Earlier today, while hacking my way with A. Saha through Gradshteyn-Rizhik in search of some clue to an integral he wanted to compute, we found a close enough approximation where, on the right-hand side, a function Dp appeared. I had no idea what it could possibly be; the lack of an index meant we had to go through the whole back section on special functions to locate it (it turns out to be a “parabolic cylinder function”, a close relative of Whittaker functions and of confluent hypergeometric functions).

This led me to wonder if one could make a whole alphabet song of special functions: is there a letter, poor thing, such that no well-known special function is named after it? (I’m allowing multi-letter names, so that “A” goes with the Airy function Ai(z)). I’m not even sure about B, though some Bessel function should fit…

As I’ve just lent my copy of G-R, I can’t look right away. But aspiring song-writers can start looking and suggesting catchy rhymes and couplets to go with the long overdue “Song of special functions”…

Title: The Computer as Crucible
Authors: Jonathan Borwein and Keith Devlin
True pp.: 154
Publisher: A K Peters Ltd
Published on: November 2008
ISBN-13: 978-1568813431
Rating: 8/10

Jonathan Borwein and Keith Devlin are well-known mathematicians who have a strong appreciation of, and expertise in, experimental mathematics. In this book they provide us with a concise, inviting introduction to the field.

The first chapter tries to succinctly explain what experimental mathematics is and why it’s a fundamental tool for the modern mathematician. The following is their definition:

Experimental mathematics is the use of a computer to run computations—sometimes no more than trial-and-error tests—to look for patterns, to identify particular numbers and sequences, to gather evidence in support of specific mathematical assertions that may themselves arise by computational means, including search. Like contemporary chemists—and before them the alchemists of old—who mix various substances together in a crucible and heat them to a high temperature to see what happens, today’s experimental mathematician puts a hopefully potent mix of numbers, formulas, and algorithms into a computer in the hope that something of interest emerges.

They immediately address some of the possible objections and illustrate how an approach that doesn’t focus on formal proof, but rather on exploration and experimentation, ultimately leads to hypotheses which can then be, in many cases, proved analytically. The authors argue that in this sense, thanks to the aid of advanced computers, mathematics is becoming more and more similar to other natural sciences.

They also make a case for how great mathematicians like Euler, Gauss, and Reimann were doing experimental mathematics well before calculators where available. Their calculations on paper were far more limited than what computers afford us these days, yet they served them well when it came to sharpening and verifying their intuitions.

The rest of the book is a continuous series of examples that show the advantages of this approach in practice. The examples are highly interesting (some of them stunning) and tend to focus on calculus, analysis and analytical number theory.

Each chapter is accompanied by a section called “Explorations”. I found this section to be particularly valuable. Within it you’ll find exercises, and further examples and considerations. The answers/solutions to the actual problems are provided in the second to last chapter, just before the brief epilogue.

Chapter 2 discusses how to calculate an arbitrary digit for irrational numbers like , in certain bases. They illustrate how the so called BBP Formula (Bailey-Borwein-Plouffe formula, co-discovered by Jonathan Borwein’s brother) came to be.

The use of a program which implements the PSQL integer relation algorithm in high-precision, floating-point arithmetic was key to its discovery. The BBP Formula in turn allowed the calculation of the quadrillionth binary digit of back in 2000.

Chapter 3 focuses on identifying numbers, digits patterns, and sequences once you obtain a numeric result through your calculations and experimentation. They introduce the subject with relatively obvious values like the approximations of or , but the chapter quickly escalates to an example where a closed form for a seemingly random sequence needs to be found.

Chapter 4 analyzes the Reimann Zeta function from the eyes of an experimental mathematician, and shows us what kind of insight we can gain from this unique perspective.

In chapter 5 we learn how by numerically evaluating definite integrals, it is sometimes possible to identify the resulting value which will help us to analytically resolve those particular integrals. The examples presented in this chapter originate for the most part from physics and are very challenging if attempted without the aid of experimental methods. To better grasp the kind of integrals discussed in this chapter, here is an example:

The explorations section provides a few more interesting integrals, including some for which a closed form is not known. The authors even include an integral that intentionally stumps Mathematica 6 and Maple 11.

