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This is the eighth “research” thread of the Polymath15 project to upper bound the de Bruijn-Newman constant , continuing this post. Discussion of the project of a non-research nature can continue for now in the existing proposal thread. Progress will be summarised at this Polymath wiki page.
Significant progress has been made since the last update; by implementing the “barrier” method to establish zero free regions for by leveraging the extensive existing numerical verification of the Riemann hypothesis (which establishes zero free regions for
), we have been able to improve our upper bound on
from 0.48 to 0.28. Furthermore, there appears to be a bit of further room to improve the bounds further by tweaking the parameters
used in the argument (we are currently using
); the most recent idea is to try to use exponential sum estimates to improve the bounds on the derivative of the approximation to
that is used in the barrier method, which currently is the most computationally intensive step of the argument.
Just a quick announcement that Dustin Mixon and Aubrey de Grey have just launched the Polymath16 project over at Dustin’s blog. The main goal of this project is to simplify the recent proof by Aubrey de Grey that the chromatic number of the unit distance graph of the plane is at least 5, thus making progress on the Hadwiger-Nelson problem. The current proof is computer assisted (in particular it requires one to control the possible 4-colorings of a certain graph with over a thousand vertices), but one of the aims of the project is to reduce the amount of computer assistance needed to verify the proof; already a number of such reductions have been found. See also this blog post where the polymath project was proposed, as well as the wiki page for the project. Non-technical discussion of the project will continue at the proposal blog post.
We now leave the topic of Riemann surfaces, and turn now to the (loosely related) topic of conformal mapping (and quasiconformal mapping). Recall that a conformal map from an open subset
of the complex plane to another open set
is a map that is holomorphic and bijective, which (by Rouché’s theorem) also forces the derivative of
to be nowhere vanishing. We then say that the two open sets
are conformally equivalent. From the Cauchy-Riemann equations we see that conformal maps are orientation-preserving and angle-preserving; from the Newton approximation
we see that they almost preserve small circles, indeed for
small the circle
will approximately map to
.
In previous quarters, we proved a fundamental theorem about this concept, the Riemann mapping theorem:
Theorem 1 (Riemann mapping theorem) Let
be a simply connected open subset of
that is not all of
. Then
is conformally equivalent to the unit disk
.
This theorem was proven in these 246A lecture notes, using an argument of Koebe. At a very high level, one can sketch Koebe’s proof of the Riemann mapping theorem as follows: among all the injective holomorphic maps from
to
that map some fixed point
to
, pick one that maximises the magnitude
of the derivative (ignoring for this discussion the issue of proving that a maximiser exists). If
avoids some point in
, one can compose
with various holomorphic maps and use Schwarz’s lemma and the chain rule to increase
without destroying injectivity; see the previous lecture notes for details. The conformal map
is unique up to Möbius automorphisms of the disk; one can fix the map by picking two distinct points
in
, and requiring
to be zero and
to be positive real.
It is a beautiful observation of Thurston that the concept of a conformal mapping has a discrete counterpart, namely the mapping of one circle packing to another. Furthermore, one can run a version of Koebe’s argument (using now a discrete version of Perron’s method) to prove the Riemann mapping theorem through circle packings. In principle, this leads to a mostly elementary approach to conformal geometry, based on extremely classical mathematics that goes all the way back to Apollonius. However, in order to prove the basic existence and uniqueness theorems of circle packing, as well as the convergence to conformal maps in the continuous limit, it seems to be necessary (or at least highly convenient) to use much more modern machinery, including the theory of quasiconformal mapping, and also the Riemann mapping theorem itself (so in particular we are not structuring these notes to provide a completely independent proof of that theorem, though this may well be possible).
To make the above discussion more precise we need some notation.
