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Given a function on the natural numbers taking values in , one can invoke the Furstenberg correspondence principle to locate a measure preserving system – a probability space together with a measure-preserving shift (or equivalently, a measure-preserving -action on ) – together with a measurable function (or “observable”) that has essentially the same statistics as in the sense that
whenever the limit on the right-hand side exists. We will refer to the system together with the designated function as a Furstenberg limit ot the sequence . These Furstenberg limits capture some, but not all, of the asymptotic behaviour of ; roughly speaking, they control the typical “local” behaviour of , involving correlations such as in the regime where are much smaller than . However, the control on error terms here is usually only qualitative at best, and one usually does not obtain non-trivial control on correlations in which the are allowed to grow at some significant rate with (e.g. like some power of ).
The correspondence principle is discussed in these previous blog posts. One way to establish the principle is by introducing a Banach limit that extends the usual limit functional on the subspace of consisting of convergent sequences while still having operator norm one. Such functionals cannot be constructed explicitly, but can be proven to exist (non-constructively and non-uniquely) using the Hahn-Banach theorem; one can also use a non-principal ultrafilter here if desired. One can then seek to construct a system and a measurable function for which one has the statistics
for all . One can explicitly construct such a system as follows. One can take to be the Cantor space with the product -algebra and the shift
with the function being the coordinate function at zero:
(so in particular for any ). The only thing remaining is to construct the invariant measure . In order to be consistent with (2), one must have
for any distinct integers and signs . One can check that this defines a premeasure on the Boolean algebra of defined by cylinder sets, and the existence of then follows from the Hahn-Kolmogorov extension theorem (or the closely related Kolmogorov extension theorem). One can then check that the correspondence (2) holds, and that is translation-invariant; the latter comes from the translation invariance of the (Banach-)Césaro averaging operation . A variant of this construction shows that the Furstenberg limit is unique up to equivalence if and only if all the limits appearing in (1) actually exist.
One can obtain a slightly tighter correspondence by using a smoother average than the Césaro average. For instance, one can use the logarithmic Césaro averages in place of the Césaro average , thus one replaces (2) by
Whenever the Césaro average of a bounded sequence exists, then the logarithmic Césaro average exists and is equal to the Césaro average. Thus, a Furstenberg limit constructed using logarithmic Banach-Césaro averaging still obeys (1) for all when the right-hand side limit exists, but also obeys the more general assertion
whenever the limit of the right-hand side exists.
In a recent paper of Frantizinakis, the Furstenberg limits of the Liouville function (with logarithmic averaging) were studied. Some (but not all) of the known facts and conjectures about the Liouville function can be interpreted in the Furstenberg limit. For instance, in a recent breakthrough result of Matomaki and Radziwill (discussed previously here), it was shown that the Liouville function exhibited cancellation on short intervals in the sense that
In terms of Furstenberg limits of the Liouville function, this assertion is equivalent to the assertion that
for all Furstenberg limits of Liouville (including those without logarithmic averaging). Invoking the mean ergodic theorem (discussed in this previous post), this assertion is in turn equivalent to the observable that corresponds to the Liouville function being orthogonal to the invariant factor of ; equivalently, the first Gowers-Host-Kra seminorm of (as defined for instance in this previous post) vanishes. The Chowla conjecture, which asserts that
for all distinct integers , is equivalent to the assertion that all the Furstenberg limits of Liouville are equivalent to the Bernoulli system ( with the product measure arising from the uniform distribution on , with the shift and observable as before). Similarly, the logarithmically averaged Chowla conjecture
is equivalent to the assertion that all the Furstenberg limits of Liouville with logarithmic averaging are equivalent to the Bernoulli system. Recently, I was able to prove the two-point version
for any Furstenberg limit of Liouville with logarithmic averaging, and any .
The situation is more delicate with regards to the Sarnak conjecture, which is equivalent to the assertion that
for any zero-entropy sequence (see this previous blog post for more discussion). Morally speaking, this conjecture should be equivalent to the assertion that any Furstenberg limit of Liouville is disjoint from any zero entropy system, but I was not able to formally establish an implication in either direction due to some technical issues regarding the fact that the Furstenberg limit does not directly control long-range correlations, only short-range ones. (There are however ergodic theoretic interpretations of the Sarnak conjecture that involve the notion of generic points; see this paper of El Abdalaoui, Lemancyk, and de la Rue.) But the situation is currently better with the logarithmically averaged Sarnak conjecture
as I was able to show that this conjecture was equivalent to the logarithmically averaged Chowla conjecture, and hence to all Furstenberg limits of Liouville with logarithmic averaging being Bernoulli; I also showed the conjecture was equivalent to local Gowers uniformity of the Liouville function, which is in turn equivalent to the function having all Gowers-Host-Kra seminorms vanishing in every Furstenberg limit with logarithmic averaging. In this recent paper of Frantzikinakis, this analysis was taken further, showing that the logarithmically averaged Chowla and Sarnak conjectures were in fact equivalent to the much milder seeming assertion that all Furstenberg limits with logarithmic averaging were ergodic.
Actually, the logarithmically averaged Furstenberg limits have more structure than just a -action on a measure preserving system with a single observable . Let denote the semigroup of affine maps on the integers with and positive. Also, let denote the profinite integers (the inverse limit of the cyclic groups ). Observe that acts on by taking the inverse limit of the obvious actions of on .
Proposition 1 (Enriched logarithmically averaged Furstenberg limit of Liouville) Let be a Banach limit. Then there exists a probability space with an action of the affine semigroup , as well as measurable functions and , with the following properties:
- (i) (Affine Furstenberg limit) For any , and any congruence class , one has
- (ii) (Equivariance of ) For any , one has
for -almost every .
- (iii) (Multiplicativity at fixed primes) For any prime , one has
for -almost every , where is the dilation map .
- (iv) (Measure pushforward) If is of the form and is the set , then the pushforward of by is equal to , that is to say one has
for every measurable .
Note that can be viewed as the subgroup of consisting of the translations . If one only keeps the -portion of the action and forgets the rest (as well as the function ) then the action becomes measure-preserving, and we recover an ordinary Furstenberg limit with logarithmic averaging. However, the additional structure here can be quite useful; for instance, one can transfer the proof of (3) to this setting, which we sketch below the fold, after proving the proposition.
The observable , roughly speaking, means that points in the Furstenberg limit constructed by this proposition are still “virtual integers” in the sense that one can meaningfully compute the residue class of modulo any natural number modulus , by first applying and then reducing mod . The action of means that one can also meaningfully multiply by any natural number, and translate it by any integer. As with other applications of the correspondence principle, the main advantage of moving to this more “virtual” setting is that one now acquires a probability measure , so that the tools of ergodic theory can be readily applied.
Given a random variable that takes on only finitely many values, we can define its Shannon entropy by the formula
with the convention that . (In some texts, one uses the logarithm to base rather than the natural logarithm, but the choice of base will not be relevant for this discussion.) This is clearly a nonnegative quantity. Given two random variables taking on finitely many values, the joint variable is also a random variable taking on finitely many values, and also has an entropy . It obeys the Shannon inequalities
so we can define some further nonnegative quantities, the mutual information
and the conditional entropies
More generally, given three random variables , one can define the conditional mutual information
and the final of the Shannon entropy inequalities asserts that this quantity is also non-negative.
The mutual information is a measure of the extent to which and fail to be independent; indeed, it is not difficult to show that vanishes if and only if and are independent. Similarly, vanishes if and only if and are conditionally independent relative to . At the other extreme, is a measure of the extent to which fails to depend on ; indeed, it is not difficult to show that if and only if is determined by in the sense that there is a deterministic function such that . In a related vein, if and are equivalent in the sense that there are deterministic functional relationships , between the two variables, then is interchangeable with for the purposes of computing the above quantities, thus for instance , , , , etc..
