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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).
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.
In this blog post, I would like to specialise the arguments of Bourgain, Demeter, and Guth from the previous post to the two-dimensional case of the Vinogradov main conjecture, namely
This particular case of the main conjecture has a classical proof using some elementary number theory. Indeed, the left-hand side can be viewed as the number of solutions to the system of equations
with . These two equations can combine (using the algebraic identity applied to ) to imply the further equation
which, when combined with the divisor bound, shows that each is associated to choices of excluding diagonal cases when two of the collide, and this easily yields Theorem 1. However, the Bourgain-Demeter-Guth argument (which, in the two dimensional case, is essentially contained in a previous paper of Bourgain and Demeter) does not require the divisor bound, and extends for instance to the the more general case where ranges in a -separated set of reals between to .
In this special case, the Bourgain-Demeter argument simplifies, as the lower dimensional inductive hypothesis becomes a simple almost orthogonality claim, and the multilinear Kakeya estimate needed is also easy (collapsing to just Fubini’s theorem). Also one can work entirely in the context of the Vinogradov main conjecture, and not turn to the increased generality of decoupling inequalities (though this additional generality is convenient in higher dimensions). As such, I am presenting this special case as an introduction to the Bourgain-Demeter-Guth machinery.
We now give the specialisation of the Bourgain-Demeter argument to Theorem 1. It will suffice to establish the bound
for all , (where we keep fixed and send to infinity), as the bound then follows by combining the above bound with the trivial bound . Accordingly, for any and , we let denote the claim that
as . Clearly, for any fixed , holds for some large , and it will suffice to establish
Proposition 2 Let , and let be such that holds. Then there exists such that holds.
Indeed, this proposition shows that for , the infimum of the for which holds is zero.
We prove the proposition below the fold, using a simplified form of the methods discussed in the previous blog post. To simplify the exposition we will be a bit cavalier with the uncertainty principle, for instance by essentially ignoring the tails of rapidly decreasing functions.
Given any finite collection of elements in some Banach space , the triangle inequality tells us that
However, when the all “oscillate in different ways”, one expects to improve substantially upon the triangle inequality. For instance, if is a Hilbert space and the are mutually orthogonal, we have the Pythagorean theorem
for any finite collection in any Banach space , where denotes the cardinality of . Thus orthogonality in a Hilbert space yields “square root cancellation”, saving a factor of or so over the trivial bound coming from the triangle inequality.
More generally, let us somewhat informally say that a collection exhibits decoupling in if one has the Pythagorean-like inequality
for any , thus one obtains almost the full square root cancellation in the norm. The theory of almost orthogonality can then be viewed as the theory of decoupling in Hilbert spaces such as . In spaces for one usually does not expect this sort of decoupling; for instance, if the are disjointly supported one has
and the right-hand side can be much larger than when . At the opposite extreme, one usually does not expect to get decoupling in , since one could conceivably align the to all attain a maximum magnitude at the same location with the same phase, at which point the triangle inequality in becomes sharp.
However, in some cases one can get decoupling for certain . For instance, suppose we are in , and that are bi-orthogonal in the sense that the products for are pairwise orthogonal in . Then we have
giving decoupling in . (Similarly if each of the is orthogonal to all but of the other .) A similar argument also gives decoupling when one has tri-orthogonality (with the mostly orthogonal to each other), and so forth. As a slight variant, Khintchine’s inequality also indicates that decoupling should occur for any fixed if one multiplies each of the by an independent random sign .
In recent years, Bourgain and Demeter have been establishing decoupling theorems in spaces for various key exponents of , in the “restriction theory” setting in which the are Fourier transforms of measures supported on different portions of a given surface or curve; this builds upon the earlier decoupling theorems of Wolff. In a recent paper with Guth, they established the following decoupling theorem for the curve parameterised by the polynomial curve
For any ball in , let denote the weight
which should be viewed as a smoothed out version of the indicator function of . In particular, the space can be viewed as a smoothed out version of the space . For future reference we observe a fundamental self-similarity of the curve : any arc in this curve, with a compact interval, is affinely equivalent to the standard arc .
of a finite Borel measure on the arc , where . Then the exhibit decoupling in for any ball of radius .