Definition 2 (Circle packing) A (finite) circle packing is a finite collection
of circles
in the complex numbers indexed by some finite set
, whose interiors are all disjoint (but which are allowed to be tangent to each other), and whose union is connected. The nerve of a circle packing is the finite graph whose vertices
are the centres of the circle packing, with two such centres connected by an edge if the circles are tangent. (In these notes all graphs are undirected, finite and simple, unless otherwise specified.)
It is clear that the nerve of a circle packing is connected and planar, since one can draw the nerve by placing each vertex (tautologically) in its location in the complex plane, and drawing each edge by the line segment between the centres of the circles it connects (this line segment will pass through the point of tangency of the two circles). Later in these notes we will also have to consider some infinite circle packings, most notably the infinite regular hexagonal circle packing.
The first basic theorem in the subject is the following converse statement:
Theorem 3 (Circle packing theorem) Every connected planar graph is the nerve of a circle packing.
Of course, there can be multiple circle packings associated to a given connected planar graph; indeed, since reflections across a line and Möbius transformations map circles to circles (or lines), they will map circle packings to circle packings (unless one or more of the circles is sent to a line). It turns out that once one adds enough edges to the planar graph, the circle packing is otherwise rigid:
Theorem 4 (Koebe-Andreev-Thurston theorem) If a connected planar graph is maximal (i.e., no further edge can be added to it without destroying planarity), then the circle packing given by the above theorem is unique up to reflections and Möbius transformations.
Exercise 5 Let
be a connected planar graph with
vertices. Show that the following are equivalent:
- (i)
is a maximal planar graph.
- (ii)
has
edges.
- (iii) Every drawing
of
divides the plane into faces that have three edges each. (This includes one unbounded face.)
- (iv) At least one drawing
of
divides the plane into faces that have three edges each.
(Hint: use Euler’s formula
, where
is the number of faces including the unbounded face.)
Thurston conjectured that circle packings can be used to approximate the conformal map arising in the Riemann mapping theorem. Here is an informal statement:
Conjecture 6 (Informal Thurston conjecture) Let
be a simply connected domain, with two distinct points
. Let
be the conformal map from
to
that maps
to the origin and
to a positive real. For any small
, let
be the portion of the regular hexagonal circle packing by circles of radius
that are contained in
, and let
be an circle packing of
with all “boundary circles” tangent to
, giving rise to an “approximate map”
defined on the subset
of
consisting of the circles of
, their interiors, and the interstitial regions between triples of mutually tangent circles. Normalise this map so that
is zero and
is a positive real. Then
converges to
as
.
A rigorous version of this conjecture was proven by Rodin and Sullivan. Besides some elementary geometric lemmas (regarding the relative sizes of various configurations of tangent circles), the main ingredients are a rigidity result for the regular hexagonal circle packing, and the theory of quasiconformal maps. Quasiconformal maps are what seem on the surface to be a very broad generalisation of the notion of a conformal map. Informally, conformal maps take infinitesimal circles to infinitesimal circles, whereas quasiconformal maps take infinitesimal circles to infinitesimal ellipses of bounded eccentricity. In terms of Wirtinger derivatives, conformal maps obey the Cauchy-Riemann equation , while (sufficiently smooth) quasiconformal maps only obey an inequality
. As such, quasiconformal maps are considerably more plentiful than conformal maps, and in particular it is possible to create piecewise smooth quasiconformal maps by gluing together various simple maps such as affine maps or Möbius transformations; such piecewise maps will naturally arise when trying to rigorously build the map
alluded to in the above conjecture. On the other hand, it turns out that quasiconformal maps still have many vestiges of the rigidity properties enjoyed by conformal maps; for instance, there are quasiconformal analogues of fundamental theorems in conformal mapping such as the Schwarz reflection principle, Liouville’s theorem, or Hurwitz’s theorem. Among other things, these quasiconformal rigidity theorems allow one to create conformal maps from the limit of quasiconformal maps in many circumstances, and this will be how the Thurston conjecture will be proven. A key technical tool in establishing these sorts of rigidity theorems will be the theory of an important quasiconformal (quasi-)invariant, the conformal modulus (or, equivalently, the extremal length, which is the reciprocal of the modulus).