One can get some initial intuition for these information-theoretic quantities by specialising to a simple situation in which all the random variables being considered come from restricting a single random (and uniformly distributed) boolean function on a given finite domain to some subset of :
In this case, has the law of a random uniformly distributed boolean function from to , and the entropy here can be easily computed to be , where denotes the cardinality of . If is the restriction of to , and is the restriction of to , then the joint variable is equivalent to the restriction of to . If one discards the normalisation factor , one then obtains the following dictionary between entropy and the combinatorics of finite sets:
|Random variables||Finite sets|
|Mutual information||Intersection cardinality|
|Conditional entropy||Set difference cardinality|
|Conditional mutual information|
|determined by||a subset of|
|conditionally independent relative to|
Every (linear) inequality or identity about entropy (and related quantities, such as mutual information) then specialises to a combinatorial inequality or identity about finite sets that is easily verified. For instance, the Shannon inequality becomes the union bound , and the definition of mutual information becomes the inclusion-exclusion formula
For a more advanced example, consider the data processing inequality that asserts that if are conditionally independent relative to , then . Specialising to sets, this now says that if are disjoint outside of , then ; this can be made apparent by considering the corresponding Venn diagram. This dictionary also suggests how to prove the data processing inequality using the existing Shannon inequalities. Firstly, if and are not necessarily disjoint outside of , then a consideration of Venn diagrams gives the more general inequality
and a further inspection of the diagram then reveals the more precise identity
Using the dictionary in the reverse direction, one is then led to conjecture the identity
which (together with non-negativity of conditional mutual information) implies the data processing inequality, and this identity is in turn easily established from the definition of mutual information.
On the other hand, not every assertion about cardinalities of sets generalises to entropies of random variables that are not arising from restricting random boolean functions to sets. For instance, a basic property of sets is that disjointness from a given set is preserved by unions:
Applying the dictionary in the reverse direction, one might now conjecture that if was independent of and was independent of , then should also be independent of , and furthermore that
but these statements are well known to be false (for reasons related to pairwise independence of random variables being strictly weaker than joint independence). For a concrete counterexample, one can take to be independent, uniformly distributed random elements of the finite field of two elements, and take to be the sum of these two field elements. One can easily check that each of and is separately independent of , but the joint variable determines and thus is not independent of .
From the inclusion-exclusion identities
one can check that (1) is equivalent to the trivial lower bound . The basic issue here is that in the dictionary between entropy and combinatorics, there is no satisfactory entropy analogue of the notion of a triple intersection . (Even the double intersection only exists information theoretically in a “virtual” sense; the mutual information allows one to “compute the entropy” of this “intersection”, but does not actually describe this intersection itself as a random variable.)
However, this issue only arises with three or more variables; it is not too difficult to show that the only linear equalities and inequalities that are necessarily obeyed by the information-theoretic quantities associated to just two variables are those that are also necessarily obeyed by their combinatorial analogues . (See for instance the Venn diagram at the Wikipedia page for mutual information for a pictorial summation of this statement.)
One can work with a larger class of special cases of Shannon entropy by working with random linear functions rather than random boolean functions. Namely, let be some finite-dimensional vector space over a finite field , and let be a random linear functional on , selected uniformly among all such functions. Every subspace of then gives rise to a random variable formed by restricting to . This random variable is also distributed uniformly amongst all linear functions on , and its entropy can be easily computed to be . Given two random variables formed by restricting to respectively, the joint random variable determines the random linear function on the union on the two spaces, and thus by linearity on the Minkowski sum as well; thus is equivalent to the restriction of to . In particular, . This implies that and also , where is the quotient map. After discarding the normalising constant , this leads to the following dictionary between information theoretic quantities and linear algebra quantities, analogous to the previous dictionary:
|Mutual information||Dimension of intersection|
|Conditional entropy||Dimension of projection|
|Conditional mutual information|
|determined by||a subspace of|
|conditionally independent relative to||, transverse.|
The combinatorial dictionary can be regarded as a specialisation of the linear algebra dictionary, by taking to be the vector space over the finite field of two elements, and only considering those subspaces that are coordinate subspaces associated to various subsets of .
As before, every linear inequality or equality that is valid for the information-theoretic quantities discussed above, is automatically valid for the linear algebra counterparts for subspaces of a vector space over a finite field by applying the above specialisation (and dividing out by the normalising factor of ). In fact, the requirement that the field be finite can be removed by applying the compactness theorem from logic (or one of its relatives, such as Los’s theorem on ultraproducts, as done in this previous blog post).
The linear algebra model captures more of the features of Shannon entropy than the combinatorial model. For instance, in contrast to the combinatorial case, it is possible in the linear algebra setting to have subspaces such that and are separately transverse to , but their sum is not; for instance, in a two-dimensional vector space , one can take to be the one-dimensional subspaces spanned by , , and respectively. Note that this is essentially the same counterexample from before (which took to be the field of two elements). Indeed, one can show that any necessarily true linear inequality or equality involving the dimensions of three subspaces (as well as the various other quantities on the above table) will also be necessarily true when applied to the entropies of three discrete random variables (as well as the corresponding quantities on the above table).
However, the linear algebra model does not completely capture the subtleties of Shannon entropy once one works with four or more variables (or subspaces). This was first observed by Ingleton, who established the dimensional inequality
for any subspaces . This is easiest to see when the three terms on the right-hand side vanish; then are transverse, which implies that ; similarly . But and are transverse, and this clearly implies that and are themselves transverse. To prove the general case of Ingleton’s inequality, one can define and use (and similarly for instead of ) to reduce to establishing the inequality
which can be rearranged using (and similarly for instead of ) and as
but this is clear since .
Returning to the entropy setting, the analogue
of (3) is true (exercise!), but the analogue
of Ingleton’s inequality is false in general. Again, this is easiest to see when all the terms on the right-hand side vanish; then are conditionally independent relative to , and relative to , and and are independent, and the claim (4) would then be asserting that and are independent. While there is no linear counterexample to this statement, there are simple non-linear ones: for instance, one can take to be independent uniform variables from , and take and to be (say) and respectively (thus are the indicators of the events and respectively). Once one conditions on either or , one of has positive conditional entropy and the other has zero entropy, and so are conditionally independent relative to either or ; also, or are independent of each other. But and are not independent of each other (they cannot be simultaneously equal to ). Somehow, the feature of the linear algebra model that is not present in general is that in the linear algebra setting, every pair of subspaces has a well-defined intersection that is also a subspace, whereas for arbitrary random variables , there does not necessarily exist the analogue of an intersection, namely a “common information” random variable that has the entropy of and is determined either by or by .
I do not know if there is any simpler model of Shannon entropy that captures all the inequalities available for four variables. One significant complication is that there exist some information inequalities in this setting that are not of Shannon type, such as the Zhang-Yeung inequality
One can however still use these simpler models of Shannon entropy to be able to guess arguments that would work for general random variables. An example of this comes from my paper on the logarithmically averaged Chowla conjecture, in which I showed among other things that
whenever was sufficiently large depending on , where is the Liouville function. The information-theoretic part of the proof was as follows. Given some intermediate scale between and , one can form certain random variables . The random variable is a sign pattern of the form where is a random number chosen from to (with logarithmic weighting). The random variable was tuple of reductions of to primes comparable to . Roughly speaking, what was implicitly shown in the paper (after using the multiplicativity of , the circle method, and the Matomaki-Radziwill theorem on short averages of multiplicative functions) is that if the inequality (5) fails, then there was a lower bound
for any , where denotes the shifted sign pattern . On the other hand, one had the entropy bounds
and from concatenating sign patterns one could see that is equivalent to the joint random variable for any . Applying these facts and using an “entropy decrement” argument, I was able to obtain a contradiction once was allowed to become sufficiently large compared to , but the bound was quite weak (coming ultimately from the unboundedness of as the interval of values of under consideration becomes large), something of the order of ; the quantity needs at various junctures to be less than a small power of , so the relationship between and becomes essentially quadruple exponential in nature, . The basic strategy was to observe that the lower bound (6) causes some slowdown in the growth rate of the mean entropy, in that this quantity decreased by as increased from to , basically by dividing into components , and observing from (6) each of these shares a bit of common information with the same variable . This is relatively clear when one works in a set model, in which is modeled by a set of size , and is modeled by a set of the form
for various sets of size (also there is some translation symmetry that maps to a shift while preserving all of the ).
