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This post is a continuation of the previous post on sieve theory, which is an ongoing part of the Polymath8 project. As the previous post was getting somewhat full, we are rolling the thread over to the current post.
In this post we will record a new truncation of the elementary Selberg sieve discussed in this previous post (and also analysed in the context of bounded prime gaps by Graham-Goldston-Pintz-Yildirim and Motohashi-Pintz) that was recently worked out by Janos Pintz, who has kindly given permission to share this new idea with the Polymath8 project. This new sieve decouples the 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.
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).
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.)
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.
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
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
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
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
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
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
.
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
(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
. 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.
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
, 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 (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
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
, 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.
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
, 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):
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:
- Going through Zhang’s argument in order to improve the value of
(perhaps by decreasing
); and
- 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.
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 :
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 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
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
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
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.

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