I am recording here some notes on a nice problem that Sorin Popa shared with me recently. To motivate the question, we begin with the basic observation that the differentiation operator and the position operator
in one dimension formally obey the commutator equation
where is the identity operator and
is the commutator. Among other things, this equation is fundamental in quantum mechanics, leading for instance to the Heisenberg uncertainty principle.
The operators are unbounded on spaces such as
. One can ask whether the commutator equation (1) can be solved using bounded operators
on a Hilbert space
rather than unbounded ones. In the finite dimensional case when
are just
matrices for some
, the answer is clearly negative, since the left-hand side of (1) has trace zero and the right-hand side does not. What about in infinite dimensions, when the trace is not available? As it turns out, the answer is still negative, as was first worked out by Wintner and Wielandt. A short proof can be given as follows. Suppose for contradiction that we can find bounded operators
obeying (1). From (1) and an easy induction argument, we obtain the identity
for all natural numbers . From the triangle inequality, this implies that
Iterating this, we conclude that
for any . Bounding
and then sending
, we conclude that
, which clearly contradicts (1). (Note the argument can be generalised without difficulty to the case when
lie in a Banach algebra, rather than be bounded operators on a Hilbert space.)
It was observed by Popa that there is a quantitative version of this result:
Theorem 1 Let
such that
Proof: By multiplying by a suitable constant and dividing
by the same constant, we may normalise
. Write
with
. Then the same induction that established (2) now shows that
and hence by the triangle inequality
We divide by and sum to conclude that
giving the claim.
Again, the argument generalises easily to any Banach algebra. Popa then posed the question of whether the quantity can be replaced by any substantially larger function of
, such as a polynomial in
. As far as I know, the above simple bound has not been substantially improved.
In the opposite direction, one can ask for constructions of operators that are not too large in operator norm, such that
is close to the identity. Again, one cannot do this in finite dimensions:
has trace zero, so at least one of its eigenvalues must outside the disk
, and therefore
for any finite-dimensional
matrices
.
However, it was shown in 1965 by Brown and Pearcy that in infinite dimensions, one can construct operators with
arbitrarily close to
in operator norm (in fact one can prescribe any operator for
as long as it is not equal to a non-zero multiple of the identity plus a compact operator). In the above paper of Popa, a quantitative version of the argument (based in part on some earlier work of Apostol and Zsido) was given as follows. The first step is to observe the following Hilbert space version of Hilbert’s hotel: in an infinite dimensional Hilbert space
, one can locate isometries
obeying the equation
where denotes the adjoint of
. For instance, if
has a countable orthonormal basis
, one could set
and
where denotes the linear functional
on
. Observe that (4) is again impossible to satisfy in finite dimension
, as the left-hand side must have trace
while the right-hand side has trace
.
Multiplying (4) on the left by and right by
, or on the left by
and right by
, then gives
From (4), (5) we see in particular that, while we cannot express as a commutator of bounded operators, we can at least express it as the sum of two commutators:
We can rewrite this somewhat strangely as
and hence there exists a bounded operator such that
Moving now to the Banach algebra of matrices with entries in
(which can be equivalently viewed as
), a short computation then gives the identity
for some bounded operator whose exact form will not be relevant for the argument. Now, by Neumann series (and the fact that
have unit operator norm), we can find another bounded operator
such that
and then another brief computation shows that
Thus we can express the operator as the commutator of two operators of norm
. Conjugating by
for any
, we may then express
as the commutator of two operators of norm
. This shows that the right-hand side of (3) cannot be replaced with anything that blows up faster than
as
. Can one improve this bound further?