However, on considering the set model recently, I realised that one can be a little more efficient by exploiting the fact (basically the Chinese remainder theorem) that the random variables are basically jointly independent as ranges over dyadic values that are much smaller than , which in the set model corresponds to the all being disjoint. One can then establish a variant
of (6), which in the set model roughly speaking asserts that each claims a portion of the of cardinality that is not claimed by previous choices of . This leads to a more efficient contradiction (relying on the unboundedness of rather than ) that looks like it removes one order of exponential growth, thus the relationship between and is now . Returning to the entropy model, one can use (7) and Shannon inequalities to establish an inequality of the form
for a small constant , which on iterating and using the boundedness of gives the claim. (A modification of this analysis, at least on the level of the back of the envelope calculation, suggests that the Matomaki-Radziwill theorem is needed only for ranges greater than or so, although at this range the theorem is not significantly simpler than the general case).
I’ve just uploaded to the arXiv my paper “Some remarks on the lonely runner conjecture“, submitted to Contributions to discrete mathematics. I had blogged about the lonely runner conjecture in this previous blog post, and I returned to the problem recently to see if I could obtain anything further. The results obtained were more modest than I had hoped, but they did at least seem to indicate a potential strategy to make further progress on the problem, and also highlight some of the difficulties of the problem.
One can rephrase the lonely runner conjecture as the following covering problem. Given any integer “velocity” and radius , define the Bohr set to be the subset of the unit circle given by the formula
where denotes the distance of to the nearest integer. Thus, for positive, is simply the union of the intervals for , projected onto the unit circle ; in the language of the usual formulation of the lonely runner conjecture, represents those times in which a runner moving at speed returns to within of his or her starting position. For any non-zero integers , let be the smallest radius such that the Bohr sets cover the unit circle:
Then define to be the smallest value of , as ranges over tuples of distinct non-zero integers. The Dirichlet approximation theorem quickly gives that
for any . The lonely runner conjecture is equivalent to the assertion that this bound is in fact optimal:
Conjecture 1 (Lonely runner conjecture) For any , one has .
This conjecture is currently known for (see this paper of Barajas and Serra), but remains open for higher .
It is natural to try to attack the problem by establishing lower bounds on the quantity . We have the following “trivial” bound, that gets within a factor of two of the conjecture:
Proposition 2 (Trivial bound) For any , one has .
we see that the covering (1) is only possible if , giving the claim.
So, in some sense, all the difficulty is coming from the need to improve upon the trivial union bound (2) by a factor of two.
which was improved a little by Chen and Cusick in 1999 to
when was prime. In a recent paper of Perarnau and Serra, the bound
was obtained for arbitrary . These bounds only improve upon the trivial bound by a multiplicative factor of . Heuristically, one reason for this is as follows. The union bound (2) would of course be sharp if the Bohr sets were all disjoint. Strictly speaking, such disjointness is not possible, because all the Bohr sets have to contain the origin as an interior point. However, it is possible to come up with a large number of Bohr sets which are almost disjoint. For instance, suppose that we had velocities that were all prime numbers between and , and that was equal to (and in particular was between and . Then each set can be split into a “kernel” interval , together with the “petal” intervals . Roughly speaking, as the prime varies, the kernel interval stays more or less fixed, but the petal intervals range over disjoint sets, and from this it is not difficult to show that
so that the union bound is within a multiplicative factor of of the truth in this case.
This does not imply that is within a multiplicative factor of of , though, because there are not enough primes between and to assign to distinct velocities; indeed, by the prime number theorem, there are only about such velocities that could be assigned to a prime. So, while the union bound could be close to tight for up to Bohr sets, the above counterexamples don’t exclude improvements to the union bound for larger collections of Bohr sets. Following this train of thought, I was able to obtain a logarithmic improvement to previous lower bounds:
Theorem 3 For sufficiently large , one has for some absolute constant .
The factors of in the denominator are for technical reasons and might perhaps be removable by a more careful argument. However it seems difficult to adapt the methods to improve the in the numerator, basically because of the obstruction provided by the near-counterexample discussed above.
Roughly speaking, the idea of the proof of this theorem is as follows. If we have the covering (1) for very close to , then the multiplicity function will then be mostly equal to , but occasionally be larger than . On the other hand, one can compute that the norm of this multiplicity function is significantly larger than (in fact it is at least ). Because of this, the norm must be very large, which means that the triple intersections must be quite large for many triples . Using some basic Fourier analysis and additive combinatorics, one can deduce from this that the velocities must have a large structured component, in the sense that there exists an arithmetic progression of length that contains of these velocities. For simplicity let us take the arithmetic progression to be , thus of the velocities lie in . In particular, from the prime number theorem, most of these velocities will not be prime, and will in fact likely have a “medium-sized” prime factor (in the precise form of the argument, “medium-sized” is defined to be “between and “). Using these medium-sized prime factors, one can show that many of the will have quite a large overlap with many of the other , and this can be used after some elementary arguments to obtain a more noticeable improvement on the union bound (2) than was obtained previously.
In my previous blog post, I showed that in order to prove the lonely runner conjecture, it suffices to do so under the additional assumption that all of the velocities are of size ; I reproduce this argument (slightly cleaned up for publication) in the current preprint. There is unfortunately a huge gap between and , so the above bound (3) does not immediately give any new bounds for . However, one could perhaps try to start attacking the lonely runner conjecture by increasing the range for which one has good results, and by decreasing the range that one can reduce to. For instance, in the current preprint I give an elementary argument (using a certain amount of case-checking) that shows that the lonely runner bound
holds if all the velocities are assumed to lie between and . This upper threshold of is only a tiny improvement over the trivial threshold of , but it seems to be an interesting sub-problem of the lonely runner conjecture to increase this threshold further. One key target would be to get up to , as there are actually a number of -tuples in this range for which (4) holds with equality. The Dirichlet approximation theorem of course gives the tuple , but there is also the double of this tuple, and furthermore there is an additional construction of Goddyn and Wong that gives some further examples such as , or more generally one can start with the standard tuple and accelerate one of the velocities to ; this turns out to work as long as shares a common factor with every integer between and . There are a few more examples of this type in the paper of Goddyn and Wong, but all of them can be placed in an arithmetic progression of length at most, so if one were very optimistic, one could perhaps envision a strategy in which the upper bound of mentioned earlier was reduced all the way to something like , and then a separate argument deployed to treat this remaining case, perhaps isolating the constructions of Goddyn and Wong (and possible variants thereof) as the only extreme cases.
Let be the divisor function. A classical application of the Dirichlet hyperbola method gives the asymptotic
where denotes the estimate as . Much better error estimates are possible here, but we will not focus on the lower order terms in this discussion. For somewhat idiosyncratic reasons I will interpret this estimate (and the other analytic number theory estimates discussed here) through the probabilistic lens. Namely, if is a random number selected uniformly between and , then the above estimate can be written as
that is to say the random variable has mean approximately . (But, somewhat paradoxically, this is not the median or mode behaviour of this random variable, which instead concentrates near , basically thanks to the Hardy-Ramanujan theorem.)
Now we turn to the pair correlations for a fixed positive integer . There is a classical computation of Ingham that shows that
The error term in (2) has been refined by many subsequent authors, as has the uniformity of the estimates in the aspect, as these topics are related to other questions in analytic number theory, such as fourth moment estimates for the Riemann zeta function; but we will not consider these more subtle features of the estimate here. However, we will look at the next term in the asymptotic expansion for (2) below the fold.
Using our probabilistic lens, the estimate (2) can be written as
From (1) (and the asymptotic negligibility of the shift by ) we see that the random variables and both have a mean of , so the additional factor of represents some arithmetic coupling between the two random variables.