Orthogonality gives the case of this theorem. The bi-orthogonality type arguments sketched earlier only give decoupling in up to the range ; the point here is that we can now get a much larger value of . The case of this theorem was previously established by Bourgain and Demeter (who obtained in fact an analogous theorem for any curved hypersurface). The exponent (and the radius ) is best possible, as can be seen by the following basic example. If
where is a bump function adapted to , then standard Fourier-analytic computations show that will be comparable to on a rectangular box of dimensions (and thus volume ) centred at the origin, and exhibit decay away from this box, with comparable to
On the other hand, is comparable to on a ball of radius comparable to centred at the origin, so is , which is just barely consistent with decoupling. This calculation shows that decoupling will fail if is replaced by any larger exponent, and also if the radius of the ball is reduced to be significantly smaller than .
This theorem has the following consequence of importance in analytic number theory:
Corollary 2 (Vinogradov main conjecture) Let be integers, and let . Then
Proof: By the Hölder inequality (and the trivial bound of for the exponential sum), it suffices to treat the critical case , that is to say to show that
We can rescale this as
As the integrand is periodic along the lattice , this is equivalent to
The left-hand side may be bounded by , where and . Since
the claim now follows from the decoupling theorem and a brief calculation.
Using the Plancherel formula, one may equivalently (when is an integer) write the Vinogradov main conjecture in terms of solutions to the system of equations
but we will not use this formulation here.
A history of the Vinogradov main conjecture may be found in this survey of Wooley; prior to the Bourgain-Demeter-Guth theorem, the conjecture was solved completely for , or for and either below or above , with the bulk of recent progress coming from the efficient congruencing technique of Wooley. It has numerous applications to exponential sums, Waring’s problem, and the zeta function; to give just one application, the main conjecture implies the predicted asymptotic for the number of ways to express a large number as the sum of fifth powers (the previous best result required fifth powers). The Bourgain-Demeter-Guth approach to the Vinogradov main conjecture, based on decoupling, is ostensibly very different from the efficient congruencing technique, which relies heavily on the arithmetic structure of the program, but it appears (as I have been told from second-hand sources) that the two methods are actually closely related, with the former being a sort of “Archimedean” version of the latter (with the intervals in the decoupling theorem being analogous to congruence classes in the efficient congruencing method); hopefully there will be some future work making this connection more precise. One advantage of the decoupling approach is that it generalises to non-arithmetic settings in which the set that is drawn from is replaced by some other similarly separated set of real numbers. (A random thought – could this allow the Vinogradov-Korobov bounds on the zeta function to extend to Beurling zeta functions?)
Below the fold we sketch the Bourgain-Demeter-Guth argument proving Theorem 1.
I thank Jean Bourgain and Andrew Granville for helpful discussions.
Let denote the Liouville function. The prime number theorem is equivalent to the estimate
as , that is to say that exhibits cancellation on large intervals such as . This result can be improved to give cancellation on shorter intervals. For instance, using the known zero density estimates for the Riemann zeta function, one can establish that
as if for some fixed ; I believe this result is due to Ramachandra (see also Exercise 21 of this previous blog post), and in fact one could obtain a better error term on the right-hand side that for instance gained an arbitrary power of . On the Riemann hypothesis (or the weaker density hypothesis), it was known that the could be lowered to .
