cardinalities, we conclude that

and hence

A remarkable theorem of Frobenius gives an unexpected amount of structure on and hence on :

Theorem 1 (Frobenius’ theorem) Let be a Frobenius group with Frobenius complement and Frobenius kernel . Then is a normal subgroup of , and hence (by (2) and the disjointness of and outside the identity) is the semidirect product of and .

I discussed Frobenius’ theorem and its proof in this recent blog post. This proof uses the theory of characters on a finite group , in particular relying on the fact that a character on a subgroup can induce a character on , which can then be decomposed into irreducible characters with natural number coefficients. Remarkably, even though a century has passed since Frobenius’ original argument, there is no proof known of this theorem which avoids character theory entirely; there are elementary proofs known when the complement has even order or when is solvable (we review both of these cases below the fold), which by the Feit-Thompson theorem does cover all the cases, but the proof of the Feit-Thompson theorem involves plenty of character theory (and also relies on Theorem 1). (The answers to this MathOverflow question give a good overview of the current state of affairs.)

I have been playing around recently with the problem of finding a character-free proof of Frobenius’ theorem. I didn’t succeed in obtaining a completely elementary proof, but I did find an argument which replaces character theory (which can be viewed as coming from the representation theory of the non-commutative group algebra ) with the Fourier analysis of class functions (i.e. the representation theory of the centre of the group algebra), thus replacing non-commutative representation theory by commutative representation theory. This is not a particularly radical depature from the existing proofs of Frobenius’ theorem, but it did seem to be a new proof which was technically “character-free” (even if it was not all that far from character-based in spirit), so I thought I would record it here.

The main ideas are as follows. The space of class functions can be viewed as a commutative algebra with respect to the convolution operation ; as the regular representation is unitary and faithful, this algebra contains no nilpotent elements. As such, (Gelfand-style) Fourier analysis suggests that one can analyse this algebra through the idempotents: class functions such that . In terms of characters, idempotents are nothing more than sums of the form for various collections of characters, but we can perform a fair amount of analysis on idempotents directly without recourse to characters. In particular, it turns out that idempotents enjoy some important integrality properties that can be established without invoking characters: for instance, by taking traces one can check that is a natural number, and more generally we will show that is a natural number whenever is a subgroup of (see Corollary 4 below). For instance, the quantity

is a natural number which we will call the rank of (as it is also the linear rank of the transformation on ).

In the case that is a Frobenius group with kernel , the above integrality properties can be used after some elementary manipulations to establish that for any idempotent , the quantity

is an integer. On the other hand, one can also show by elementary means that this quantity lies between and . These two facts are not strong enough on their own to impose much further structure on , unless one restricts attention to minimal idempotents . In this case spectral theory (or Gelfand theory, or the fundamental theorem of algebra) tells us that has rank one, and then the integrality gap comes into play and forces the quantity (3) to always be either zero or one. This can be used to imply that the convolution action of every minimal idempotent either preserves or annihilates it, which makes itself an idempotent, which makes normal.