Ingham’s formula can be established in a number of ways. Firstly, one can expand out and use the hyperbola method (splitting into the cases and and removing the overlap). If one does so, one soon arrives at the task of having to estimate sums of the form
for various . For much less than this can be achieved using a further application of the hyperbola method, but for comparable to things get a bit more complicated, necessitating the use of non-trivial estimates on Kloosterman sums in order to obtain satisfactory control on error terms. A more modern approach proceeds using automorphic form methods, as discussed in this previous post. A third approach, which unfortunately is only heuristic at the current level of technology, is to apply the Hardy-Littlewood circle method (discussed in this previous post) to express (2) in terms of exponential sums for various frequencies . The contribution of “major arc” can be computed after a moderately lengthy calculation which yields the right-hand side of (2) (as well as the correct lower order terms that are currently being suppressed), but there does not appear to be an easy way to show directly that the “minor arc” contributions are of lower order, although the methods discussed previously do indirectly show that this is ultimately the case.
Each of the methods outlined above requires a fair amount of calculation, and it is not obvious while performing them that the factor will emerge at the end. One can at least explain the as a normalisation constant needed to balance the factor (at a heuristic level, at least). To see this through our probabilistic lens, introduce an independent copy of , then
using symmetry to order (discarding the diagonal case ) and making the change of variables , we see that (4) is heuristically consistent with (3) as long as the asymptotic mean of in is equal to . (This argument is not rigorous because there was an implicit interchange of limits present, but still gives a good heuristic “sanity check” of Ingham’s formula.) Indeed, if denotes the asymptotic mean in , then we have (heuristically at least)
and we obtain the desired consistency after multiplying by .
This still however does not explain the presence of the factor. Intuitively it is reasonable that if has many prime factors, and has a lot of factors, then will have slightly more factors than average, because any common factor to and will automatically be acquired by . But how to quantify this effect?
One heuristic way to proceed is through analysis of local factors. Observe from the fundamental theorem of arithmetic that we can factor
where the product is over all primes , and is the local version of at (which in this case, is just one plus the –valuation of : ). Note that all but finitely many of the terms in this product will equal , so the infinite product is well-defined. In a similar fashion, we can factor
(or in terms of valuations, ). Heuristically, the Chinese remainder theorem suggests that the various factors behave like independent random variables, and so the correlation between and should approximately decouple into the product of correlations between the local factors and . And indeed we do have the following local version of Ingham’s asymptotics:
From the Euler formula
we see that
and so one can “explain” the arithmetic factor in Ingham’s asymptotic as the product of the arithmetic factors in the (much easier) local Ingham asymptotics. Unfortunately we have the usual “local-global” problem in that we do not know how to rigorously derive the global asymptotic from the local ones; this problem is essentially the same issue as the problem of controlling the minor arc contributions in the circle method, but phrased in “physical space” language rather than “frequency space”.
Remark 2 The relation between the local means and the global mean can also be seen heuristically through the application
Let us now prove this proposition. One could brute-force the computations by observing that for any fixed , the valuation is equal to with probability , and with a little more effort one can also compute the joint distribution of and , at which point the proposition reduces to the calculation of various variants of the geometric series. I however find it cleaner to proceed in a more recursive fashion (similar to how one can prove the geometric series formula by induction); this will also make visible the vague intuition mentioned previously about how common factors of and force to have a factor also.
It is first convenient to get rid of error terms by observing that in the limit , the random variable converges vaguely to a uniform random variable on the profinite integers , or more precisely that the pair converges vaguely to . Because of this (and because of the easily verified uniform integrability properties of and their powers), it suffices to establish the exact formulae
We begin with (5). Observe that is coprime to with probability , in which case is equal to . Conditioning to the complementary probability event that is divisible by , we can factor where is also uniformly distributed over the profinite integers, in which event we have . We arrive at the identity
As and have the same distribution, the quantities and are equal, and (5) follows by a brief amount of high-school algebra.
We use a similar method to treat (6). First treat the case when is coprime to . Then we see that with probability , and are simultaneously coprime to , in which case . Furthermore, with probability , is divisible by and is not; in which case we can write as before, with and . Finally, in the remaining event with probability , is divisible by and is not; we can then write , so that and . Putting all this together, we obtain
Now suppose that is divisible by , thus for some integer . Then with probability , and are simultaneously coprime to , in which case . In the remaining event, we can write , and then and . Putting all this together we have
which by (5) (and replacing by ) leads to the recursive relation
and (6) then follows by induction on the number of powers of .
for certain complicated but explicit coefficients . For instance, is given by the formula
where is the Euler-Mascheroni constant,
The formula for is similar but even more complicated. The error term was improved by Heath-Brown to ; it is conjectured (for instance by Conrey and Gonek) that one in fact has square root cancellation here, but this is well out of reach of current methods.
These lower order terms are traditionally computed either from a Dirichlet series approach (using Perron’s formula) or a circle method approach. It turns out that a refinement of the above heuristics can also predict these lower order terms, thus keeping the calculation purely in physical space as opposed to the “multiplicative frequency space” of the Dirichlet series approach, or the “additive frequency space” of the circle method, although the computations are arguably as messy as the latter computations for the purposes of working out the lower order terms. We illustrate this just for the term below the fold.
The twin prime conjecture, still unsolved, asserts that there are infinitely many primes such that is also prime. A more precise form of this conjecture is (a special case) of the Hardy-Littlewood prime tuples conjecture, which asserts that
Because is almost entirely supported on the primes, it is not difficult to see that (1) implies the twin prime conjecture.
One can give a heuristic justification of the asymptotic (1) (and hence the twin prime conjecture) via sieve theoretic methods. Recall that the von Mangoldt function can be decomposed as a Dirichlet convolution
or (to simplify things by removing the logarithm)
for odd. Summing by parts, one then expects
and so we heuristically have
The Dirichlet series
has an Euler product factorisation
for ; comparing this with the Euler product factorisation
for the Riemann zeta function, and recalling that has a simple pole of residue at , we see that
has a simple zero at with first derivative
From this and standard multiplicative number theory manipulations, one can calculate the asymptotic
which concludes the heuristic justification of (1).
What prevents us from making the above heuristic argument rigorous, and thus proving (1) and the twin prime conjecture? Note that the variable in (2) ranges to be as large as . On the other hand, the prime number theorem in arithmetic progressions (3) is not expected to hold for anywhere that large (for instance, the left-hand side of (3) vanishes as soon as exceeds ). The best unconditional result known of the type (3) is the Siegel-Walfisz theorem, which allows to be as large as . Even the powerful generalised Riemann hypothesis (GRH) only lets one prove an estimate of the form (3) for up to about .
However, because of the averaging effect of the summation in in (2), we don’t need the asymptotic (3) to be true for all in a particular range; having it true for almost all in that range would suffice. Here the situation is much better; the celebrated Bombieri-Vinogradov theorem (sometimes known as “GRH on the average”) implies, roughly speaking, that the approximation (3) is valid for almost all for any fixed . While this is not enough to control (2) or (1), the Bombieri-Vinogradov theorem can at least be used to control variants of (1) such as
for various sieve weights whose associated divisor function is supposed to approximate the von Mangoldt function , although that theorem only lets one do this when the weights are supported on the range . This is still enough to obtain some partial results towards (1); for instance, by selecting weights according to the Selberg sieve, one can use the Bombieri-Vinogradov theorem to establish the upper bound
It has been difficult to improve upon the Bombieri-Vinogradov theorem in its full generality, although there are various improvements to certain restricted versions of the Bombieri-Vinogradov theorem, for instance in the famous work of Zhang on bounded gaps between primes. Nevertheless, it is believed that the Elliott-Halberstam conjecture (EH) holds, which roughly speaking would mean that (3) now holds for almost all for any fixed . (Unfortunately, the factor cannot be removed, as investigated in a series of papers by Friedlander, Granville, and also Hildebrand and Maier.) This comes tantalisingly close to having enough distribution to control all of (1). Unfortunately, it still falls short. Using this conjecture in place of the Bombieri-Vinogradov theorem leads to various improvements to sieve theoretic bounds; for instance, the factor of in (4) can now be improved to .