Early this year, there was a major breakthrough by Matomaki and Radziwill, who (among other things) showed that the asymptotic (1) was in fact valid for any with that went to infinity as , thus yielding cancellation on extremely short intervals. This has many further applications; for instance, this estimate, or more precisely its extension to other “non-pretentious” bounded multiplicative functions, was a key ingredient in my recent solution of the Erdös discrepancy problem, as well as in obtaining logarithmically averaged cases of Chowla’s conjecture, such as
It is of interest to twist the above estimates by phases such as the linear phase . In 1937, Davenport showed that
from which one can see that this is another averaged form of Chowla’s conjecture (stronger than the one I was able to prove with Matomaki and Radziwill, but a consequence of the unaveraged Chowla conjecture). If one inserted such a bound into the machinery I used to solve the Erdös discrepancy problem, it should lead to further averaged cases of Chowla’s conjecture, such as
though I have not fully checked the details of this implication. It should also have a number of new implications for sign patterns of the Liouville function, though we have not explored these in detail yet.
One can write (4) equivalently in the form
uniformly for all -dependent phases . In contrast, (3) is equivalent to the subcase of (6) when the linear phase coefficient is independent of . This dependency of on seems to necessitate some highly nontrivial additive combinatorial analysis of the function in order to establish (4) when is small. To date, this analysis has proven to be elusive, but I would like to record what one can do with more classical methods like Vaughan’s identity, namely:
The values of in this range are far too large to yield implications such as new cases of the Chowla conjecture, but it appears that the exponent is the limit of “classical” methods (at least as far as I was able to apply them), in the sense that one does not do any combinatorial analysis on the function , nor does one use modern equidistribution results on “Type III sums” that require deep estimates on Kloosterman-type sums. The latter may shave a little bit off of the exponent, but I don’t see how one would ever hope to go below without doing some non-trivial combinatorics on the function . UPDATE: I have come across this paper of Zhan which uses mean-value theorems for L-functions to lower the exponent to .
Let me now sketch the proof of the proposition, omitting many of the technical details. We first remark that known estimates on sums of the Liouville function (or similar functions such as the von Mangoldt function) in short arithmetic progressions, based on zero-density estimates for Dirichlet -functions, can handle the “major arc” case of (4) (or (6)) where is restricted to be of the form for (the exponent here being of the same numerology as the exponent in the classical result of Ramachandra, tied to the best zero density estimates currently available); for instance a modification of the arguments in this recent paper of Koukoulopoulos would suffice. Thus we can restrict attention to “minor arc” values of (or , using the interpretation of (6)).
Next, one breaks up (or the closely related Möbius function) into Dirichlet convolutions using one of the standard identities (e.g. Vaughan’s identity or Heath-Brown’s identity), as discussed for instance in this previous post (which is focused more on the von Mangoldt function, but analogous identities exist for the Liouville and Möbius functions). The exact choice of identity is not terribly important, but the upshot is that can be decomposed into terms, each of which is either of the “Type I” form
for some coefficients that are roughly of logarithmic size on the average, and scales with and , or else of the “Type II” form
for some coefficients that are roughly of logarithmic size on the average, and scales with and . As discussed in the previous post, the exponent is a natural barrier in these identities if one is unwilling to also consider “Type III” type terms which are roughly of the shape of the third divisor function .
A Type I sum makes a contribution to that can be bounded (via Cauchy-Schwarz) in terms of an expression such as
The inner sum exhibits a lot of cancellation unless is within of an integer. (Here, “a lot” should be loosely interpreted as “gaining many powers of over the trivial bound”.) Since is significantly larger than , standard Vinogradov-type manipulations (see e.g. Lemma 13 of these previous notes) show that this bad case occurs for many only when is “major arc”, which is the case we have specifically excluded. This lets us dispose of the Type I contributions.
A Type II sum makes a contribution to roughly of the form
We can break this up into a number of sums roughly of the form
for ; note that the range is non-trivial because is much larger than . Applying the usual bilinear sum Cauchy-Schwarz methods (e.g. Theorem 14 of these notes) we conclude that there is a lot of cancellation unless one has for some . But with , is well below the threshold for the definition of major arc, so we can exclude this case and obtain the required cancellation.
A basic estimate in multiplicative number theory (particularly if one is using the Granville-Soundararajan “pretentious” approach to this subject) is the following inequality of Halasz (formulated here in a quantitative form introduced by Montgomery and Tenenbaum).