Chapter 6 is dedicated to serendipitous discoveries (“proof by serendipity”) with a few interesting examples of how “luck” met preparation, ultimately enriching the body of mathematical knowledge almost by chance.

In chapter 7 the authors go back to talk about , this time in base 10, to calculate its digits with efficient, fast converging formulas and methods. The chapter wraps up with a discussion about the normality of , which hasn’t been proved of course, but appears to be empirically supported by the statistical analysis of the first trillion digits. In the explorations section there is a nice discussion about the implementation of fast arithmetic through the Karatsuba multiplication, and the subject of Montecarlo simulations (a very inefficient method of calculating , but a great way to show the idea behind Montecarlo simulations).

Chapter 8 has a bold title, “The computer knows more math than you do”. This provocative title is quickly diminished to put it in context though. The authors start by approaching a tough problem posed by Donald Knuth (of TeX and The Art of Computer Programming fame) to the readers of the American Mathematical Monthly:

In an attempt to solve this the authors invite us to go on a journey involving the Lambert W function, the Pochhammer function, and Abel’s limit theorem. The rest of the chapter illustrates another difficult problem whose solution obtained through the aid of Maple has important implications not only for mathematics, but also for quantum field theory and statistical mechanics.

In chapter 9 a few infinite series are calculated in order to show how CAS systems and experimental methodology can still be useful when dealing with problems that involve infinite sequences, series, and products.

Chapter 10 is dedicated to the limits and the dangers of this approach. Several examples showcase how one can be misled into making assumptions, and how to avoid this from happening. The ad hoc example below is correct to over half a billion digits:

After having calculated a few hundred digits, it would be natural to assume that the series converges to a natural number, when in reality it’s an irrational and transcendental number.

In chapter 11, conscious of the selective focus on analysis and analytical number theory throughout the book, Borwein and Devlin introduce other examples such as a topology problem whose proof was reached thanks to a deeper insight gained through computer visualization of a surface, a knot theory problem, the Four Color Theorem, the Robbins Conjecture, the computation of , and so on.

In truth, I feel that such a thin book could have used more examples like the ones in chapter 11, in order to make a stronger case for the applicability of experimental mathematics to areas outside of analysis.

The book is well written and the tone is never heavy, despite the advanced mathematical examples within it. The authors include historical background and anecdotes which makes for a more interesting read and provides a human perspective behind the formulas presented. The (at times) funny illustrations and occasional jokes are definitely a pleasant addition.

This book is relatively tool agnostic; Maple and Mathematica are referenced throughout, and so are a few online tools to identify number sequences and known numeric values. Overall though, the emphasis in on the methodology rather than a particular CAS (Computer Algebra System) or programming language. In fact, with the exception of a snippet of Maple code in one of the explorations in the first chapter, the book describe the examples from a mathematical and algorithmic standpoint. You won’t find source code for the examples illustrated.

The ideal target audience for The Computer as Crucible is graduate students and researchers. A bright, motivated high-school student will get the gist of this book, but a more mature mathematical audience will actually be able to follow the steps within the examples and fully appreciate the insight on how an experimental approach can aid their research.

Despite the numerous examples employed to make their case, the authors start the book by explaining that it is not intended to be comprehensive. It’s meant to be thought provoking and to whet your appetite as to what is now possible in mathematical research thanks to computers.

As a computer programmer who’s passionate about mathematics, experimental mathematics fascinates me greatly. As such, I hope to work my way through the actual textbooks that are generally suggested as a follow up to this book. Namely, I’ve already started reading Mathematics by Experiment: Plausible Reasoning in the 21st Century (Second Edition), which is co-authored by Jonathan Borwein himself. Other textbooks referenced in this introduction are Experimental Mathematics in Action and Experimentation in Mathematics: Computational Paths to Discovery.

In conclusion, The Computer as Crucible is a lovely little book which builds a strong case for experimental mathematics. Any practicing mathematician or serious amateur should consider checking out this introduction to a topic that will no doubt transform mathematics.

Full disclosure: We received this book for free from the publisher, but we’re under no obligation to review or endorse it. We routinely receive a fair number of books from several publishers that never make the cut for an actual review. The links have our Amazon referral id which gives us a tiny percentage if you buy a book. In turn this helps support this site.

