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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
Theorem 1 (Two-dimensional Vinogradov main conjecture) One has
as .
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 (depending continuously on ) 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 sake of comparison, from the triangle inequality and Cauchy-Schwarz one has the general inequality
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 .
Theorem 1 (Decoupling theorem) Let . Subdivide the unit interval into equal subintervals of length , and for each such , let be the Fourier transform
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
which of course improves the prime number theorem. Recently with Matomaki and Radziwill, we obtained a common generalisation of this estimate with (1), showing that
as , for any that went to infinity as . We were able to use this estimate to obtain an averaged form of Chowla's conjecture.
In that paper, we asked whether one could improve this estimate further by moving the supremum inside the integral, that is to say to establish the bound
as , for any that went to infinity as . This bound is asserting that is locally Fourier-uniform on most short intervals; it can be written equivalently in terms of the "local Gowers norm" as
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:
Proposition 1 The estimate (4) (or equivalently (6)) holds in the range for any fixed . (In fact one can improve the right-hand side by an arbitrary power of in this case.)
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).
Theorem 1 (Halasz inequality) Let be a multiplicative function bounded in magnitude by , and suppose that , , and are such that
for all real numbers with . Then
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.
The Chowla conjecture asserts, among other things, that one has the asymptotic
as for any distinct integers , where is the Liouville function. (The usual formulation of the conjecture also allows one to consider more general linear forms than the shifts , but for sake of discussion let us focus on the shift case.) This conjecture remains open for , though there are now some partial results when one averages either in or in the , as discussed in this recent post.
A natural generalisation of the Chowla conjecture is the Elliott conjecture. Its original formulation was basically as follows: one had
whenever were bounded completely multiplicative functions and were distinct integers, and one of the was "non-pretentious" in the sense that
for all Dirichlet characters and real numbers . It is easy to see that some condition like (2) is necessary; for instance if and has period then can be verified to be bounded away from zero as .
In a previous paper with Matomaki and Radziwill, we provided a counterexample to the original formulation of the Elliott conjecture, and proposed that (2) be replaced with the stronger condition
as for any Dirichlet character . To support this conjecture, we proved an averaged and non-asymptotic version of this conjecture which roughly speaking showed a bound of the form
whenever was an arbitrarily slowly growing function of , was sufficiently large (depending on and the rate at which grows), and one of the obeyed the condition
for some that was sufficiently large depending on , and all Dirichlet characters of period at most . As further support of this conjecture, I recently established the bound
under the same hypotheses, where is an arbitrarily slowly growing function of .
In view of these results, it is tempting to conjecture that the condition (4) for one of the should be sufficient to obtain the bound
when is large enough depending on . This may well be the case for . However, the purpose of this blog post is to record a simple counterexample for . Let's take for simplicity. Let be a quantity much larger than but much smaller than (e.g. ), and set
For , Taylor expansion gives
and hence
and hence
On the other hand one can easily verify that all of the obey (4) (the restriction there prevents from getting anywhere close to ). So it seems the correct non-asymptotic version of the Elliott conjecture is the following:
Conjecture 1 (Non-asymptotic Elliott conjecture) Let be a natural number, and let be integers. Let , let be sufficiently large depending on , and let be sufficiently large depending on . Let be bounded multiplicative functions such that for some , one has
for all Dirichlet characters of conductor at most . Then
The case of this conjecture follows from the work of Halasz; in my recent paper a logarithmically averaged version of the case of this conjecture is established. The requirement to take to be as large as does not emerge in the averaged Elliott conjecture in my previous paper with Matomaki and Radziwill; it thus seems that this averaging has concealed some of the subtler features of the Elliott conjecture. (However, this subtlety does not seem to affect the asymptotic version of the conjecture formulated in that paper, in which the hypothesis is of the form (3), and the conclusion is of the form (1).)
A similar subtlety arises when trying to control the maximal integral
In my previous paper with Matomaki and Radziwill, we could show that easier expression
was small (for a slowly growing function of ) if was bounded and completely multiplicative, and one had a condition of the form
for some large . However, to obtain an analogous bound for (5) it now appears that one needs to strengthen the above condition to
in order to address the counterexample in which for some between and . This seems to suggest that proving (5) (which is closely related to the case of the Chowla conjecture) could in fact be rather difficult; the estimation of (6) relied primarily of prior work of Matomaki and Radziwill which used the hypothesis (7), but as this hypothesis is not sufficient to conclude (5), some additional input must also be used.
I recently learned about a curious operation on square matrices known as sweeping, which is used in numerical linear algebra (particularly in applications to statistics), as a useful and more robust variant of the usual Gaussian elimination operations seen in undergraduate linear algebra courses. Given an matrix (with, say, complex entries) and an index , with the entry non-zero, the sweep of at is the matrix given by the formulae
for all . Thus for instance if , and is written in block form as
for some row vector , column vector , and minor , one has
The inverse sweep operation is given by a nearly identical set of formulae:
for some row vector , column vector , and minor , one has
The inverse sweep operation is given by a nearly identical set of formulae:
for all . One can check that these operations invert each other. Actually, each sweep turns out to have order , so that : an inverse sweep performs the same operation as three forward sweeps. Sweeps also preserve the space of symmetric matrices (allowing one to cut down computational run time in that case by a factor of two), and behave well with respect to principal minors; a sweep of a principal minor is a principal minor of a sweep, after adjusting indices appropriately.
Remarkably, the sweep operators all commute with each other: . If and we perform the first sweeps (in any order) to a matrix
with a minor, a matrix, a matrix, and a matrix, one obtains the new matrix
Note the appearance of the Schur complement in the bottom right block. Thus, for instance, one can essentially invert a matrix by performing all sweeps:
If a matrix has the form
for a minor , column vector , row vector , and scalar , then performing the first sweeps gives
and all the components of this matrix are usable for various numerical linear algebra applications in statistics (e.g. in least squares regression). Given that sweeps behave well with inverses, it is perhaps not surprising that sweeps also behave well under determinants: the determinant of can be factored as the product of the entry and the determinant of the matrix formed from by removing the row and column. As a consequence, one can compute the determinant of fairly efficiently (so long as the sweep operations don't come close to dividing by zero) by sweeping the matrix for in turn, and multiplying together the entry of the matrix just before the sweep for to obtain the determinant.
It turns out that there is a simple geometric explanation for these seemingly magical properties of the sweep operation. Any matrix creates a graph (where we think of as the space of column vectors). This graph is an -dimensional subspace of . Conversely, most subspaces of arises as graphs; there are some that fail the vertical line test, but these are a positive codimension set of counterexamples.
We use to denote the standard basis of , with the standard basis for the first factor of and the standard basis for the second factor. The operation of sweeping the entry then corresponds to a ninety degree rotation in the plane, that sends to (and to ), keeping all other basis vectors fixed: thus we have
for generic (more precisely, those with non-vanishing entry ). For instance, if and is of the form (1), then is the set of tuples obeying the equations
The image of under is . Since we can write the above system of equations (for ) as
we see from (2) that is the graph of . Thus the sweep operation is a multidimensional generalisation of the high school geometry fact that the line in the plane becomes after applying a ninety degree rotation.