In two papers from the 1970s (which can be found online here and here respectively, the latter starting on page 255 of the pdf), Bombieri developed what is now known as the Bombieri asymptotic sieve to clarify the situation more precisely. First, he showed that on the Elliott-Halberstam conjecture, while one still could not establish the asymptotic (1), one could prove the generalised asymptotic
These functions behave like the von Mangoldt function, but are concentrated on -almost primes (numbers with at most prime factors) rather than primes. The right-hand side of (5) corresponds to what one would expect if one ran the same heuristics used to justify (1). Sadly, the case of (5), which is just (1), is just barely excluded from Bombieri’s analysis.
for any fixed and any tuple of natural numbers other than , where
is a further generalisation of the von Mangoldt function (now concentrated on -almost primes). By combining these asymptotics with some elementary identities involving the , together with the Weierstrass approximation theorem, Bombieri was able to control a wide family of sums including (1), except for one undetermined scalar . Namely, he was able to show (again on EH) that for any fixed and any continuous function on the simplex that had suitable vanishing at the boundary, the sum
and the twin prime conjecture would be proved if one could show that is bounded away from zero, while (1) is equivalent to the assertion that is equal to . Unfortunately, no additional bound beyond the inequalities provided by the Bombieri asymptotic sieve is known, even if one assumes all other major conjectures in number theory than the prime tuples conjecture and its variants (e.g. GRH, GEH, GUE, abc, Chowla, …).
for and some fixed , with vanishing elsewhere and for some continuous (symmetric) functions obeying some vanishing at the boundary, so long as the parity condition
is obeyed (informally: gives the same weight to products of an odd number of primes as to products of an even number of primes, or to put it another way, is asymptotically orthogonal to the Möbius function ). But when violates the parity condition, the asymptotic involves the unknown . This scalar thus embodies the “parity problem” for the twin prime conjecture (discussed in these previous blog posts).
Because the obstruction to the parity problem is only one-dimensional (on EH), one can replace any parity-violating weight (such as ) with any other parity-violating weight and obtain a logically equivalent estimate. For instance, to prove the twin prime conjecture on EH, it would suffice to show that
for some fixed , or equivalently that there are solutions to the equation in primes with and . (In some cases, this sort of reduction can also be made using other sieves than the Bombieri asymptotic sieve, as was observed by Ng.) As another example, the Bombieri asymptotic sieve can be used to show that the asymptotic (1) is equivalent to the asymptotic
where is the set of numbers that are rough in the sense that they have no prime factors less than for some fixed (the function clearly correlates with and so must violate the parity condition). One can replace with similar sieve weights (e.g. a Selberg sieve) that concentrate on almost primes if desired.
As it turns out, if one is willing to strengthen the assumption of the Elliott-Halberstam (EH) conjecture to the assumption of the generalised Elliott-Halberstam (GEH) conjecture (as formulated for instance in Claim 2.6 of the Polymath8b paper), one can also swap the factor in the above asymptotics with other parity-violating weights and obtain a logically equivalent estimate, as the Bombieri asymptotic sieve also applies to weights such as under the assumption of GEH. For instance, on GEH one can use two such applications of the Bombieri asymptotic sieve to show that the twin prime conjecture would follow if one could show that there are solutions to the equation
in primes with and , for some . Similarly, on GEH the asymptotic (1) is equivalent to the asymptotic
for some fixed , and similarly with replaced by other sieves. This form of the quantitative twin primes conjecture is appealingly similar to the (special case)
of the Chowla conjecture, for which there has been some recent progress (discussed for instance in these recent posts). Informally, the Bombieri asymptotic sieve lets us (on GEH) view the twin prime conjecture as a sort of Chowla conjecture restricted to almost primes. Unfortunately, the recent progress on the Chowla conjecture relies heavily on the multiplicativity of at small primes, which is completely destroyed by inserting a weight such as , so this does not yet yield a viable path towards the twin prime conjecture even assuming GEH. Still, the similarity is striking, and one can hope that further ways to attack the Chowla conjecture may emerge that could impact the twin prime conjecture. (Alternatively, if one assumes a sufficiently optimistic version of the GEH, one could perhaps relax the notion of “almost prime” to the extent that one could start usefully using multiplicativity at smallish primes, though this seems rather wishful at present, particularly since the most optimistic versions of GEH are known to be false.)
The Bombieri asymptotic sieve is already well explained in the original two papers of Bombieri; there is also a slightly different treatment of the sieve by Friedlander and Iwaniec, as well as a simplified version in the book of Friedlander and Iwaniec (in which the distribution hypothesis is strengthened in order to shorten the arguments. I’ve decided though to write up my own notes on the sieve below the fold; this is primarily for my own benefit, but may be useful to some readers also. I largely follow the treatment of Bombieri, with the one idiosyncratic twist of replacing the usual “elementary” Selberg sieve with the “analytic” Selberg sieve used in particular in many of the breakthrough works in small gaps between primes; I prefer working with the latter due to its Fourier-analytic flavour.
— 1. Controlling generalised von Mangoldt sums —
To prove (5), we shall first generalise it, by replacing the sequence by a more general sequence obeying the following axioms:
- (i) (Non-negativity) One has for all .
- (ii) (Crude size bound) One has for all , where is the divisor function.
- (iii) (Size) We have for some constant .
- (iv) (Elliott-Halberstam type conjecture) For any , one has
where is a multiplicative function with for all primes and .
These axioms are a little bit stronger than what is actually needed to make the Bombieri asymptotic sieve work, but we will not attempt to work with the weakest possible axioms here.
We introduce the function
which is analytic for ; in particular it can be evaluated at to yield
There are two model examples of data to keep in mind. The first, discussed in the introduction, is when , then and is as in the introduction; one of course needs EH to justify axiom (iv) in this case. The other is when , in which case and for all . We will later take advantage of the second example to avoid doing some (routine, but messy) main term computations.
The main result of this section is then
as , where .
Note that this recovers (5) (on EH) as a special case.
We now begin the proof of this theorem. Henceforth we allow implied constants in the or notation to depend on and .
It will be convenient to replace the range by a shorter range by the following standard localisation trick. Let be a large quantity depending on to be chosen later, and let denote the interval . We will show the estimate
for any .
Write for the logarithm function , thus for any . Without loss of generality we may assume that ; we then factor , where
This function is just when . When the function is more complicated, but we at least have the following crude bound:
Proof: We induct on . The case is obvious, so suppose and the claim has already been proven for . Since , we see from induction hypothesis and the triangle inequality that
Since by Möbius inversion, the claim follows.
We can write
In the region , we have . Thus
for . The contribution of the error term to to (10) is easily seen to be negligible if is large enough, so we may freely replace with with little difficulty.
If we insert this replacement directly into the left-hand side of (10) and rearrange, we get
One could in principle compute explicitly from the proof of (13), but one can avoid doing so by the following comparison trick. In the special case , standard multiplicative number theory (noting that the Dirichlet series has a pole of order at , with top Laurent coefficient ) gives the asymptotic
which when compared with (14) for (recalling that in this case) gives the formula
As it turns out, the estimate (13) is easy to establish, but the estimate (12) is not, roughly speaking because the typical number in has too many divisors in the range , each of which gives a contribution to the error term. (In the book of Friedlander and Iwaniec, the estimate (13) is established anyway, but only after assuming a stronger version of (iv), roughly speaking in which is allowed to be as large as .) To resolve this issue, we will insert a preliminary sieve that will remove most of the potential divisors i the range (leaving only about such divisors on the average for typical ), making the analogue of (12) easier to prove (at the cost of making the analogue of (13) more difficult). Namely, if one can find a function for which one has the estimates
for some quantity that depends on but not on , then by repeating the previous arguments we will again be able to establish (10).
The key estimate is (16). As we shall see, when comparing with , the weight will cost us a factor of , but the term in the definitions of and will recover a factor of , which will give the desired bound since we are assuming .
One has some flexibility in how to select the weight : basically any standard sieve that uses divisors of size at most to localise (at least approximately) to numbers that are rough in the sense that they have no (or at least very few) factors less than , will do. We will use the analytic Selberg sieve choice
where denotes the derivative of . Note the loss of that had previously been pointed out. In the arguments that follows I will be a little brief with the details, as they are standard (see e.g. this previous post).