The fundamental object of study in real differential geometry are the real manifolds: Hausdorff topological spaces that locally look like open subsets of a Euclidean space
, and which can be equipped with an atlas
of coordinate charts
from open subsets
covering
to open subsets
in
, which are homeomorphisms; in particular, the transition maps
defined by
are all continuous. (It is also common to impose the requirement that the manifold
be second countable, though this will not be important for the current discussion.) A smooth real manifold is a real manifold in which the transition maps are all smooth.
In a similar fashion, the fundamental object of study in complex differential geometry are the complex manifolds, in which the model space is rather than
, and the transition maps
are required to be holomorphic (and not merely smooth or continuous). In the real case, the one-dimensional manifolds (curves) are quite simple to understand, particularly if one requires the manifold to be connected; for instance, all compact connected one-dimensional real manifolds are homeomorphic to the unit circle (why?). However, in the complex case, the connected one-dimensional manifolds – the ones that look locally like subsets of
– are much richer, and are known as Riemann surfaces. For sake of completeness we give the (somewhat lengthy) formal definition:
Definition 1 (Riemann surface) If
is a Hausdorff connected topological space, a (one-dimensional complex) atlas is a collection
of homeomorphisms from open subsets
of
that cover
to open subsets
of the complex numbers
, such that the transition maps
defined by
are all holomorphic. Here
is an arbitrary index set. Two atlases
,
on
are said to be equivalent if their union is also an atlas, thus the transition maps
and their inverses are all holomorphic. A Riemann surface is a Hausdorff connected topological space
equipped with an equivalence class of one-dimensional complex atlases.
A map
from one Riemann surface
to another
is holomorphic if the maps
are holomorphic for any charts
,
of an atlas of
and
respectively; it is not hard to see that this definition does not depend on the choice of atlas. It is also clear that the composition of two holomorphic maps is holomorphic (and in fact the class of Riemann surfaces with their holomorphic maps forms a category).
Here are some basic examples of Riemann surfaces.
Example 2 (Quotients of
) The complex numbers
clearly form a Riemann surface (using the identity map
as the single chart for an atlas). Of course, maps
that are holomorphic in the usual sense will also be holomorphic in the sense of the above definition, and vice versa, so the notion of holomorphicity for Riemann surfaces is compatible with that of holomorphicity for complex maps. More generally, given any discrete additive subgroup
of
, the quotient
is a Riemann surface. There are an infinite number of possible atlases to use here; one such is to pick a sufficiently small neighbourhood
of the origin in
and take the atlas
where
and
for all
. In particular, given any non-real complex number
, the complex torus
formed by quotienting
by the lattice
is a Riemann surface.
Example 3 Any open connected subset
of
is a Riemann surface. By the Riemann mapping theorem, all simply connected open
, other than
itself, are isomorphic (as Riemann surfaces) to the unit disk (or, equivalently, to the upper half-plane).
Example 4 (Riemann sphere) The Riemann sphere
, as a topological manifold, is the one-point compactification of
. Topologically, this is a sphere and is in particular connected. One can cover the Riemann sphere by the two open sets
and
, and give these two open sets the charts
and
defined by
for
,
for
, and
. This is a complex atlas since the
is holomorphic on
.
An alternate way of viewing the Riemann sphere is as the projective line
. Topologically, this is the punctured complex plane
quotiented out by non-zero complex dilations, thus elements of this space are equivalence classes
with the usual quotient topology. One can cover this space by two open sets
and
and give these two open sets the charts
and
defined by
for
,
. This is a complex atlas, basically because
for
and
is holomorphic on
.
Exercise 5 Verify that the Riemann sphere is isomorphic (as a Riemann surface) to the projective line.
Example 6 (Smooth algebraic plane curves) Let
be a complex polynomial in three variables which is homogeneous of some degree
, thus
Define the complex projective plane
to be the punctured space
quotiented out by non-zero complex dilations, with the usual quotient topology. (There is another important topology to place here of fundamental importance in algebraic geometry, namely the Zariski topology, but we will ignore this topology here.) This is a compact space, whose elements are equivalence classes
. Inside this plane we can define the (projective, degree
) algebraic curve
this is well defined thanks to (1). It is easy to verify that
is a closed subset of
and hence compact; it is non-empty thanks to the fundamental theorem of algebra.