I’ll be taking a break from blogging and other work for a few weeks.

Filed under: admin

Apologies for the attention-seeking title, but that really is the purpose of this post. I want to draw attention to some ideas that are buried in more comments that most people are likely to want to read, because I think there is a chance that all that stands between where we are now and a solution to EDP is a few soluble technical problems. It goes without saying that that chance is distinctly less than 100%, but I think it is high enough for it to be worth my going to some trouble to lay out as precisely as I can what the current approach is and what remains to be done. I’ll try to write it in such a way that it explains what is going on even to somebody who has not read any of the posts or comments so far. The exception to that is that I shall not repeat very basic things such as what the Erdős discrepancy problem is, what a homogeneous arithmetic progression is, etc. For that kind of information, I refer the reader to the front page of the Polymath5 wiki.

Representing the identity.

Let me now discuss an approach that doesn’t work. (If you have been keeping up with the discussion, then this will be familiar material explained in a slightly different way.) Let be a large integer, and if and are two HAPs contained in then write for the matrix that is 1 at if and and 0 otherwise. In other words, it’s the characteristic function of Note that if then Let us write as etc.

Suppose that we could find for every pair of HAPs contained in a coefficient in such a way that and Then for every real sequence we have

It follows by averaging that there exist and such that

In particular, if each then there must exist and such that from which it follows that either or is at least So if we can get to tend to zero as tends to infinity then we are done.

Unfortunately, it is impossible to get to tend to zero. The reason is that the above argument would imply that if when mod 3, then for every there exists a HAP such that But that is not true: it can never be greater than 1.

Because of this example, we have been trying a different approach, which is to look for a more general diagonal matrix and write that as a linear combination of matrices of the form If one generalizes the approach in this way, then it is no longer clear that it cannot work — indeed, it seems likely that it can. However, it also seems to be hard to find a suitable diagonal matrix, and hard to think how one might decompose it once it is found.

Working over the rationals instead.

The single main point of this post is to suggest a way of overcoming this last difficulty. And that is to resurrect an idea that was first raised right near the beginning of this project, which is to look at the problem for functions defined on the positive rationals rather than the positive integers. (It is a straightforward exercise to show that the two problems are equivalent. For details go to the Polymath5 wiki and look at the section on the first page with simple observations about EDP.)

The point is that the counterexamples that show that the approach cannot work for the integers all make crucial use of the fact that some numbers are more divisible by small factors than others. But over the positive rationals all numbers are equally divisible. Or to put it another way, multiplying by a positive rational is an automorphism of This suggests that perhaps over the rationals it would be possible to use the identity matrix.

Dealing with infinite sets.

Now a problem arises if we try to do this, which is that the rationals are infinite. So what are we supposed to say about the sum of coefficients when we decompose the identity into a linear combination ?

Let me answer this question in two stages. First I’ll say what happens if we decompose the identity when the ground set is which shows a way of dealing with infinite sets, and then I’ll move on to where some additional problems arise.

Suppose, then, that the infinite identity matrix (that is, the function where and range over ) has been expressed as a linear combination Now let be a sequence. We’d like to show that it has unbounded discrepancy: that is, we’d like to show that for every there exists a HAP such that

Our problem is that is going to be infinite. We’d somehow like to show that it has “density” at most where is an arbitrary positive constant, or perhaps show that it has density zero. One way we might do this is as follows. For each positive integer Let be the sum of the over all such that both and have non-empty intersection with Then define the upper density of the coefficients to be If this is zero we can say that the coefficients have density zero. And if it is at most then we can say that they have density at most (In fact, even the would be OK — we just want the density to be small infinitely often.)

Let’s suppose that is at most Then if we truncate the sequence at , by changing all values after to zero, we find that

where I have written for the set of all HAPs that have non-empty intersection with Since the same averaging argument as before gives us a HAP (either or — WLOG ) such that

I fully admit that this is not very infinitary, but it is simple, and I’m not sure it matters too much that it is not infinitary. I’ll just briefly mention that one can use it to express the proof of Roth’s theorem (about AP discrepancy rather than HAP discrepancy). One expresses the infinite identity matrix as the following integral:

where One then expresses each function as a linear combination of HAPs of length (an arbitrary positive integer) and common difference at most One then obtains some cancellation in the coefficients, and proves that the density of the coefficients is at most (up to a constant factor). For details of how this calculation works, see this write-up, and in particular the third section.