It is then an instructive exercise to use this geometric interpretation of the sweep operator to recover all the remarkable properties about these operations listed above. It is also useful to compare the geometric interpretation of sweeping as rotation of the graph to that of Gaussian elimination, which instead shears and reflects the graph by various elementary transformations (this is what is going on geometrically when one performs Gaussian elimination on an augmented matrix). Rotations are less distorting than shears, so one can see geometrically why sweeping can produce fewer numerical artefacts than Gaussian elimination.
Let and be two random variables taking values in the same (discrete) range , and let be some subset of , which we think of as the set of "bad" outcomes for either or . If and have the same probability distribution, then clearly
In particular, if it is rare for to lie in , then it is also rare for to lie in .
If and do not have exactly the same probability distribution, but their probability distributions are close to each other in some sense, then we can expect to have an approximate version of the above statement. For instance, from the definition of the total variation distance between two random variables (or more precisely, the total variation distance between the probability distributions of two random variables), we see that
for any . In particular, if it is rare for to lie in , and are close in total variation, then it is also rare for to lie in .
for any . In particular, if it is rare for to lie in , and are close in total variation, then it is also rare for to lie in .
A basic inequality in information theory is Pinsker's inequality
where the Kullback-Leibler divergence is defined by the formula
(See this previous blog post for a proof of this inequality.) A standard application of Jensen's inequality reveals that is non-negative (Gibbs' inequality), and vanishes if and only if , have the same distribution; thus one can think of as a measure of how close the distributions of and are to each other, although one should caution that this is not a symmetric notion of distance, as in general. Inserting Pinsker's inequality into (1), we see for instance that
Thus, if is close to in the Kullback-Leibler sense, and it is rare for to lie in , then it is rare for to lie in as well.
We can specialise this inequality to the case when a uniform random variable on a finite range of some cardinality , in which case the Kullback-Leibler divergence simplifies to
is the Shannon entropy of . Again, a routine application of Jensen's inequality shows that , with equality if and only if is uniformly distributed on . The above inequality then becomes
Thus, if is a small fraction of (so that it is rare for to lie in ), and the entropy of is very close to the maximum possible value of , then it is rare for to lie in also.
Thus, if is a small fraction of (so that it is rare for to lie in ), and the entropy of is very close to the maximum possible value of , then it is rare for to lie in also.
The inequality (2) is only useful when the entropy is close to in the sense that , otherwise the bound is worse than the trivial bound of . In my recent paper on the Chowla and Elliott conjectures, I ended up using a variant of (2) which was still non-trivial when the entropy was allowed to be smaller than . More precisely, I used the following simple inequality, which is implicit in the arguments of that paper but which I would like to make more explicit in this post:
Lemma 1 (Pinsker-type inequality) Let be a random variable taking values in a finite range of cardinality , let be a uniformly distributed random variable in , and let be a subset of . Then
Proof: Consider the conditional entropy . On the one hand, we have
by Jensen's inequality. On the other hand, one has
where we have again used Jensen's inequality. Putting the two inequalities together, we obtain the claim.
Remark 2 As noted in comments, this inequality can be viewed as a special case of the more general inequality
for arbitrary random variables taking values in the same discrete range , which follows from the data processing inequality
for arbitrary functions , applied to the indicator function . Indeed one has
where is the entropy function.
Thus, for instance, if one has
for some much larger than (so that ), then
More informally: if the entropy of is somewhat close to the maximum possible value of , and it is exponentially rare for a uniform variable to lie in , then it is still somewhat rare for to lie in . The estimate given is close to sharp in this regime, as can be seen by calculating the entropy of a random variable which is uniformly distributed inside a small set with some probability and uniformly distributed outside of with probability , for some parameter .
It turns out that the above lemma combines well with concentration of measure estimates; in my paper, I used one of the simplest such estimates, namely Hoeffding's inequality, but there are of course many other estimates of this type (see e.g. this previous blog post for some others). Roughly speaking, concentration of measure inequalities allow one to make approximations such as
with exponentially high probability, where is a uniform distribution and is some reasonable function of . Combining this with the above lemma, we can then obtain approximations of the form
with somewhat high probability, if the entropy of is somewhat close to maximum. This observation, combined with an "entropy decrement argument" that allowed one to arrive at a situation in which the relevant random variable did have a near-maximum entropy, is the key new idea in my recent paper; for instance, one can use the approximation (3) to obtain an approximation of the form
with somewhat high probability, if the entropy of is somewhat close to maximum. This observation, combined with an "entropy decrement argument" that allowed one to arrive at a situation in which the relevant random variable did have a near-maximum entropy, is the key new idea in my recent paper; for instance, one can use the approximation (3) to obtain an approximation of the form
for "most" choices of and a suitable choice of (with the latter being provided by the entropy decrement argument). The left-hand side is tied to Chowla-type sums such as through the multiplicativity of , while the right-hand side, being a linear correlation involving two parameters rather than just one, has "finite complexity" and can be treated by existing techniques such as the Hardy-Littlewood circle method. One could hope that one could similarly use approximations such as (3) in other problems in analytic number theory or combinatorics.
The Chowla conjecture asserts that all non-trivial correlations of the Liouville function are asymptotically negligible; for instance, it asserts that
as for any fixed natural number . This conjecture remains open, though there are a number of partial results (e.g. these two previous results of Matomaki, Radziwill, and myself).
A natural generalisation of Chowla's conjecture was proposed by Elliott. For simplicity we will only consider Elliott's conjecture for the pair correlations
For such correlations, the conjecture was that one had
as for any natural number , as long as was a completely multiplicative function with magnitude bounded by , and such that
for any Dirichlet character and any real number . In the language of "pretentious number theory", as developed by Granville and Soundararajan, the hypothesis (2) asserts that the completely multiplicative function does not "pretend" to be like the completely multiplicative function for any character and real number . A condition of this form is necessary; for instance, if is precisely equal to and has period , then is equal to as and (1) clearly fails. The prime number theorem in arithmetic progressions implies that the Liouville function obeys (2), and so the Elliott conjecture contains the Chowla conjecture as a special case.
as for any natural number , as long as was a completely multiplicative function with magnitude bounded by , and such that
for any Dirichlet character and any real number . In the language of "pretentious number theory", as developed by Granville and Soundararajan, the hypothesis (2) asserts that the completely multiplicative function does not "pretend" to be like the completely multiplicative function for any character and real number . A condition of this form is necessary; for instance, if is precisely equal to and has period , then is equal to as and (1) clearly fails. The prime number theorem in arithmetic progressions implies that the Liouville function obeys (2), and so the Elliott conjecture contains the Chowla conjecture as a special case.