Suppose that
is irreducible, which means that it is not the product of polynomials of smaller degree. As we shall show in the appendix, this makes the algebraic curve connected. (Actually, algebraic curves remain connected even in the reducible case, thanks to Bezout’s theorem, but we will not prove that theorem here.) We will in fact make the stronger nonsingularity hypothesis: there is no triple
such that the four numbers
simultaneously vanish for
. (This looks like four constraints, but is in fact essentially just three, due to the Euler identity
that arises from differentiating (1) in
. The fact that nonsingularity implies irreducibility is another consequence of Bezout’s theorem, which is not proven here.) For instance, the polynomial
is irreducible but singular (there is a “cusp” singularity at
). With this hypothesis, we call the curve
smooth.
Now suppose
is a point in
; without loss of generality we may take
non-zero, and then we can normalise
. Now one can think of
as an inhomogeneous polynomial in just two variables
, and by nondegeneracy we see that the gradient
is non-zero whenever
. By the (complexified) implicit function theorem, this ensures that the affine algebraic curve
is a Riemann surface in a neighbourhood of
; we leave this as an exercise. This can be used to give a coordinate chart for
in a neighbourhood of
when
. Similarly when
is non-zero. This can be shown to give an atlas on
, which (assuming the connectedness claim that we will prove later) gives
the structure of a Riemann surface.
Exercise 7 State and prove a complex version of the implicit function theorem that justifies the above claim that the charts in the above example form an atlas, and an algebraic curve associated to a non-singular polynomial is a Riemann surface.
Exercise 8
- (i) Show that all (irreducible plane projective) algebraic curves of degree
are isomorphic to the Riemann sphere. (Hint: reduce to an explicit linear polynomial such as
.)
- (ii) Show that all (irreducible plane projective) algebraic curves of degree
are isomorphic to the Riemann sphere. (Hint: to reduce computation, first use some linear algebra to reduce the homogeneous quadratic polynomial to a standard form, such as
or
.)
Exercise 9 If
are complex numbers, show that the projective cubic curve
is nonsingular if and only if the discriminant
is non-zero. (When this occurs, the curve is called an elliptic curve (in Weierstrass form), which is a fundamentally important example of a Riemann surface in many areas of mathematics, and number theory in particular. One can also define the discriminant for polynomials of higher degree, but we will not do so here.)
A recurring theme in mathematics is that an object is often best studied by understanding spaces of “good” functions on
. In complex analysis, there are two basic types of good functions:
Definition 10 Let
be a Riemann surface. A holomorphic function on
is a holomorphic map from
to
; the space of all such functions will be denoted
. A meromorphic function on
is a holomorphic map from
to the Riemann sphere
, that is not identically equal to
; the space of all such functions will be denoted
.
One can also define holomorphicity and meromorphicity in terms of charts: a function is holomorphic if and only if, for any chart
, the map
is holomorphic in the usual complex analysis sense; similarly, a function
is meromorphic if and only if the preimage
is discrete (otherwise, by analytic continuation and the connectedness of
,
will be identically equal to
) and for any chart
, the map
becomes a meromorphic function in the usual complex analysis sense, after removing the discrete set of complex numbers where this map is infinite. One consequence of this alternate definition is that the space
of holomorphic functions is a commutative complex algebra (a complex vector space closed under pointwise multiplication), while the space
of meromorphic functions is a complex field (a commutative complex algebra where every non-zero element has an inverse). Another consequence is that one can define the notion of a zero of given order
, or a pole of order
, for a holomorphic or meromorphic function, by composing with a chart map and using the usual complex analysis notions there, noting (from the holomorphicity of transition maps and their inverses) that this does not depend on the choice of chart. (However, one cannot similarly define the residue of a meromorphic function on
this way, as the residue turns out to be chart-dependent thanks to the chain rule. Residues should instead be applied to meromorphic
-forms, a concept we will introduce later.) A third consequence is analytic continuation: if two holomorphic or meromorphic functions on
agree on a non-empty open set, then they agree everywhere.