The importance of the restriction on the length and common difference is that the edge effects (that is, APs that intersect without being contained in ) are negligible for large It is this feature that is slightly trickier to obtain for the rationals, to which I now turn.

Transferring to HAPs and rationals.

One useful feature of the set of APs of length and common difference at most is that each number greater than or equal to is contained in precisely such APs. A first question to ask ourselves is whether we can find a set of HAPs that covers the rationals in a similarly nice way. To start with, observe that if is a rational, then we can easily describe every HAP of length that contains Indeed, for every between and we have the HAP consisting of the first multiples of and that is all of them (since must be in the th place of the HAP for some and that and the length determine the HAP). So we have the extremely undeep result that every is contained in precisely HAPs of length (Note, however, how untrue this is if we work in the positive integers rather than the positive rationals.)

This looks promising, but we now need an analogue of the “increasing system of neighbourhoods” that was provided for us by the sets (It might have been more natural to work in and take the sets ) What is a sensible collection of finite sets with union equal to ?

One way of thinking about the sets is as follows. Using our system of APs, we can define a graph: we join to if there is an AP of length and common difference at most that contains both and The sets are quite close to increasing neighbourhoods in this graph: start with the number 1 and then take all points of distance at most from it. If we work with rather than then this graph is a Cayley graph, and after a while the neighbourhoods grow linearly with which is why the boundary effects are small.

What happens if we define a similar graph using HAPs in ? Now we are joining to if there exists and such that That is precisely the condition that there exists a HAP of length that contains both and In other words, we take the multiplicative group of and inside it we take the Cayley graph with generators all numbers with

This feels like the right graph to take, but it has the slight drawback that it is not connected: it is impossible to get from to if is a rational such that in its lowest terms either its numerator or denominator is divisible by a prime greater than The connected component of 1 in the graph is the set of all rationals where both and are products of primes less than or equal to But this is not really a problem: we’ll just work in that component of the graph.

Let’s write for the component of we are working in, and for the set of all points at distance at most from 1 in the graph. Now we can say how it would in principle be possible to prove EDP. We would like to find a way of writing the identity (this time thought of as the function where and range over ) in the form

where and are HAPs of length at most that are contained in For each let be the set of all such HAPs that have non-empty intersection with the neighbourhood Then we can define to be and we can define the upper density of the coefficients to be

Now let me show that if is at most for some sufficiently large then the HAP-discrepancy of every function on is at least This is by almost exactly the same argument that worked in The first step is to consider the restriction of to Then we know that

Now any HAP of length at most that intersects is contained in from which it follows that any HAP of length at most that intersects but is not contained in must intersect Since is a finitely generated Abelian group, the sets grow polynomially, which implies that the ratio tends to zero with

I’ll now be very slightly sketchy. We are supposing that is at most It follows that either or is noticeably smaller than In the second case we can change to and start again — we won’t be able to do this too many times so eventually we’ll reach the first case, where

In that case we have that

and that

After that, the argument really is the same as before (give or take the small approximations). [Remark: I have not checked the details of the above sketch, but I'm confident that something along these lines can be done if this doesn't quite work. It's slightly more difficult than in because it isn't obvious that the intersection of a HAP with is a HAP, whereas the intersection of an AP with is an AP.]

Characters on the rationals.

Now we must think about how to express the identity as a linear combination of HAP products with coefficients of density at most where is some arbitrary positive constant. Taking our cue from the case, it would be natural to express the identity on in terms of characters, and then to decompose the characters. So a preliminary task is to work out what the characters are.

This is (not surprisingly) a piece of known mathematics. I shall discuss it in a completely bare-hands way, but readers who don’t like that kind of discussion may prefer to look at the comment by BCnrd to this Mathoverflow question.