As it turns out, Elliott's conjecture is false as stated, with the counterexample having the property that "pretends" locally to be the function for in various intervals , where and go to infinity in a certain prescribed sense. See this paper of Matomaki, Radziwill, and myself for details. However, we view this as a technicality, and continue to believe that certain "repaired" versions of Elliott's conjecture still hold. For instance, our counterexample does not apply when is restricted to be real-valued rather than complex, and we believe that Elliott's conjecture is valid in this setting. Returning to the complex-valued case, we still expect the asymptotic (1) provided that the condition (2) is replaced by the stronger condition
as for all fixed Dirichlet characters . In our paper we supported this claim by establishing a certain "averaged" version of this conjecture; see that paper for further details. (See also this recent paper of Frantzikinakis and Host which establishes a different averaged version of this conjecture.)
One can make a stronger "non-asymptotic" version of this corrected Elliott conjecture, in which the parameter does not go to infinity, or equivalently that the function is permitted to depend on :
Conjecture 1 (Non-asymptotic Elliott conjecture) Let , let be sufficiently large depending on , and let be sufficiently large depending on . Suppose that is a completely multiplicative function with magnitude bounded by , such that
for all Dirichlet characters of period at most . Then one has
for all natural numbers .
The -dependent factor in the constraint is necessary, as can be seen by considering the completely multiplicative function (for instance). Again, the results in my previous paper with Matomaki and Radziwill can be viewed as establishing an averaged version of this conjecture.
Meanwhile, we have the following conjecture that is the focus of the Polymath5 project:
Conjecture 2 (Erdös discrepancy conjecture) For any function , the discrepancy
is infinite.
It is instructive to compute some near-counterexamples to Conjecture 2 that illustrate the difficulty of the Erdös discrepancy problem. The first near-counterexample is that of a non-principal Dirichlet character that takes values in rather than . For this function, one has from the complete multiplicativity of that
If denotes the period of , then has mean zero on every interval of length , and thus
Thus has bounded discrepancy.
Of course, this is not a true counterexample to Conjecture 2 because can take the value . Let us now consider the following variant example, which is the simplest member of a family of examples studied by Borwein, Choi, and Coons. Let be the non-principal Dirichlet character of period (thus equals when , when , and when ), and define the completely multiplicative function by setting when and . This is about the simplest modification one can make to the previous near-counterexample to eliminate the zeroes. Now consider the sum
with for some large . Writing with coprime to and at most , we can write this sum as
Now observe that . The function has mean zero on every interval of length three, and is equal to mod , and thus
for every , and thus
Thus also has unbounded discrepancy, but only barely so (it grows logarithmically in ). These examples suggest that the main "enemy" to proving Conjecture 2 comes from completely multiplicative functions that somehow "pretend" to be like a Dirichlet character but do not vanish at the zeroes of that character. (Indeed, the special case of Conjecture 2 when is completely multiplicative is already open, appears to be an important subcase.)
All of these conjectures remain open. However, I would like to record in this blog post the following striking connection, illustrating the power of the Elliott conjecture (particularly in its nonasymptotic formulation):
Theorem 3 (Elliott conjecture implies unbounded discrepancy) Conjecture 1 implies Conjecture 2.
The argument relies heavily on two observations that were previously made in connection with the Polymath5 project. The first is a Fourier-analytic reduction that replaces the Erdos Discrepancy Problem with an averaged version for completely multiplicative functions . An application of Cauchy-Schwarz then shows that any counterexample to that version will violate the conclusion of Conjecture 1, so if one assumes that conjecture then must pretend to be like a function of the form . One then uses (a generalisation) of a second argument from Polymath5 to rule out this case, basically by reducing matters to a more complicated version of the Borwein-Choi-Coons analysis. Details are provided below the fold.
There is some hope that the Chowla and Elliott conjectures can be attacked, as the parity barrier which is so impervious to attack for the twin prime conjecture seems to be more permeable in this setting. (For instance, in my previous post I raised a possible approach, based on establishing expander properties of a certain random graph, which seems to get around the parity problem, in principle at least.)
(Update, Sep 25: fixed some treatment of error terms, following a suggestion of Andrew Granville.)
The twin prime conjecture is one of the oldest unsolved problems in analytic number theory. There are several reasons why this conjecture remains out of reach of current techniques, but the most important obstacle is the parity problem which prevents purely sieve-theoretic methods (or many other popular methods in analytic number theory, such as the circle method) from detecting pairs of prime twins in a way that can distinguish them from other twins of almost primes. The parity problem is discussed in these previous blog posts; this obstruction is ultimately powered by the Möbius pseudorandomness principle that asserts that the Möbius function is asymptotically orthogonal to all "structured" functions (and in particular, to the weight functions constructed from sieve theory methods).
However, there is an intriguing "alternate universe" in which the Möbius function is strongly correlated with some structured functions, and specifically with some Dirichlet characters, leading to the existence of the infamous "Siegel zero". In this scenario, the parity problem obstruction disappears, and it becomes possible, in principle, to attack problems such as the twin prime conjecture. In particular, we have the following result of Heath-Brown:
Theorem 1 At least one of the following two statements are true:
- (Twin prime conjecture) There are infinitely many primes such that is also prime.
- (No Siegel zeroes) There exists a constant such that for every real Dirichlet character of conductor , the associated Dirichlet -function has no zeroes in the interval .
Informally, this result asserts that if one had an infinite sequence of Siegel zeroes, one could use this to generate infinitely many twin primes. See this survey of Friedlander and Iwaniec for more on this "illusory" or "ghostly" parallel universe in analytic number theory that should not actually exist, but is surprisingly self-consistent and to date proven to be impossible to banish from the realm of possibility.
The strategy of Heath-Brown's proof is fairly straightforward to describe. The usual starting point is to try to lower bound
for some large value of , where is the von Mangoldt function. Actually, in this post we will work with the slight variant
is the second von Mangoldt function, and denotes Dirichlet convolution, and is an (unsquared) Selberg sieve that damps out small prime factors. This sum also detects twin primes, but will lead to slightly simpler computations. For technical reasons we will also smooth out the interval and remove very small primes from , but we will skip over these steps for the purpose of this informal discussion. (In Heath-Brown's original paper, the Selberg sieve is essentially replaced by the more combinatorial restriction for some large , where is the primorial of , but I found the computations to be slightly easier if one works with a Selberg sieve, particularly if the sieve is not squared to make it nonnegative.)
If there is a Siegel zero with close to and a Dirichlet character of conductor , then multiplicative number theory methods can be used to show that the Möbius function "pretends" to be like the character in the sense that for "most" primes near (e.g. in the range for some small and large ). Traditionally, one uses complex-analytic methods to demonstrate this, but one can also use elementary multiplicative number theory methods to establish these results (qualitatively at least), as will be shown below the fold.
The fact that pretends to be like can be used to construct a tractable approximation (after inserting the sieve weight ) in the range (where for some large ) for the second von Mangoldt function , namely the function
Roughly speaking, we think of the periodic function and the slowly varying function as being of about the same "complexity" as the constant function , so that is roughly of the same "complexity" as the divisor function
which is considerably simpler to obtain asymptotics for than the von Mangoldt function as the Möbius function is no longer present. (For instance, note from the Dirichlet hyperbola method that one can estimate to accuracy with little difficulty, whereas to obtain a comparable level of accuracy for or is essentially the Riemann hypothesis.)