As a qualitative corollary, we conclude (by standard compactness arguments) that if
as . In the more recent work of this paper of Granville and Soundararajan, the sharper bound
is obtained (with a more precise description of the term).
The usual proofs of Halasz’s theorem are somewhat lengthy (though there has been a recent simplification, in forthcoming work of Granville, Harper, and Soundarajan). Below the fold I would like to give a relatively short proof of the following “cheap” version of the inequality, which has slightly weaker quantitative bounds, but still suffices to give qualitative conclusions such as (2).
Theorem 2 (Cheap Halasz inequality) Let be a multiplicative function bounded in magnitude by . Let and , and suppose that is sufficiently large depending on . If (1) holds for all , then
The non-optimal exponent can probably be improved a bit by being more careful with the exponents, but I did not try to optimise it here. A similar bound appears in the first paper of Halasz on this topic.
The idea of the argument is to split as a Dirichlet convolution where is the portion of coming from “small”, “medium”, and “large” primes respectively (with the dividing line between the three types of primes being given by various powers of ). Using a Perron-type formula, one can express this convolution in terms of the product of the Dirichlet series of respectively at various complex numbers with . One can use based estimates to control the Dirichlet series of , while using the hypothesis (1) one can get estimates on the Dirichlet series of . (This is similar to the Fourier-analytic approach to ternary additive problems, such as Vinogradov’s theorem on representing large odd numbers as the sum of three primes.) This idea was inspired by a similar device used in the work of Granville, Harper, and Soundarajan. A variant of this argument also appears in unpublished work of Adam Harper.
I thank Andrew Granville for helpful comments which led to significant simplifications of the argument.
Kevin Ford, James Maynard, and I have uploaded to the arXiv our preprint “Chains of large gaps between primes“. This paper was announced in our previous paper with Konyagin and Green, which was concerned with the largest gap
between consecutive primes up to , in which we improved the Rankin bound of
for large (where we use the abbreviations , , and ). Here, we obtain an analogous result for the quantity
which measures how far apart the gaps between chains of consecutive primes can be. Our main result is
whenever is sufficiently large depending on , with the implied constant here absolute (and effective). The factor of is inherent to the method, and related to the basic probabilistic fact that if one selects numbers at random from the unit interval , then one expects the minimum gap between adjacent numbers to be about (i.e. smaller than the mean spacing of by an additional factor of ).
for an infinite sequence of going to infinity. (Maier needed to restrict to an infinite sequence to avoid Siegel zeroes, but we are able to resolve this issue by the now standard technique of simply eliminating a prime factor of an exceptional conductor from the sieve-theoretic portion of the argument. As a byproduct, this also makes all of the estimates in our paper effective.)
As its name suggests, the Maier matrix method is usually presented by imagining a matrix of numbers, and using information about the distribution of primes in the columns of this matrix to deduce information about the primes in at least one of the rows of the matrix. We found it convenient to interpret this method in an equivalent probabilistic form as follows. Suppose one wants to find an interval which contained a block of at least primes, each separated from each other by at least (ultimately, will be something like and something like ). One can do this by the probabilistic method: pick to be a random large natural number (with the precise distribution to be chosen later), and try to lower bound the probability that the interval contains at least primes, no two of which are within of each other.
By carefully choosing the residue class of with respect to small primes, one can eliminate several of the from consideration of being prime immediately. For instance, if is chosen to be large and even, then the with even have no chance of being prime and can thus be eliminated; similarly if is large and odd, then cannot be prime for any odd . Using the methods of our previous paper, we can find a residue class (where is a product of a large number of primes) such that, if one chooses to be a large random element of (that is, for some large random integer ), then the set of shifts for which still has a chance of being prime has size comparable to something like ; furthermore this set is fairly well distributed in in the sense that it does not concentrate too strongly in any short subinterval of . The main new difficulty, not present in the previous paper, is to get lower bounds on the size of in addition to upper bounds, but this turns out to be achievable by a suitable modification of the arguments.