On the complex numbers , there are of course many holomorphic functions and meromorphic functions; for instance any power series with an infinite radius of convergence will give a holomorphic function, and the quotient of any two such functions (with non-zero denominator) will give a meromorphic function. Furthermore, we have extremely wide latitude in how to specify the zeroes of the holomorphic function, or the zeroes and poles of the meromorphic function, thanks to tools such as the Weierstrass factorisation theorem or the Mittag-Leffler theorem (covered in previous quarters).
It turns out, however, that the situation changes dramatically when the Riemann surface is compact, with the holomorphic and meromorphic functions becoming much more rigid. First of all, compactness eliminates all holomorphic functions except for the constants:
Lemma 11 Let
be a holomorphic function on a compact Riemann surface
. Then
is constant.
This result should be seen as a close sibling of Liouville’s theorem that all bounded entire functions are constant. (Indeed, in the case of a complex torus, this lemma is a corollary of Liouville’s theorem.)
Proof: As is continuous and
is compact,
must attain a maximum at some point
. Working in a chart around
and applying the maximum principle, we conclude that
is constant in a neighbourhood of
, and hence is constant everywhere by analytic continuation.
This dramatically cuts down the number of possible meromorphic functions – indeed, for an abstract Riemann surface, it is not immediately obvious that there are any non-constant meromorphic functions at all! As the poles are isolated and the surface is compact, a meromorphic function can only have finitely many poles, and if one prescribes the location of the poles and the maximum order at each pole, then we shall see that the space of meromorphic functions is now finite dimensional. The precise dimensions of these spaces are in fact rather interesting, and obey a basic duality law known as the Riemann-Roch theorem. We will give a mostly self-contained proof of the Riemann-Roch theorem in these notes, omitting only some facts about genus and Euler characteristic, as well as construction of certain meromorphic -forms (also known as Abelian differentials).
This is the seventh “research” thread of the Polymath15 project to upper bound the de Bruijn-Newman constant , continuing this post. Discussion of the project of a non-research nature can continue for now in the existing proposal thread. Progress will be summarised at this Polymath wiki page.
The most recent news is that we appear to have completed the verification that is free of zeroes when
and
, which implies that
. For very large
(for instance when the quantity
is at least
) this can be done analytically; for medium values of
(say when
is between
and
) this can be done by numerically evaluating a fast approximation
to
and using the argument principle in a rectangle; and most recently it appears that we can also handle small values of
, in part due to some new, and significantly faster, numerical ways to evaluate
in this range.
One obvious thing to do now is to experiment with lowering the parameters and
and see what happens. However there are two other potential ways to bound
which may also be numerically feasible. One approach is based on trying to exclude zeroes of
in a region of the form
,
and
for some moderately large
(this acts as a “barrier” to prevent zeroes from flowing into the region
at time
, assuming that they were not already there at time
). This require significantly less numerical verification in the
aspect, but more numerical verification in the
aspect, so it is not yet clear whether this is a net win.
Another, rather different approach, is to study the evolution of statistics such as over time. One has fairly good control on such quantities at time zero, and their time derivative looks somewhat manageable, so one may be able to still have good control on this quantity at later times
. However for this approach to work, one needs an effective version of the Riemann-von Mangoldt formula for
, which at present is only available asymptotically (or at time
). This approach may be able to avoid almost all numerical computation, except for numerical verification of the Riemann hypothesis, for which we can appeal to existing literature.
Participants are also welcome to add any further summaries of the situation in the comments below.