One expects to be a good approximant to if is of size and has no prime factors less than for some large constant . The Selberg sieve will be mostly supported on numbers with no prime factor less than . As such, one can hope to approximate (1) by the expression
as it turns out, the error between this expression and (1) is easily controlled by sieve-theoretic techniques. Let us ignore the Selberg sieve for now and focus on the slightly simpler sum
As discussed above, this sum should be thought of as a slightly more complicated version of the sum
Accordingly, let us look (somewhat informally) at the task of estimating the model sum (3). One can think of this problem as basically that of counting solutions to the equation with in various ranges; this is clearly related to understanding the equidistribution of the hyperbola in . Taking Fourier transforms, the latter problem is closely related to estimation of the Kloosterman sums
where denotes the inverse of in . One can then use the Weil bound
where is the greatest common divisor of (with the convention that this is equal to if vanish), and the decays to zero as . The Weil bound yields good enough control on error terms to estimate (3), and as it turns out the same method also works to estimate (2) (provided that with large enough).
Actually one does not need the full strength of the Weil bound here; any power savings over the trivial bound of will do. In particular, it will suffice to use the weaker, but easier to prove, bounds of Kloosterman:
Lemma 2 (Kloosterman bound) One has
whenever and are coprime to , where the is with respect to the limit (and is uniform in ).
Proof: Observe from change of variables that the Kloosterman sum is unchanged if one replaces with for . For fixed , the number of such pairs is at least , thanks to the divisor bound. Thus it will suffice to establish the fourth moment bound
The left-hand side can be rearranged as
which by Fourier summation is equal to
Observe from the quadratic formula and the divisor bound that each pair has at most solutions to the system of equations . Hence the number of quadruples of the desired form is , and the claim follows.
We will also need another easy case of the Weil bound to handle some other portions of (2):
Lemma 3 (Easy Weil bound) Let be a primitive real Dirichlet character of conductor , and let . Then
Proof: As is the conductor of a primitive real Dirichlet character, is equal to times a squarefree odd number for some . By the Chinese remainder theorem, it thus suffices to establish the claim when is an odd prime. We may assume that is not divisible by this prime , as the claim is trivial otherwise. If vanishes then does not vanish, and the claim follows from the mean zero nature of ; similarly if vanishes. Hence we may assume that do not vanish, and then we can normalise them to equal . By completing the square it now suffices to show that
whenever . As is on the quadratic residues and on the non-residues, it now suffices to show that
But by making the change of variables , the left-hand side becomes , and the claim follows.
While the basic strategy of Heath-Brown's argument is relatively straightforward, implementing it requires a large amount of computation to control both main terms and error terms. I experimented for a while with rearranging the argument to try to reduce the amount of computation; I did not fully succeed in arriving at a satisfactorily minimal amount of superfluous calculation, but I was able to at least reduce this amount a bit, mostly by replacing a combinatorial sieve with a Selberg-type sieve (which was not needed to be positive, so I dispensed with the squaring aspect of the Selberg sieve to simplify the calculations a little further; also for minor reasons it was convenient to retain a tiny portion of the combinatorial sieve to eliminate extremely small primes). Also some modest reductions in complexity can be obtained by using the second von Mangoldt function in place of . These exercises were primarily for my own benefit, but I am placing them here in case they are of interest to some other readers.
The Poincaré upper half-plane (with a boundary consisting of the real line together with the point at infinity ) carries an action of the projective special linear group
via fractional linear transformations:
Here and in the rest of the post we will abuse notation by identifying elements of the special linear group with their equivalence class in ; this will occasionally create or remove a factor of two in our formulae, but otherwise has very little effect, though one has to check that various definitions and expressions (such as (1)) are unaffected if one replaces a matrix by its negation . In particular, we recommend that the reader ignore the signs that appear from time to time in the discussion below.
Here and in the rest of the post we will abuse notation by identifying elements of the special linear group with their equivalence class in ; this will occasionally create or remove a factor of two in our formulae, but otherwise has very little effect, though one has to check that various definitions and expressions (such as (1)) are unaffected if one replaces a matrix by its negation . In particular, we recommend that the reader ignore the signs that appear from time to time in the discussion below.
As the action of on is transitive, and any given point in (e.g. ) has a stabiliser isomorphic to the projective rotation group , we can view the Poincaré upper half-plane as a homogeneous space for , and more specifically the quotient space of of a maximal compact subgroup . In fact, we can make the half-plane a symmetric space for , by endowing with the Riemannian metric
(using Cartesian coordinates ), which is invariant with respect to the action. Like any other Riemannian metric, the metric on generates a number of other important geometric objects on , such as the distance function which can be computed to be given by the formula
the volume measure , which can be computed to be
the volume measure , which can be computed to be
and the Laplace-Beltrami operator, which can be computed to be (here we use the negative definite sign convention for ). As the metric was -invariant, all of these quantities arising from the metric are similarly -invariant in the appropriate sense.
The Gauss curvature of the Poincaré half-plane can be computed to be the constant , thus is a model for two-dimensional hyperbolic geometry, in much the same way that the unit sphere in is a model for two-dimensional spherical geometry (or is a model for two-dimensional Euclidean geometry). (Indeed, is isomorphic (via projection to a null hyperplane) to the upper unit hyperboloid in the Minkowski spacetime , which is the direct analogue of the unit sphere in Euclidean spacetime or the plane in Galilean spacetime .)
One can inject arithmetic into this geometric structure by passing from the Lie group to the full modular group
or congruence subgroups such as
for natural number , or to the discrete stabiliser of the point at infinity:
These are discrete subgroups of , nested by the subgroup inclusions
for natural number , or to the discrete stabiliser of the point at infinity:
These are discrete subgroups of , nested by the subgroup inclusions
There are many further discrete subgroups of (known collectively as Fuchsian groups) that one could consider, but we will focus attention on these three groups in this post.
Any discrete subgroup of generates a quotient space , which in general will be a non-compact two-dimensional orbifold. One can understand such a quotient space by working with a fundamental domain – a set consisting of a single representative of each of the orbits of in . This fundamental domain is by no means uniquely defined, but if the fundamental domain is chosen with some reasonable amount of regularity, one can view as the fundamental domain with the boundaries glued together in an appropriate sense. Among other things, fundamental domains can be used to induce a volume measure on from the volume measure on (restricted to a fundamental domain). By abuse of notation we will refer to both measures simply as when there is no chance of confusion.
For instance, a fundamental domain for is given (up to null sets) by the strip , with identifiable with the cylinder formed by gluing together the two sides of the strip. A fundamental domain for is famously given (again up to null sets) by an upper portion , with the left and right sides again glued to each other, and the left and right halves of the circular boundary glued to itself. A fundamental domain for can be formed by gluing together
copies of a fundamental domain for in a rather complicated but interesting fashion.