First I’ll work out what the characters on are, and then I’ll look at Recall that a character is a function from to the unit circle in such that for every That is, it is a homomorphism from to

Suppose we know the value of at a rational That tells us what is for every integer However, it does not tell us what is, say. All we know is that which gives us two possibilities for So in order to specify we need to specify its values at enough rationals that we can write every other rational as a multiple of one of them. And the choices we make at those rationals have to be compatible with each other.

A simple way of doing that is to choose the value of at for every positive integer making sure that That is, we choose to be any point in and then for each we have choices for given our choices up to that point.

An equivalent way of thinking about this is that we choose a sequence of real numbers satisfying the conditions that and is a multiple of Then the corresponding character is defined at to be for any Note that this is well-defined, since if and are both at least then so

Now because we chose an element of and then made a sequence of finite choices, it is easy to put a probability measure on the set of characters. We can therefore make sense of the expression

and prove that it is Let me briefly sketch this. Suppose that Then for every so we get 1. If then let when written in its lowest terms. If is the (random) sequence that determines then each is uniformly distributed in the interval and But is uniformly distributed in the interval so this expectation is zero (as ).

We have therefore shown that

where is the identity indexed by

What do we do if we want to modify this to work for ? Well, an initial complication that (I hope) turns out not to be a serious complication is that is not an additive group: it contains 1 and it does not contain any prime greater than However, it generates a subgroup of which consists of all rationals with denominators that are products of primes less than or equal to When I refer to “characters on ” I will really mean characters on

To describe these, we no longer need a sequence of reciprocals such that every rational is a multiple of one of them: we just want to capture all rationals in But that is straightforward: instead of taking the sequence we could take the sequence or we could replace by the product of all the primes up to There are any number of things that we could do.

Decomposing a character into “non-local” HAPs.

The thing that seems to me to make this approach very promising is that for any character on it is possible to partition into long HAPs on each of which is approximately constant. As this result suggests, it is possible to decompose in an efficient way as a linear combination of HAPs, which is very much the kind of thing we need to do in order to imitate the Roth proof.

I should warn in advance that it is not quite good enough for our purposes, because the HAPs we use are not “local” enough: they are sets of the form such that is small, but we do not also know that and are small. Without that, each number is contained in infinitely many HAPs, so we no longer have the condition that enabled us to define the “density” of a set of coefficients. Later I shall present a different idea that does use “local” HAPs, but fails for a different reason. My gut instinct is that these difficulties are not fundamental to the approach, but whether that is mathematical intuition or wishful thinking is hard to say.

Before I go any further, here is an easy lemma.

Lemma. Let be a character on and let Then there exists such that for every there exists a positive integer such that

Proof. Without loss of generality Let and let be the lowest common multiple of the numbers from 1 to [Thanks to David Speyer for pointing out at Mathoverflow that the l.c.m. is around a non-negligible improvement over .] Then by the usual pigeonhole argument we can find a positive integer such that so we can take

For the next result we define a HAP to be a set of the form

Corollary. Let be a character on let and let be a positive integer. Then we can partition into HAPs of lengths between and on each of which varies by at most

Proof. We begin by covering the integers. Find a positive integer such that Then on any HAP with common difference and length at most varies by at most We can partition the multiples of into HAPs of length and they will cover all the integers. (Indeed, they will cover all the multiples of )

We now want to fill in some gaps. Let us write and let us pick an integer a multiple of and greater than such that Between any two multiples of there are at least multiples of forming a HAP. This HAP can be partitioned into HAPs of common difference and lengths between and If we continue this process and make sure that every positive integer divides at least one (which is easy to do), then we are done.

Now a character restricted to a HAP is just a trigonometric function. If the character varies very slowly as you progress along the HAP, then convolving it with an interval (as in the Roth argument, but now we are talking about a chunk with the same common difference as the HAP) we obtain a multiple very close to 1 of the character itself. With the help of this observation, we can actually decompose the character efficiently as a linear combination of HAPs. Since this does not obviously help us, I will leave the details as an exercise.

What can we do with local HAPs?

Let us fix a positive integer and define a HAP to be local if it is of the form where (Thus, the definition of “local” depends on ) What happens if we have a character and try to decompose its restriction to as a linear combination of local HAPs (of reasonable length) on each of which varies very little?