We now prove (19). The left-hand side can be expanded as
where denotes the least common multiple of and . From the support of we see that the summand is only non-vanishing when . We now use axiom (iv) and split the left-hand side into a main term
so from axiom (iv) and Cauchy-Schwarz we see that the error term (20) is acceptable. Thus it will suffice to establish the bound
and so the left-hand side of (21) can be rearranged using Fubini’s theorem as
We can factorise as an Euler product:
Taking absolute values and using Mertens’ theorem leads to the crude bound
which when combined with the rapid decrease of , allows us to restrict the region of integration in (23) to the square (say) with negligible error. Next, we use the Euler product
for to factorise
For with nonnegative real part, one has
and so by the Weierstrass -test, is continuous at . Since
we thus have
Also, since has a pole of order at with residue , we have
The quantity (23) can thus be written, up to errors of , as
Using the rapid decrease of , we may remove the restriction on , and it will now suffice to prove the identity
But on differentiating and then squaring (22) we have
and the claim follows by integrating in from zero to infinity (noting that vanishes for ).
We have the following variant of (19):
Roughly speaking, the above estimates assert that is concentrated on those numbers with no prime factors much less than , but factors without such small prime divisors occur with about the same relative density as they do in the integers.
Proof: The left-hand side of (24) can be expanded as
If we define
then the previous expression can be written as
while one has
From Mertens’ theorem we have
when , so the contribution of the terms where can be absorbed into the error (after increasing that error slightly). For the remaining contributions, we see that
where if does not divide , and
if divides times for some . In the latter case, Taylor expansion gives the bounds
and the claim (28) follows. When and we have
Now we can prove (15), (16), (17). We begin with (15). Using the Leibniz rule applied to the identity and using and Möbius inversion (and the associativity and commutativity of Dirichlet convolution) we see that
Next, by applying the Leibniz rule to for some and using (29) we see that
In particular, from induction we see that is supported on numbers with at most distinct prime factors, and hence is supported on numbers with at most distinct prime factors. In particular, from (18) we see that on the support of . Thus it will suffice to show that
If and , then has at most distinct prime factors , with . If we factor , where is the contribution of those with , and is the contribution of those with , then at least one of the following two statements hold:
- (a) (and hence ) is divisible by a square number of size at least .
- (b) .
The contribution of case (a) is easily seen to be acceptable by axiom (ii). For case (b), we observe from (30) and induction that
and so it will suffice to show that
where ranges over numbers bounded by with at most distinct prime factors, the smallest of which is at most , and consists of those numbers with no prime factor less than or equal to . Applying (26) (with replaced by ) gives the bound
so by (25) it suffices to show that
subject to the same constraints on as before. The contribution of those with distinct prime factors can be bounded by
applying Mertens’ theorem and summing over , one obtains the claim.
From the support of , the summand on the left-hand side is only non-zero when , which makes , where we use the crucial hypothesis to gain enough powers of to make the argument here work. Applying Lemma 2, we reduce to showing that
We can make the change of variables to flip the sum
and then swap the sums to reduce to showing that
By Lemma 3, it suffices to show that
To prove this, we use the Rankin trick, bounding the implied weight by . We can then bound the left-hand side by the Euler product
which can be bounded by
and the claim follows from Mertens’ theorem.
We let be a small constant to be chosen later. We divide the outer sum into two ranges, depending on whether only has prime factors greater than or not. In the former case, we can apply (27) to write this contribution as
plus a negligible error, where the is implicitly restricted to numbers with all prime factors greater than . The main term is messy, but it is of the required form up to an acceptable error, so there is no need to compute it any further. It remains to consider those that have at least one prime factor less than . Here we use (24) instead of (27) as well as Lemma 3 to dominate this contribution by
up to negligible errors, where is now restricted to have at least one prime factor less than . This makes at least one of the factors to be at most . A routine application of Rankin’s trick shows that
and so the total contribution of this case is . Since can be made arbitrarily small, (17) follows.
— 2. Weierstrass approximation —
Let , , , be as in that theorem. It will be convenient to normalise the weights by to make their mean value comparable to . From Theorem 1 and summation by parts we have
We now take a closer look at what happens when does consist entirely of ones. Let denote the -tuple . Convolving the case of (30) with copies of for some and using the Leibniz rule, we see that
Multiplying by and summing over , and using (31) to control the term, one has
If we define (up to an error of ) by the formula
then an induction then shows that
for odd , and
for even . In particular, after adjusting by if necessary, we have since the left-hand sides are non-negative.
If we now define the comparison sequence , standard multiplicative number theory shows that the above estimates also hold when is replaced by ; thus
for both odd and even . The bound (31) also holds for when does not consist entirely of ones, and hence
for any fixed (which may or may not consist entirely of ones).
Next, from induction (on ), the Leibniz rule, and (30), we see that for any and , , the function
whenever is one of these functions (32). Specialising to the case , we thus have
where . The contribution of those that are powers of primes can be easily seen to be negligible, leading to
where now . The contribution of the case where two of the primes agree can also be seen to be negligible, as can the error when replacing with , and then by symmetry
By linearity, this implies that
for any polynomial that vanishes on the coordinate hyperplanes . The right-hand side can also be evaluated by Mertens’ theorem as
when is odd and
when is even. Using the Weierstrass approximation theorem, we then have
Remark 4 The Bombieri asymptotic sieve has to use the full power of EH (or GEH); there are constructions due to Ford that show that if one only has a distributional hypothesis up to for some fixed constant , then the asymptotics of sums such as (5), or more generally (9), are not determined by a single scalar parameter , but can also vary in other ways as well. Thus the Bombieri asymptotic sieve really is asymptotic; in order to get type error terms one needs the level of distribution to be asymptotically equal to as . Related to this, the quantitative decay of the error terms in the Bombieri asymptotic sieve are extremely poor; in particular, they depend on the dependence of implied constant in axiom (iv) on the parameters , for which there is no consensus on what one should conjecturally expect.
I’ve just uploaded to the arXiv my paper “Equivalence of the logarithmically averaged Chowla and Sarnak conjectures“, submitted to the Festschrift “Number Theory – Diophantine problems, uniform distribution and applications” in honour of Robert F. Tichy. This paper is a spinoff of my previous paper establishing a logarithmically averaged version of the Chowla (and Elliott) conjectures in the two-point case. In that paper, the estimate
as was demonstrated, where was any positive integer and denoted the Liouville function. The proof proceeded using a method I call the “entropy decrement argument”, which ultimately reduced matters to establishing a bound of the form
It is natural to see to what extent the arguments can be adapted to attack the higher-point cases of the logarithmically averaged Chowla conjecture (ignoring for this post the more general Elliott conjecture for other bounded multiplicative functions than the Liouville function). That is to say, one would like to prove that
as for any fixed distinct integers . As it turns out (and as is detailed in the current paper), the entropy decrement argument extends to this setting (after using some known facts about linear equations in primes), and allows one to reduce the above estimate to an estimate of the form
for a slowly growing function of and some fixed (in fact we can take for ), where is the (normalised) local Gowers uniformity norm. (In the case , , this becomes the Fourier-uniformity conjecture discussed in this previous post.) If one then applied the (now proven) inverse conjecture for the Gowers norms, this estimate is in turn equivalent to the more complicated looking assertion
where the supremum is over all possible choices of nilsequences of controlled step and complexity (see the paper for definitions of these terms).
The main novelty in the paper (elaborating upon a previous comment I had made on this blog) is to observe that this latter estimate in turn follows from the logarithmically averaged form of Sarnak’s conjecture (discussed in this previous post), namely that
whenever is a zero entropy (i.e. deterministic) sequence. Morally speaking, this follows from the well-known fact that nilsequences have zero entropy, but the presence of the supremum in (1) means that we need a little bit more; roughly speaking, we need the class of nilsequences of a given step and complexity to have “uniformly zero entropy” in some sense.