While fundamental domains can be a convenient choice of coordinates to work with for some computations (as well as for drawing appropriate pictures), it is geometrically more natural to avoid working explicitly on such domains, and instead work directly on the quotient spaces . In order to analyse functions on such orbifolds, it is convenient to lift such functions back up to and identify them with functions which are -automorphic in the sense that for all and . Such functions will be referred to as -automorphic forms, or automorphic forms for short (we always implicitly assume all such functions to be measurable). (Strictly speaking, these are the automorphic forms with trivial factor of automorphy; one can certainly consider other factors of automorphy, particularly when working with holomorphic modular forms, which corresponds to sections of a more non-trivial line bundle over than the trivial bundle that is implicitly present when analysing scalar functions . However, we will not discuss this (important) more general situation here.)
An important way to create a -automorphic form is to start with a non-automorphic function obeying suitable decay conditions (e.g. bounded with compact support will suffice) and form the Poincaré series defined by
which is clearly -automorphic. (One could equivalently write in place of here; there are good argument for both conventions, but I have ultimately decided to use the convention, which makes explicit computations a little neater at the cost of making the group actions work in the opposite order.) Thus we naturally see sums over associated with -automorphic forms. A little more generally, given a subgroup of and a -automorphic function of suitable decay, we can form a relative Poincaré series by
where is any fundamental domain for , that is to say a subset of consisting of exactly one representative for each right coset of . As is -automorphic, we see (if has suitable decay) that does not depend on the precise choice of fundamental domain, and is -automorphic. These operations are all compatible with each other, for instance . A key example of Poincaré series are the Eisenstein series, although there are of course many other Poincaré series one can consider by varying the test function .
For future reference we record the basic but fundamental unfolding identities
for any function with sufficient decay, and any -automorphic function of reasonable growth (e.g. bounded and compact support, and bounded, will suffice). Note that is viewed as a function on on the left-hand side, and as a -automorphic function on on the right-hand side. More generally, one has
whenever are discrete subgroups of , is a -automorphic function with sufficient decay on , and is a -automorphic (and thus also -automorphic) function of reasonable growth. These identities will allow us to move fairly freely between the three domains , , and in our analysis.
for any function with sufficient decay, and any -automorphic function of reasonable growth (e.g. bounded and compact support, and bounded, will suffice). Note that is viewed as a function on on the left-hand side, and as a -automorphic function on on the right-hand side. More generally, one has
whenever are discrete subgroups of , is a -automorphic function with sufficient decay on , and is a -automorphic (and thus also -automorphic) function of reasonable growth. These identities will allow us to move fairly freely between the three domains , , and in our analysis.
When computing various statistics of a Poincaré series , such as its values at special points , or the quantity , expressions of interest to analytic number theory naturally emerge. We list three basic examples of this below, discussed somewhat informally in order to highlight the main ideas rather than the technical details.
The first example we will give concerns the problem of estimating the sum
where is the divisor function. This can be rewritten (by factoring and ) as
which is basically a sum over the full modular group . At this point we will "cheat" a little by moving to the related, but different, sum
This sum is not exactly the same as (8), but will be a little easier to handle, and it is plausible that the methods used to handle this sum can be modified to handle (8). Observe from (2) and some calculation that the distance between and is given by the formula
where is the divisor function. This can be rewritten (by factoring and ) as
which is basically a sum over the full modular group . At this point we will "cheat" a little by moving to the related, but different, sum
This sum is not exactly the same as (8), but will be a little easier to handle, and it is plausible that the methods used to handle this sum can be modified to handle (8). Observe from (2) and some calculation that the distance between and is given by the formula
and so one can express the above sum as
(the factor of coming from the quotient by in the projective special linear group); one can express this as , where and is the indicator function of the ball . Thus we see that expressions such as (7) are related to evaluations of Poincaré series. (In practice, it is much better to use smoothed out versions of indicator functions in order to obtain good control on sums such as (7) or (9), but we gloss over this technical detail here.)
The second example concerns the relative
of the sum (7). Note from multiplicativity that (7) can be written as , which is superficially very similar to (10), but with the key difference that the polynomial is irreducible over the integers.
of the sum (7). Note from multiplicativity that (7) can be written as , which is superficially very similar to (10), but with the key difference that the polynomial is irreducible over the integers.
As with (7), we may expand (10) as
At first glance this does not look like a sum over a modular group, but one can manipulate this expression into such a form in one of two (closely related) ways. First, observe that any factorisation of into Gaussian integers gives rise (upon taking norms) to an identity of the form , where and . Conversely, by using the unique factorisation of the Gaussian integers, every identity of the form gives rise to a factorisation of the form , essentially uniquely up to units. Now note that is of the form if and only if , in which case . Thus we can essentially write the above sum as something like
and one the modular group is now manifest. An equivalent way to see these manipulations is as follows. A triple of natural numbers with gives rise to a positive quadratic form of normalised discriminant equal to with integer coefficients (it is natural here to allow to take integer values rather than just natural number values by essentially doubling the sum). The group acts on the space of such quadratic forms in a natural fashion (by composing the quadratic form with the inverse of an element of ). Because the discriminant has class number one (this fact is equivalent to the unique factorisation of the gaussian integers, as discussed in this previous post), every form in this space is equivalent (under the action of some element of ) with the standard quadratic form . In other words, one has
and one the modular group is now manifest. An equivalent way to see these manipulations is as follows. A triple of natural numbers with gives rise to a positive quadratic form of normalised discriminant equal to with integer coefficients (it is natural here to allow to take integer values rather than just natural number values by essentially doubling the sum). The group acts on the space of such quadratic forms in a natural fashion (by composing the quadratic form with the inverse of an element of ). Because the discriminant has class number one (this fact is equivalent to the unique factorisation of the gaussian integers, as discussed in this previous post), every form in this space is equivalent (under the action of some element of ) with the standard quadratic form . In other words, one has
which (up to a harmless sign) is exactly the representation , , introduced earlier, and leads to the same reformulation of the sum (10) in terms of expressions like (11). Similar considerations also apply if the quadratic polynomial is replaced by another quadratic, although one has to account for the fact that the class number may now exceed one (so that unique factorisation in the associated quadratic ring of integers breaks down), and in the positive discriminant case the fact that the group of units might be infinite presents another significant technical problem.
Note that has real part and imaginary part . Thus (11) is (up to a factor of two) the Poincaré series as in the preceding example, except that is now the indicator of the sector .
Sums involving subgroups of the full modular group, such as , often arise when imposing congruence conditions on sums such as (10), for instance when trying to estimate the expression when and are large. As before, one then soon arrives at the problem of evaluating a Poincaré series at one or more special points, where the series is now over rather than .