The short answer is that it is easy to cover with HAPs of this kind, but it doesn’t seem to be easy to partition into them. In order to achieve the partition into non-local HAPs in we helped ourselves to smaller and smaller common differences, and correspondingly larger and larger values of and Another problem with that method was that what we are really searching for is a very nice and uniform way of decomposing characters, such as we had in the Roth proof. There the niceness was absolutely essential to getting enough cancellation for the proof to work, but it wasn’t essential to represent the identity — we could allow ourselves a bit extra as long as that extra was positive semidefinite.

So let’s not even try to partition Instead, we could simply take our character and use it as a guide to the coefficients that we will give to our HAPs.

The rough idea would be something like this. Given a character and a HAP we choose a coefficient in some nice way that makes it large when is small. It could for example be for some suitable And then we could use convolution to represent times the restriction of to as a linear combination of sub-HAPs of of length smaller than but not too much smaller, and common difference

We would know from the lemma above that every would be contained in at least some HAPs with large coefficients, so the restriction of every would in some sense be catered for. I think there would be some that were catered for too much (the precise relationship between those and remains to be worked out, but I think this will be straightforward), but I hope that the whole decomposition can be defined in a nice enough way for the function that results to be the pointwise product of the original character with a “nice” non-negative real function that’s bounded away from zero. More speculatively, one can hope that the coefficients have small density and that it is possible to subtract a not too small multiple of the identity from the matrix and still be left with something that’s non-negative definite.

An attempt to be more precise.

In this final section I want simply to guess at a matrix decomposition that might potentially prove EDP. As I write this sentence I do not know what the result will be, but the two most likely outcomes are that it will fail for some easily identifiable reason or that the calculations will be such that I cannot tell whether it fails or succeeds.

Actually, to make the guess just a little bit more systematic, let’s suppose that for each HAP in the class of HAPs we are considering and for each character we have a coefficient That is, corresponding to we are taking the function Given two HAPs and what will the coefficient of be in the decomposition ?

Expanding this out gives us

If we reverse the order of expectation and summation then we see that the coefficient of is

Now let us think what we are trying to achieve with the HAPs and Given a HAP of the form we want to use it to contribute to the representation of characters for which is small. If is such a character, then we know that the numbers vary only slowly. Therefore, if we take short subprogressions of the form then on each one will be roughly constant. If we fix a length (which may have to be logarithmic in or something like that), then we can represent the restriction of to as a linear combination of the HAPs of length give or take some problems at the two ends.

Roughly speaking, the coefficient of will be That is, if we write for the HAP then it is approximately true to say that the restriction of to is

Now we want to do this only if is close to 1. So the coefficient of in the decomposition of will in general be where is some nice function (which, if the Roth proof is anything to go by, means that it has non-negative and nicely summable Fourier coefficients) that is zero except at points that are close to 1.

What would we expect to be like? Is there any chance that it is close to a multiple of ?

Since is roughly constant on whenever is non-zero, the value of this function at is roughly the same as it is if we take just singletons — that is, if we set and define to be So the function we get should have a value at that is close to

In other words, we get something that has the same argument as but that has big modulus at a rational if it so happens that is close to 1 for unexpectedly many positive integers Now this can happen, but I would think that for nearly all and if is not too small, then would be about its expected value of So I am hoping that we will have some kind of rough proportionality statement.

Going back to the coefficient of in the decomposition, let us take the two HAPs and We have decided to define to be so is equal to

It is a significant problem that we are forced to consider the case where in the Roth proof, we did not have to look at products of APs with different common differences. But if we are to have any chance of decomposing into “local” HAPs, then necessarily we must use completely different common differences to deal with rational numbers that are a long way apart in the graph. This is such a significant difference from the Roth proof that it may be a reason for this approach not working at all. However, it does look as though there is plenty of cancellation.

I think the expression we would be trying to bound is

which is perhaps better thought of as

And our main strength would be that we would be free to choose as it suited us. There would be two challenges: to obtain a good bound on the above expression, and to prove that we could subtract a reasonable-sized multiple of the identity and still be left with a positive semi-definite matrix.

I think the above counts as a calculation for which I cannot tell whether it fails or succeeds. But I hope it may be enough to provoke a few thoughts.

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