Using a version of the prime number theorem in arithmetic progressions due to Gallagher, one can show that for each remaining shift , is going to be prime with probability comparable to , so one expects about primes in the set . An upper bound sieve (e.g. the Selberg sieve) also shows that for any distinct , the probability that and are both prime is . Using this and some routine second moment calculations, one can then show that with large probability, the set will indeed contain about primes, no two of which are closer than to each other; with no other numbers in this interval being prime, this gives a lower bound on .
Klaus Roth, who made fundamental contributions to analytic number theory, died this Tuesday, aged 90.
I never met or communicated with Roth personally, but was certainly influenced by his work; he wrote relatively few papers, but they tended to have outsized impact. For instance, he was one of the key people (together with Bombieri) to work on simplifying and generalising the large sieve, taking it from the technically formidable original formulation of Linnik and Rényi to the clean and general almost orthogonality principle that we have today (discussed for instance in these lecture notes of mine). The paper of Roth that had the most impact on my own personal work was his three-page paper proving what is now known as Roth’s theorem on arithmetic progressions:
Theorem 1 (Roth’s theorem on arithmetic progressions) Let be a set of natural numbers of positive upper density (thus ). Then contains infinitely many arithmetic progressions of length three (with non-zero of course).
At the heart of Roth’s elegant argument was the following (surprising at the time) dichotomy: if had some moderately large density within some arithmetic progression , either one could use Fourier-analytic methods to detect the presence of an arithmetic progression of length three inside , or else one could locate a long subprogression of on which had increased density. Iterating this dichotomy by an argument now known as the density increment argument, one eventually obtains Roth’s theorem, no matter which side of the dichotomy actually holds. This argument (and the many descendants of it), based on various “dichotomies between structure and randomness”, became essential in many other results of this type, most famously perhaps in Szemerédi’s proof of his celebrated theorem on arithmetic progressions that generalised Roth’s theorem to progressions of arbitrary length. More recently, my recent work on the Chowla and Elliott conjectures that was a crucial component of the solution of the Erdös discrepancy problem, relies on an entropy decrement argument which was directly inspired by the density increment argument of Roth.
The Erdös discrepancy problem also is connected with another well known theorem of Roth:
Theorem 2 (Roth’s discrepancy theorem for arithmetic progressions) Let be a sequence in . Then there exists an arithmetic progression in with positive such that
for an absolute constant .
In fact, Roth proved a stronger estimate regarding mean square discrepancy, which I am not writing down here; as with the Roth theorem in arithmetic progressions, his proof was short and Fourier-analytic in nature (although non-Fourier-analytic proofs have since been found, for instance the semidefinite programming proof of Lovasz). The exponent is known to be sharp (a result of Matousek and Spencer).
As a particular corollary of the above theorem, for an infinite sequence of signs, the sums are unbounded in . The Erdös discrepancy problem asks whether the same statement holds when is restricted to be zero. (Roth also established discrepancy theorems for other sets, such as rectangles, which will not be discussed here.)
Finally, one has to mention Roth’s most famous result, cited for instance in his Fields medal citation:
Theorem 3 (Roth’s theorem on Diophantine approximation) Let be an irrational algebraic number. Then for any there is a quantity such that
From the Dirichlet approximation theorem (or from the theory of continued fractions) we know that the exponent in the denominator cannot be reduced to or below. A classical and easy theorem of Liouville gives the claim with the exponent replaced by the degree of the algebraic number ; work of Thue and Siegel reduced this exponent, but Roth was the one who obtained the near-optimal result. An important point is that the constant is ineffective – it is a major open problem in Diophantine approximation to produce any bound significantly stronger than Liouville’s theorem with effective constants. This is because the proof of Roth’s theorem does not exclude any single rational from being close to , but instead very ingeniously shows that one cannot have two different rationals , that are unusually close to , even when the denominators are very different in size. (I refer to this sort of argument as a “dueling conspiracies” argument; they are strangely prevalent throughout analytic number theory.)