The third and final example concerns averages of Kloosterman sums
where and is the inverse of in the multiplicative group . It turns out that the norms of Poincaré series or are closely tied to such averages. Consider for instance the quantity
where is a natural number and is a -automorphic form that is of the form
where and is the inverse of in the multiplicative group . It turns out that the norms of Poincaré series or are closely tied to such averages. Consider for instance the quantity
where is a natural number and is a -automorphic form that is of the form
for some integer and some test function , which for sake of discussion we will take to be smooth and compactly supported. Using the unfolding formula (6), we may rewrite (13) as
To compute this, we use the double coset decomposition
where for each , are arbitrarily chosen integers such that . To see this decomposition, observe that every element in outside of can be assumed to have by applying a sign , and then using the row and column operations coming from left and right multiplication by (that is, shifting the top row by an integer multiple of the bottom row, and shifting the right column by an integer multiple of the left column) one can place in the interval and to be any specified integer pair with . From this we see that
and so from further use of the unfolding formula (5) we may expand (13) as
The first integral is just . The second expression is more interesting. We have
so we can write
which on shifting by simplifies a little to
and then on scaling by simplifies a little further to
Note that as , we have modulo . Comparing the above calculations with (12), we can thus write (13) as
is a certain integral involving and a parameter , but which does not depend explicitly on parameters such as . Thus we have indeed expressed the expression (13) in terms of Kloosterman sums. It is possible to invert this analysis and express varius weighted sums of Kloosterman sums in terms of expressions (possibly involving inner products instead of norms) of Poincaré series, but we will not do so here; see Chapter 16 of Iwaniec and Kowalski for further details.
Traditionally, automorphic forms have been analysed using the spectral theory of the Laplace-Beltrami operator on spaces such as or , so that a Poincaré series such as might be expanded out using inner products of (or, by the unfolding identities, ) with various generalised eigenfunctions of (such as cuspidal eigenforms, or Eisenstein series). With this approach, special functions, and specifically the modified Bessel functions of the second kind, play a prominent role, basically because the -automorphic functions
for and non-zero are generalised eigenfunctions of (with eigenvalue ), and are almost square-integrable on (the norm diverges only logarithmically at one end of the cylinder , while decaying exponentially fast at the other end ).
However, as discussed in this previous post, the spectral theory of an essentially self-adjoint operator such as is basically equivalent to the theory of various solution operators associated to partial differential equations involving that operator, such as the Helmholtz equation , the heat equation , the Schrödinger equation , or the wave equation . Thus, one can hope to rephrase many arguments that involve spectral data of into arguments that instead involve resolvents , heat kernels , Schrödinger propagators , or wave propagators , or involve the PDE more directly (e.g. applying integration by parts and energy methods to solutions of such PDE). This is certainly done to some extent in the existing literature; resolvents and heat kernels, for instance, are often utilised. In this post, I would like to explore the possibility of reformulating spectral arguments instead using the inhomogeneous wave equation
Actually it will be a bit more convenient to normalise the Laplacian by , and look instead at the automorphic wave equation
This equation somewhat resembles a "Klein-Gordon" type equation, except that the mass is imaginary! This would lead to pathological behaviour were it not for the negative curvature, which in principle creates a spectral gap of that cancels out this factor.
This equation somewhat resembles a "Klein-Gordon" type equation, except that the mass is imaginary! This would lead to pathological behaviour were it not for the negative curvature, which in principle creates a spectral gap of that cancels out this factor.
The point is that the wave equation approach gives access to some nice PDE techniques, such as energy methods, Sobolev inequalities and finite speed of propagation, which are somewhat submerged in the spectral framework. The wave equation also interacts well with Poincaré series; if for instance and are -automorphic solutions to (15) obeying suitable decay conditions, then their Poincaré series and will be -automorphic solutions to the same equation (15), basically because the Laplace-Beltrami operator commutes with translations. Because of these facts, it is possible to replicate several standard spectral theory arguments in the wave equation framework, without having to deal directly with things like the asymptotics of modified Bessel functions. The wave equation approach to automorphic theory was introduced by Faddeev and Pavlov (using the Lax-Phillips scattering theory), and developed further by by Lax and Phillips, to recover many spectral facts about the Laplacian on modular curves, such as the Weyl law and the Selberg trace formula. Here, I will illustrate this by deriving three basic applications of automorphic methods in a wave equation framework, namely
- Using the Weil bound on Kloosterman sums to derive Selberg's 3/16 theorem on the least non-trivial eigenvalue for on (discussed previously here);
- Conversely, showing that Selberg's eigenvalue conjecture (improving Selberg's bound to the optimal ) implies an optimal bound on (smoothed) sums of Kloosterman sums; and
- Using the same bound to obtain pointwise bounds on Poincaré series similar to the ones discussed above. (Actually, the argument here does not use the wave equation, instead it just uses the Sobolev inequality.)
This post originated from an attempt to finally learn this part of analytic number theory properly, and to see if I could use a PDE-based perspective to understand it better. Ultimately, this is not that dramatic a depature from the standard approach to this subject, but I found it useful to think of things in this fashion, probably due to my existing background in PDE.
I thank Bill Duke and Ben Green for helpful discussions. My primary reference for this theory was Chapters 15, 16, and 21 of Iwaniec and Kowalski.


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46 CFR 131.565 - Monthly tests and inspections.
(a) Each lifesaving appliance, including lifeboat equipment, must be inspected monthly against the checklist required by § 131.545(a)(1) of this subpart to ensure that it is aboard and in good order. A report of the inspection, including a statement on the condition of the appliance, must be entered in the vessel's logbook.
(b) Each Emergency Position Indicating Radio Beacon (EPIRB) and each Search and Rescue Transponder (SART), other than an EPIRB or SART in an inflatable liferaft, must be tested monthly. The EPIRB must be tested using the integrated test circuit and the output indicator (test button) to determine that it works.
Title 46 published on 2015-12-02.
No entries appear in the Federal Register after this date, for 46 CFR Part 131.