On the other hand, it was already known (see previous post) that the Chowla conjecture implied the Sarnak conjecture, and similarly for the logarithmically averaged form of the two conjectures. Putting all these implications together, we obtain the pleasant fact that the logarithmically averaged Sarnak and Chowla conjectures are equivalent, which is the main result of the current paper. There have been a large number of special cases of the Sarnak conjecture worked out (when the deterministic sequence involved came from a special dynamical system), so these results can now also be viewed as partial progress towards the Chowla conjecture also (at least with logarithmic averaging). However, my feeling is that the full resolution of these conjectures will not come from these sorts of special cases; instead, conjectures like the Fourier-uniformity conjecture in this previous post look more promising to attack.
It would also be nice to get rid of the pesky logarithmic averaging, but this seems to be an inherent requirement of the entropy decrement argument method, so one would probably have to find a way to avoid that argument if one were to remove the log averaging.
Over the last few years, a large group of mathematicians have been developing an online database to systematically collect the known facts, numerical data, and algorithms concerning some of the most central types of objects in modern number theory, namely the L-functions associated to various number fields, curves, and modular forms, as well as further data about these modular forms. This of course includes the most famous examples of L-functions and modular forms respectively, namely the Riemann zeta function and the discriminant modular form , but there are countless other examples of both. The connections between these classes of objects lie at the heart of the Langlands programme.
As of today, the “L-functions and modular forms database” is now out of beta, and open to the public; at present the database is mostly geared towards specialists in computational number theory, but will hopefully develop into a more broadly useful resource as time develops. An article by John Cremona summarising the purpose of the database can be found here.
Tamar Ziegler and I have just uploaded to the arXiv two related papers: “Concatenation theorems for anti-Gowers-uniform functions and Host-Kra characteoristic factors” and “polynomial patterns in primes“, with the former developing a “quantitative Bessel inequality” for local Gowers norms that is crucial in the latter.
We use the term “concatenation theorem” to denote results in which structural control of a function in two or more “directions” can be “concatenated” into structural control in a joint direction. A trivial example of such a concatenation theorem is the following: if a function is constant in the first variable (thus is constant for each ), and also constant in the second variable (thus is constant for each ), then it is constant in the joint variable . A slightly less trivial example: if a function is affine-linear in the first variable (thus, for each , there exist such that for all ) and affine-linear in the second variable (thus, for each , there exist such that for all ) then is a quadratic polynomial in ; in fact it must take the form
The same phenomenon extends to higher degree polynomials. Given a function from one additive group to another, we say that is of degree less than along a subgroup of if all the -fold iterated differences of along directions in vanish, that is to say
for all and , where is the difference operator
(We adopt the convention that the only of degree less than is the zero function.)
We then have the following simple proposition:
Proposition 1 (Concatenation of polynomiality) Let be of degree less than along one subgroup of , and of degree less than along another subgroup of , for some . Then is of degree less than along the subgroup of .
Note the previous example was basically the case when , , , , and .
Proof: The claim is trivial for or (in which is constant along or respectively), so suppose inductively and the claim has already been proven for smaller values of .
We take a derivative in a direction along to obtain
where is the shift of by . Then we take a further shift by a direction to obtain
leading to the cocycle equation
Since has degree less than along and degree less than along , has degree less than along and less than along , so is degree less than along by induction hypothesis. Similarly is also of degree less than along . Combining this with the cocycle equation we see that is of degree less than along for any , and hence is of degree less than along , as required.
While this proposition is simple, it already illustrates some basic principles regarding how one would go about proving a concatenation theorem:
- (i) One should perform induction on the degrees involved, and take advantage of the recursive nature of degree (in this case, the fact that a function is of less than degree along some subgroup of directions iff all of its first derivatives along are of degree less than ).
- (ii) Structure is preserved by operations such as addition, shifting, and taking derivatives. In particular, if a function is of degree less than along some subgroup , then any derivative of is also of degree less than along , even if does not belong to .
Here is another simple example of a concatenation theorem. Suppose an at most countable additive group acts by measure-preserving shifts on some probability space ; we call the pair (or more precisely ) a -system. We say that a function is a generalised eigenfunction of degree less than along some subgroup of and some if one has
almost everywhere for all , and some functions of degree less than along , with the convention that a function has degree less than if and only if it is equal to . Thus for instance, a function is an generalised eigenfunction of degree less than along if it is constant on almost every -ergodic component of , and is a generalised function of degree less than along if it is an eigenfunction of the shift action on almost every -ergodic component of . A basic example of a higher order eigenfunction is the function on the skew shift with action given by the generator for some irrational . One can check that for every integer , where is a generalised eigenfunction of degree less than along , so is of degree less than along .
We then have
Proposition 2 (Concatenation of higher order eigenfunctions) Let be a -system, and let be a generalised eigenfunction of degree less than along one subgroup of , and a generalised eigenfunction of degree less than along another subgroup of , for some . Then is a generalised eigenfunction of degree less than along the subgroup of .
The argument is almost identical to that of the previous proposition and is left as an exercise to the reader. The key point is the point (ii) identified earlier: the space of generalised eigenfunctions of degree less than along is preserved by multiplication and shifts, as well as the operation of “taking derivatives” even along directions that do not lie in . (To prove this latter claim, one should restrict to the region where is non-zero, and then divide by to locate .)
A typical example of this proposition in action is as follows: consider the -system given by the -torus with generating shifts
for some irrational , which can be checked to give a action
The function can then be checked to be a generalised eigenfunction of degree less than along , and also less than along , and less than along . One can view this example as the dynamical systems translation of the example (1) (see this previous post for some more discussion of this sort of correspondence).
The main results of our concatenation paper are analogues of these propositions concerning a more complicated notion of “polynomial-like” structure that are of importance in additive combinatorics and in ergodic theory. On the ergodic theory side, the notion of structure is captured by the Host-Kra characteristic factors of a -system along a subgroup . These factors can be defined in a number of ways. One is by duality, using the Gowers-Host-Kra uniformity seminorms (defined for instance here) . Namely, is the factor of defined up to equivalence by the requirement that
An equivalent definition is in terms of the dual functions of along , which can be defined recursively by setting and
where denotes the ergodic average along a Følner sequence in (in fact one can also define these concepts in non-amenable abelian settings as per this previous post). The factor can then be alternately defined as the factor generated by the dual functions for .
In the case when and is -ergodic, a deep theorem of Host and Kra shows that the factor is equivalent to the inverse limit of nilsystems of step less than . A similar statement holds with replaced by any finitely generated group by Griesmer, while the case of an infinite vector space over a finite field was treated in this paper of Bergelson, Ziegler, and myself. The situation is more subtle when is not -ergodic, or when is -ergodic but is a proper subgroup of acting non-ergodically, when one has to start considering measurable families of directional nilsystems; see for instance this paper of Austin for some of the subtleties involved (for instance, higher order group cohomology begins to become relevant!).
One of our main theorems is then
Proposition 3 (Concatenation of characteristic factors) Let be a -system, and let be measurable with respect to the factor and with respect to the factor for some and some subgroups of . Then is also measurable with respect to the factor .
We give two proofs of this proposition in the paper; an ergodic-theoretic proof using the Host-Kra theory of “cocycles of type (along a subgroup )”, which can be used to inductively describe the factors , and a combinatorial proof based on a combinatorial analogue of this proposition which is harder to state (but which roughly speaking asserts that a function which is nearly orthogonal to all bounded functions of small norm, and also to all bounded functions of small norm, is also nearly orthogonal to alll bounded functions of small norm). The combinatorial proof parallels the proof of Proposition 2. A key point is that dual functions obey a property analogous to being a generalised eigenfunction, namely that
where and is a “structured function of order ” along . (In the language of this previous paper of mine, this is an assertion that dual functions are uniformly almost periodic of order .) Again, the point (ii) above is crucial, and in particular it is key that any structure that has is inherited by the associated functions and . This sort of inheritance is quite easy to accomplish in the ergodic setting, as there is a ready-made language of factors to encapsulate the concept of structure, and the shift-invariance and -algebra properties of factors make it easy to show that just about any “natural” operation one performs on a function measurable with respect to a given factor, returns a function that is still measurable in that factor. In the finitary combinatorial setting, though, encoding the fact (ii) becomes a remarkably complicated notational nightmare, requiring a huge amount of “epsilon management” and “second-order epsilon management” (in which one manages not only scalar epsilons, but also function-valued epsilons that depend on other parameters). In order to avoid all this we were forced to utilise a nonstandard analysis framework for the combinatorial theorems, which made the arguments greatly resemble the ergodic arguments in many respects (though the two settings are still not equivalent, see this previous blog post for some comparisons between the two settings). Unfortunately the arguments are still rather complicated.