This is the sixth “research” thread of the Polymath15 project to upper bound the de Bruijn-Newman constant , continuing this post. Discussion of the project of a non-research nature can continue for now in the existing proposal thread. Progress will be summarised at this Polymath wiki page.
The last two threads have been focused primarily on the test problem of showing that whenever
. We have been able to prove this for most regimes of
, or equivalently for most regimes of the natural number parameter
. In many of these regimes, a certain explicit approximation
to
was used, together with a non-zero normalising factor
; see the wiki for definitions. The explicit upper bound
has been proven for certain explicit expressions (see here) depending on
. In particular, if
satisfies the inequality
then is non-vanishing thanks to the triangle inequality. (In principle we have an even more accurate approximation
available, but it is looking like we will not need it for this test problem at least.)
We have explicit upper bounds on ,
,
; see this wiki page for details. They are tabulated in the range
here. For
, the upper bound
for
is monotone decreasing, and is in particular bounded by
, while
and
are known to be bounded by
and
respectively (see here).
Meanwhile, the quantity can be lower bounded by
for certain explicit coefficients and an explicit complex number
. Using the triangle inequality to lower bound this by
we can obtain a lower bound of for
, which settles the test problem in this regime. One can get more efficient lower bounds by multiplying both Dirichlet series by a suitable Euler product mollifier; we have found
for
to be good choices to get a variety of further lower bounds depending only on
, see this table and this wiki page. Comparing this against our tabulated upper bounds for the error terms we can handle the range
.
In the range , we have been able to obtain a suitable lower bound
(where
exceeds the upper bound for
) by numerically evaluating
at a mesh of points for each choice of
, with the mesh spacing being adaptive and determined by
and an upper bound for the derivative of
; the data is available here.
This leaves the final range (roughly corresponding to
). Here we can numerically evaluate
to high accuracy at a fine mesh (see the data here), but to fill in the mesh we need good upper bounds on
. It seems that we can get reasonable estimates using some contour shifting from the original definition of
(see here). We are close to finishing off this remaining region and thus solving the toy problem.
Beyond this, we need to figure out how to show that for
as well. General theory lets one do this for
, leaving the region
. The analytic theory that handles
and
should also handle this region; for
presumably the argument principle will become relevant.
The full argument also needs to be streamlined and organised; right now it sprawls over many wiki pages and github code files. (A very preliminary writeup attempt has begun here). We should also see if there is much hope of extending the methods to push much beyond the bound of that we would get from the above calculations. This would also be a good time to start discussing whether to move to the writing phase of the project, or whether there are still fruitful research directions for the project to explore.
Participants are also welcome to add any further summaries of the situation in the comments below.
Next quarter (starting Monday, April 2) I will be teaching Math 246C (complex analysis) here at UCLA. This is the third in a three-series graduate course on complex analysis; a few years ago I taught the first course in this series (246A), so this course can be thought of in some sense as a sequel to that one (and would certainly assume knowledge of the material in that course as a prerequisite), although it also assumes knowledge of material from the second course 246B (which covers such topics as Weierstrass factorization and the theory of harmonic functions).
246C is primarily a topics course, and tends to be a somewhat miscellaneous collection of complex analysis subjects that were not covered in the previous two installments of the series. The initial topics I have in mind to cover are
- Elliptic functions;
- The Riemann-Roch theorem;
- Circle packings;
- The Bieberbach conjecture (proven by de Branges); and
- the Schramm-Loewner equation (SLE).
- This list is however subject to change (it is the first time I will have taught on any of these topics, and I am not yet certain on the most logical way to arrange them; also I am not completely certain that I will be able to cover all the above topics in ten weeks). I welcome reference recommendations and other suggestions from readers who have taught on one or more of these topics.
As usual, I will be posting lecture notes on this blog as the course progresses.
[Update: Mar 13: removed elliptic functions, as I have just learned that this was already covered in the prior 246B course.]