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text: World War I National World War I Museum And Memorial To Add More Exhibition Space By Laura Spencer • May 5, 2016 courtesy: National World War I Museum and Memorial Weeks after the end of World War I in 1918, Kansas Citians started fundraising for a memorial. A community fund drive raised more than $2.5 million, and Liberty Memorial opened on Nov. 11, 1926. In 2006, the National World War I Museum, a $102 million project "dedicated to remembering, interpreting and understanding the Great War" opened to the public. Men In Uniform By Gina Kaufmann & Matthew Long-Middleton & Coy Dugger • Mar 28, 2016 According to Pellom McDaniels, when African-Americans served in World War I donning uniforms, the experience empowered them, not just as Americans but as men. On the homefront, they relived that dignity in their baseball careers. Guest: Pellom McDaniels, curator of African-American collections, Emory University, panelist, Wide Open Town: Kansas City During the Pendergast Era National World War I Museum's 100-Year-Old Letters Provide Inspiration For A Folk Song By Laura Spencer • Feb 26, 2016 Laura Spencer / KCUR 89.3 Listen Listening... / 5:26 Florence Hemphill grew up in a small town in Kansas, and saw the horrors of World War I up close when she served as a nurse in France. She wrote more than a hundred letters, sharing her experiences with family members. Singer-songwriter Joe Crookston recently teamed up with the National World War I Museum and Memorial to tell her story – through art and music — at the Folk Alliance International Conference in Kansas City, Missouri. Nurse, Please By Gina Kaufmann & Matthew Long-Middleton & Kyle Smith • Jan 4, 2016 The history of nursing started on the battlefield. The profession that emerged is still with us, but in a totally transformed medical landscape. Using an exhibit at the World War I Museum as a jumping off point, this discussion explores how the origins of nursing have shaped both the realities and misconceptions of the field today. Guests: Video Gurus: World War I By Steve Kraske & Beth Lipoff • Aug 28, 2015 Whether you want to view World War I from the trenches, explore the war's Christmas truce or cruise the skies, our Video Gurus have something to feed your historical need. Check out what they had to say on this edition of Up to Date. All Quiet on the Western Front, unrated (before current ratings) Author Erik Larson On The Last Voyage Of The Lusitania By Steve Kraske & Eliza Spertus • Mar 17, 2015 In May 1915, a German U-boat sunk one of the world's greatest ocean liners, the Lusitania. Erik Larson's new book, Dead Wake: the Last Crossing of the Lusitania, maps the tale known to many as the event that launched America into the Great War. On this edition of Up to Date, Steve Kraske talks with Erik Larson about his research process, the captains behind the ships involved, and the mystery of Room 40. Guest: 100 Years On, Kansas City Arts Organizations Explore Impact Of WWI By Laura Spencer • Jan 9, 2015 Laura Spencer / KCUR In 1914, at the outbreak of World War I, many artists put their art-making on hold, leaving their studios for the battlefield. Some in the United States waited for years for their country to enter the conflict, and others forged a new path in neutral Switzerland. It was a time of radical approaches in music, visual arts and literature. And now, local arts organizations are marking the centennial of the Great War. Music reflects change Journalism In World War I Through The Pen Of A Kansas City Reporter By Steve Kraske, Danie Alexander & Eliza Spertus • Dec 17, 2014 National World War I Museum For many families in America during World War I, newspaper reports were their only connection with loved ones serving in the trenches. On this edition of Up to Date, Steve Kraske talks with a journalism historian and an archivist from the World War I museum about the life of journalists reporting from the front lines during the Great War. Remembering The Legacy Of WWI's Buffalo Soldiers By Steve Kraske & Beth Lipoff • Nov 3, 2014 Western Newspaper Union / NARA/Wikimedia Commons They served with distinction in World War I but the Buffalo Soldiers are not always remembered for their contributions during the Great War. On Monday's Up to Date, we look at these regiments of African-American soldiers, their heroism and the racism they faced. Guest: The 'Kansas Flu' Of 1918 And The City's Response To It By Gina Kaufmann, Matthew Long-Middleton, Sylvia Maria Gross & Patrick Quick • Oct 22, 2014 In the wake of swirling fears about the spread of Ebola as well as Kansas cases of pertussis and measles, we look back on a pandemic that hit home for Kansas City: the Influenza pandemic of 1918. The death rate in Kansas City outpaced that in other places, and some say the city's politics and public health infrastructure were largely to blame. Stubby: The Hero Dog of World War One By Steve Kraske & Eliza Spertus • Sep 30, 2014 In World War I, he served alongside American forces in 17 battles. He had a unique talent for locating wounded soldiers, and he often alerted his unit to incoming gas attacks. This unlikely hero was Sergeant Stubby a stray stump-tailed terrier mutt who became a national hero. After WWI, The Pilgrimage Of Gold Star Mothers By Steve Kraske & Danie Alexander • Sep 24, 2014 Random House Of the 4.7 million Americans who took part in World War I, over 116,000 of them died. Many were given a final resting place in American military cemeteries in Europe. After the Great War a program was begun to give Gold Star mothers and widows (those whose son or husband had served during the conflict) the opportunity to cross the Atlantic to visit their loved one's grave. WWI: Changing The Dynamics Of Espionage By Steve Kraske & Beth Lipoff • Aug 20, 2014 Archives New Zealand / Flickr-CC World War I brought a new kind of warfare to the battlefield in many ways. The world of espionage got a facelift, especially in America, with aerial photography, code breaking and more. On Wednesday's Up to Date, we talk with a former intelligence analyst about these strategies and how they changed the way spies work. Guest: How Missouri's German-Americans Survived World War I By Cody Newill • Aug 8, 2014 When World War I broke out in Europe a century ago, more than one in 10 Missourians was German-American. Host Monroe Dodd is joined by Petra DeWitt, author of a book about the struggles that Missouri's German population faced during the war. Guest: Petra DeWitt, Assistant Teaching Professor at Missouri S&T and author of Degrees of Allegiance: Harassment and Loyalty in Missouri's German-American Community during World War I. Eddie Rickenbacker: WWI Ace By Matthew Long-Middleton • May 9, 2014 A century ago, America got hooked on speed. On the ground, speed meant motor cars and in the air, it meant planes. All that speed was delivered by the internal combustion engine, and no one represented the new world of motor speed better than Eddie Rickenbacker. He was not only a champion race-car driver, but also the greatest of World War I flying aces. Kansas City's World War I Monuments By Steve Kraske & Jenny Whitty • May 5, 2014 Whoever Credit Goes To / Flickr--CC Everyone is familiar with the National World War I Monument in Kansas City, but there are others. On Monday, we'll hear the stories behind some of the most prominent WWI monuments and memorials in Kansas City. James J. Heiman the author of Voices In the Bronze and Stone: Kansas City's World War I Monuments and Memorials joins us. Guest: James J. Heiman is the author of Voices In Bronze and Stone: Kansas City's World War I Monuments and Memorials. The Role Of Food During World War I By Gina Kaufmann, Matthew Long-Middleton & Kate Baxendale • Apr 29, 2014 The U.S. National Archives / Flickr / Creative Commons On today's Central Standard, culinary historian Andrea Broomfield joins us to discuss the importance of food during the first World War. Broomfield explains what the food industry was like during that time at War Fare: Chow Challenge on April 30. Chefs from area restaurants will compete in an Iron Chef-style event using food available during World War I. Guest: 'War Horse' Salutes World War I's Equine Brigade By Steve Walker • Mar 19, 2014 @broadway.com The National World War I Museum, housed at the base of the Liberty Memorial, is this year marking the 100th anniversary of the start of that war. By pure coincidence, the national tour of the Tony Award-winning play War Horse arrives at the Music Hall next month, creating a rare convergence of history and theatricality in Kansas City. Confounding the skeptics New Exhibit At National WWI Museum Highlights International Perspectives By Maria Carter • Mar 17, 2014 Courtesy: National World War I Museum This summer marks the 100th anniversary of the start of World War I, and a new exhibit at the National World War I Museum at Liberty Memorial in Kansas City, Mo., details the events that led up to the war — from the assassination of Archduke Franz Ferdinand to the declaration of war a month later on July 28, 1914. The exhibit presents archival newspaper articles and diplomatic communications from around the world. WWI: Who Lit The Fire? By Steve Kraske & Beth Lipoff • Jan 29, 2014 When you think of World War I, you may picture soldiers fighting in the trenches, but the whole conflict started with the assassination of an Austrian archduke. On Wednesday's Up to Date, we talk with historian Sean McMeekin, who says it was a group of corrupt statesmen who held the match that lit the European powder keg. Guest: New Appointee At National WWI Museum To Spearhead Centennial Celebration By Laura Ziegler • Nov 19, 2013 International Relations Council Linda Trout, long-time Executive Director of International Relations Council (IRC) in Kansas City, Mo. has been named Director of Strategic Engagement at the National World War I Museum. In the new post for the museum, Trout will oversee planning and implementation of the centennial celebration of WWI, beginning next year. WWI: Making History Come Alive By Steve Kraske & Beth Lipoff • Oct 22, 2013 Making history come alive is tricky, but some historians manage to paint people like the czars of Russia or Kaiser Wilhelm of Germany not just as dusty textbook figures. In the first part of Wednesday's Up to Date, we talk with historian Robert Massie about his vivid descriptions of the powerful people pulling the strings behind World War I. Guest: 1913: Before The Great War By Steve Kraske, Beth Lipoff & Cody Newill • Oct 20, 2013 By 1919, much of continental Europe lay in ruins in the aftermath of World War I. Prior to that conflict, with three European empires ruled by the "Kingly cousins," most people thought a war was nearly impossible. How Music Affected World War I By Beth Lipoff & Steve Kraske • Feb 15, 2013 All wasn't quiet on the homefront during World War I. National Designation For Liberty Memorial Makes It To Full House By Dan Verbeck • Dec 7, 2012 User: Charvex / Wikipedia Kansas City Congressman Emmanuel Cleaver says he feels good about the designation of Liberty Memorial as the nation's official World War One Memorial. The bill made it out of committee this week without any House opposition, but similar measures have had a checkered history. Great War Memorial Plan Inches Forward By Dan Verbeck • Dec 6, 2012 Dan Verbeck / KCUR Kansas City's Congressman won't guarantee passage of a bill that would make Liberty Memorial the nation's official World War One Museum and Memorial.