For combinatorial applications, dual formulations of the concatenation theorem are more useful. A direct dualisation of the theorem yields the following decomposition theorem: a bounded function which is small in norm can be split into a component that is small in norm, and a component that is small in norm. (One may wish to understand this type of result by first proving the following baby version: any function that has mean zero on every coset of , can be decomposed as the sum of a function that has mean zero on every coset, and a function that has mean zero on every coset. This is dual to the assertion that a function that is constant on every coset and constant on every coset, is constant on every coset.) Combining this with some standard “almost orthogonality” arguments (i.e. Cauchy-Schwarz) give the following Bessel-type inequality: if one has a lot of subgroups and a bounded function is small in norm for most , then it is also small in norm for most . (Here is a baby version one may wish to warm up on: if a function has small mean on for some large prime , then it has small mean on most of the cosets of most of the one-dimensional subgroups of .)
There is also a generalisation of the above Bessel inequality (as well as several of the other results mentioned above) in which the subgroups are replaced by more general coset progressions (of bounded rank), so that one has a Bessel inequailty controlling “local” Gowers uniformity norms such as by “global” Gowers uniformity norms such as . This turns out to be particularly useful when attempting to compute polynomial averages such as
(actually one ends up with more complicated expressions than this, but let’s use this example for sake of discussion). This can be viewed as an average of various Gowers uniformity norms of along arithmetic progressions of the form for various . Using the above Bessel inequality, this can be controlled in turn by an average of various Gowers uniformity norms along rank two generalised arithmetic progressions of the form for various . But for generic , this rank two progression is close in a certain technical sense to the “global” interval (this is ultimately due to the basic fact that two randomly chosen large integers are likely to be coprime, or at least have a small gcd). As a consequence, one can use the concatenation theorems from our first paper to control expressions such as (2) in terms of global Gowers uniformity norms. This is important in number theoretic applications, when one is interested in computing sums such as
where and are the Möbius and von Mangoldt functions respectively. This is because we are able to control global Gowers uniformity norms of such functions (thanks to results such as the proof of the inverse conjecture for the Gowers norms, the orthogonality of the Möbius function with nilsequences, and asymptotics for linear equations in primes), but much less control is currently available for local Gowers uniformity norms, even with the assistance of the generalised Riemann hypothesis (see this previous blog post for some further discussion).
By combining these tools and strategies with the “transference principle” approach from our previous paper (as improved using the recent “densification” technique of Conlon, Fox, and Zhao, discussed in this previous post), we are able in particular to establish the following result:
Theorem 4 (Polynomial patterns in the primes) Let be polynomials of degree at most , whose degree coefficients are all distinct, for some . Suppose that is admissible in the sense that for every prime , there are such that are all coprime to . Then there exist infinitely many pairs of natural numbers such that are prime.
Furthermore, we obtain an asymptotic for the number of such pairs in the range , (actually for minor technical reasons we reduce the range of to be very slightly less than ). In fact one could in principle obtain asymptotics for smaller values of , and relax the requirement that the degree coefficients be distinct with the requirement that no two of the differ by a constant, provided one had good enough local uniformity results for the Möbius or von Mangoldt functions. For instance, we can obtain an asymptotic for triplets of the form unconditionally for , and conditionally on GRH for all , using known results on primes in short intervals on average.
The case of this theorem was obtained in a previous paper of myself and Ben Green (using the aforementioned conjectures on the Gowers uniformity norm and the orthogonality of the Möbius function with nilsequences, both of which are now proven). For higher , an older result of Tamar and myself was able to tackle the case when (though our results there only give lower bounds on the number of pairs , and no asymptotics). Both of these results generalise my older theorem with Ben Green on the primes containing arbitrarily long arithmetic progressions. The theorem also extends to multidimensional polynomials, in which case there are some additional previous results; see the paper for more details. We also get a technical refinement of our previous result on narrow polynomial progressions in (dense subsets of) the primes by making the progressions just a little bit narrower in the case of the density of the set one is using is small.
. This latter Bessel type inequality is particularly useful in combinatorial and number-theoretic applications, as it allows one to convert “global” Gowers uniformity norm (basically, bounds on norms such as ) to “local” Gowers uniformity norm control.
There is a very nice recent paper by Lemke Oliver and Soundararajan (complete with a popular science article about it by the consistently excellent Erica Klarreich for Quanta) about a surprising (but now satisfactorily explained) bias in the distribution of pairs of consecutive primes when reduced to a small modulus .
This phenomenon is superficially similar to the more well known Chebyshev bias concerning the reduction of a single prime to a small modulus , but is in fact a rather different (and much stronger) bias than the Chebyshev bias, and seems to arise from a completely different source. The Chebyshev bias asserts, roughly speaking, that a randomly selected prime of a large magnitude will typically (though not always) be slightly more likely to be a quadratic non-residue modulo than a quadratic residue, but the bias is small (the difference in probabilities is only about for typical choices of ), and certainly consistent with known or conjectured positive results such as Dirichlet’s theorem or the generalised Riemann hypothesis. The reason for the Chebyshev bias can be traced back to the von Mangoldt explicit formula which relates the distribution of the von Mangoldt function modulo with the zeroes of the -functions with period . This formula predicts (assuming some standard conjectures like GRH) that the von Mangoldt function is quite unbiased modulo . The von Mangoldt function is mostly concentrated in the primes, but it also has a medium-sized contribution coming from squares of primes, which are of course all located in the quadratic residues modulo . (Cubes and higher powers of primes also make a small contribution, but these are quite negligible asymptotically.) To balance everything out, the contribution of the primes must then exhibit a small preference towards quadratic non-residues, and this is the Chebyshev bias. (See this article of Rubinstein and Sarnak for a more technical discussion of the Chebyshev bias, and this survey of Granville and Martin for an accessible introduction. The story of the Chebyshev bias is also related to Skewes’ number, once considered the largest explicit constant to naturally appear in a mathematical argument.)
The paper of Lemke Oliver and Soundararajan considers instead the distribution of the pairs for small and for large consecutive primes , say drawn at random from the primes comparable to some large . For sake of discussion let us just take . Then all primes larger than are either or ; Chebyshev’s bias gives a very slight preference to the latter (of order , as discussed above), but apart from this, we expect the primes to be more or less equally distributed in both classes. For instance, assuming GRH, the probability that lands in would be , and similarly for .
In view of this, one would expect that up to errors of or so, the pair should be equally distributed amongst the four options , , , , thus for instance the probability that this pair is would naively be expected to be , and similarly for the other three tuples. These assertions are not yet proven (although some non-trivial upper and lower bounds for such probabilities can be obtained from recent work of Maynard).
However, Lemke Oliver and Soundararajan argue (backed by both plausible heuristic arguments (based ultimately on the Hardy-Littlewood prime tuples conjecture), as well as substantial numerical evidence) that there is a significant bias away from the tuples and – informally, adjacent primes don’t like being in the same residue class! For instance, they predict that the probability of attaining is in fact
with similar predictions for the other three pairs (in fact they give a somewhat more precise prediction than this). The magnitude of this bias, being comparable to , is significantly stronger than the Chebyshev bias of .
One consequence of this prediction is that the prime gaps are slightly less likely to be divisible by than naive random models of the primes would predict. Indeed, if the four options , , , all occurred with equal probability , then should equal with probability , and and with probability each (as would be the case when taking the difference of two random numbers drawn from those integers not divisible by ); but the Lemke Oliver-Soundararajan bias predicts that the probability of being divisible by three should be slightly lower, being approximately .
Below the fold we will give a somewhat informal justification of (a simplified version of) this phenomenon, based on the Lemke Oliver-Soundararajan calculation using the prime tuples conjecture.