This is the fifth “research” thread of the Polymath15 project to upper bound the de Bruijn-Newman constant , continuing this post. Discussion of the project of a non-research nature can continue for now in the existing proposal thread. Progress will be summarised at this Polymath wiki page.
We have almost finished off the test problem of showing that whenever
. We have two useful approximations for
, which we have denoted
and
, and a normalising quantity
that is asymptotically equal to the above expressions; see the wiki page for definitions. In practice, the
approximation seems to be accurate within about one or two significant figures, whilst the
approximation is accurate to about three or four. We have an effective upper bound
where the expressions are quite small in practice (
is typically about two orders of magnitude smaller than the main term
once
is moderately large, and the error terms
are even smaller). See this page for details. In principle we could also obtain an effective upper bound for
(the
term would be replaced by something smaller).
The ratio takes the form of a difference
of two Dirichlet series, where
is a phase whose value is explicit but perhaps not terribly important, and the coefficients
are explicit and relatively simple (
is
, and
is approximately
). To bound this away from zero, we have found it advantageous to mollify this difference by multiplying by an Euler product
to cancel much of the initial oscillation; also one can take advantage of the fact that the
are real and the
are (approximately) real. See this page for details. The upshot is that we seem to be getting good lower bounds for the size of this difference of Dirichlet series starting from about
or so. The error terms
are already quite small by this stage, so we should soon be able to rigorously keep
from vanishing at this point. We also have a scheme for lower bounding the difference of Dirichlet series below this range, though it is not clear at present how far we can continue this before the error terms
become unmanageable. For very small
we may have to explore some faster ways to compute the expression
, which is still difficult to compute directly with high accuracy. One will also need to bound the somewhat unwieldy expressions
by something more manageable. For instance, right now these quantities depend on the continuous variable
; it would be preferable to have a quantity that depends only on the parameter
, as this could be computed numerically for all
in the remaining range of interest quite quickly.
As before, any other mathematical discussion related to the project is also welcome here, for instance any summaries of previous discussion that was not covered in this post.
This is the fourth “research” thread of the Polymath15 project to upper bound the de Bruijn-Newman constant , continuing https://terrytao.wordpress.com/2018/01/24/polymath-proposal-upper-bounding-the-de-bruijn-newman-constant/. Progress will be summarised at this Polymath wiki page.
We are getting closer to finishing off the following test problem: can one show that whenever
,
? This would morally show that
. A wiki page for this problem has now been created here. We have obtained a number of approximations
to
(see wiki page), though numeric evidence indicates that the approximations are all very close to each other. (Many of these approximations come with a correction term
, but thus far it seems that we may be able to avoid having to use this refinement to the approximations.) The effective approximation
also comes with an effective error bound
for some explicit (but somewhat messy) error terms : see this wiki page for details. The original approximations
can be considered deprecated at this point in favour of the (slightly more complicated) approximation
; the approximation
is a simplified version of
which is not quite as accurate but might be useful for testing purposes.
It is convenient to normalise everything by an explicit non-zero factor . Asymptotically,
converges to 1; numerically, it appears that its magnitude (and also its real part) stays roughly between 0.4 and 3 in the range
, and we seem to be able to keep it (or at least the toy counterpart
) away from zero starting from about
(here it seems that there is a useful trick of multiplying by Euler-type factors like
to cancel off some of the oscillation). Also, the bounds on the error
seem to be of size about 0.1 or better in these ranges also. So we seem to be on track to be able to rigorously eliminate zeroes starting from about
or so. We have not discussed too much what to do with the small values of
; at some point our effective error bounds will become unusable, and we may have to find some more faster ways to compute
.
In addition to this main direction of inquiry, there have been additional discussions on the dynamics of zeroes, and some numerical investigations of the behaviour Lehmer pairs under heat flow. Contributors are welcome to summarise any findings from these discussions from previous threads (or on any other related topic, e.g. improvements in the code) in the comments below.
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