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Watch American Ninja 4: The Annihilation
The movie American Ninja 4: The Annihilation brings together two ninjas from the United States. They have to work together to fight a group of terrorists that have their own group of ninjas for defense. It is up to them in order to save the lives of many people.
| 1991 | 1 hr 39 min | 10 /
- Cast
- Michael Dudikoff, David Bradley, James Booth, Dwayne Alexandre, Ken Gampu
- Studio
- Cannon Home Video
- Director
- Cedric Sundstrom
- Language
- English


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Russia would never invade Ukraine? Get ready for $5 per gallon of gas? Iran will have a nuclear weapon?
Minimum wage hikes will cripple job growth? Long-term unemployment is tied to unemployment benefits? Obamacare will be a drain on the economy?
The Massachusetts senator isn't letting her Democratic colleagues off the hook


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Sweet Tea Glazed Ribs and Lemony Potato Salad
Sweet Tea Ribs with Lemony Potato Salad
recipe courtesy Food Network Magazine
Black tea (6 bags of any kind)
1/4 cup plus 2 tbsp. packed light brown sugar
Kosher salt & freshly ground black pepper
1 orange
2 racks baby back ribs (about 2 lbs)
2 lbs russet potatoes, peeled and cut into 3/4 inch chunks
1/2 cup mayonnaise
1 tablespoon chopped fresh parsley
1 lemon
Empty 3 tea bags into a bowl and combine the loose tea with 1/4 cup brown sugar, 2 teaspoons salt and 1/4 teaspoons pepper. Grate in the orange zest. Pat the ribs dry and trim the membrane from the underside. Rub the tea mixture over the ribs; place in a roasting pan, meat-side up, and bring to room temperature, about 20 minutes.
Preheat the oven to 275°F Steep the remaining 3 tea bags in 2 cups boiling water, about 5 minutes. Discard the bags and stir in the remaining 2 tablespoons brown sugar and the juice of half the orange. Pour the mixture around the ribs in the pan. Cover with foil and roast until tender, about 1 hour 30 minutes.
Meanwhile, put the potatoes in a pot of salted water; bring to a simmer and cook until tender, about 5 minutes. Drain and cool slightly. Mix the mayonnaise and parsley in a large bowl. Grate in the zest of half the lemon, and the juice of the lemon half. Fold in the potatoes and add 1 1/2 teaspoons salt, and pepper to taste; refrigerate until ready to serve.
Remove the ribs from the oven and increase the temperature to 450°F Pour the cooking liquid into a saucepan and bring to a simmer over medium-high heat. Baste the ribs with some of the liquid, then return to the oven and cook uncovered, basting a few times, until the ribs are dark and glazed, 20 to 30 minutes.
When the remaining liquid in the saucepan is syrupy, add the lemon juice to taste. Brush the ribs with the glaze. Cut the racks into pieces and serve with the potato salad.


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What do Mick Jagger, President Vaclav Havel of the Czech Republic, La Musee Moderne in Paris, and Vanity Fair magazine's headquarters in New York have in common? For that matter, how about Swan Lake, the Blues Brothers, and "Lord of the Dance" Michael Flatley? The answer: LD Patrick Woodroffe, who in the past 20 years has worked with all these people and projects, when not on the road lighting acts like the Rolling Stones, AC/DC, Tina Turner, Phil Collins, and Garth Brooks, including the ...
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A. Harrison Barnes is a well-known attorney throughout the United States. As a member of the Malibu community, Harrison is committed to serving the needs of Malibu residents and businesses in legal matters.


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Tuesday, Nov. 4 is Election Day.
If you can't make it to the polls for the General Election you can still take part by casting an early ballot. The deadline for early voting is 5 p.m. or the close of the clerk's office on Monday, Nov. 3 — the day before voting at the polls.
BRISTOL — Bristol will be awash in patriotic fervor Saturday when the town hosts the 16th annual Scouting Salute to Veterans Parade. The event will kick off at 11 a.m. with a parade featuring Scouts and veterans from many parts of the state and wrap up with ceremony and remarks by Maj. Gen. Steven A. Cray, the adjutant general of the state of Vermont and the senior uniformed officer in the state.
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BRISTOL — Voters in the five towns that comprise the Addison Northeast Supervisory Union will decide Tuesday whether to approve a $32.6 million bond to finance an ambitious renovation of Mount Abraham Union High School.
The project would make major structural and aesthetic improvements to the building, which has not undergone a substantial renovation since it was constructed in the late 1960s.
MIDDLEBURY — Middlebury residents beginning next April will be able to choose their own haulers to collect curbside recyclables from their homes and businesses. That's because the Middlebury selectboard on Tuesday voted unanimously to discontinue the townwide recycling contract once the current pact with Casella expires on March 31, 2015.
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text: Featuring the same proven, late-rolling Neutron core utilized in both the Rising Star and Shooting Star, the Shatter is a model of consistency, producing good length and strong motion down-lane. The 51ML pearl reactive coverstock was created to go toe-to-toe with medium to lighter oil conditions. This reliable shell allows the ball to easily clear the front part of the lane and provide you with more stored energy down-lane. The Shatter produces more length and backend on medium to lighter oil conditions than the Rising Star and Shooting star, creating the perfect combination when paired with either of its predecessors.