The Summer 2017 Mathematics A To Z: Zeta Function

Today Gaurish, of For the love of Mathematics, gives me the last subject for my Summer 2017 A To Z sequence. And also my greatest challenge: the Zeta function. The subject comes to all pop mathematics blogs. It comes to all mathematics blogs. It’s not difficult to say something about a particular zeta function. But to say something at all original? Let’s watch.

Summer 2017 Mathematics A to Z, featuring a coati (it's kind of the Latin American raccoon) looking over alphabet blocks, with a lot of equations in the background.
Art courtesy of Thomas K Dye, creator of the web comic Newshounds. He has a Patreon for those able to support his work. He’s also open for commissions, starting from US$10.

Zeta Function.

The spring semester of my sophomore year I had Intro to Complex Analysis. Monday Wednesday 7:30; a rare evening class, one of the few times I’d eat dinner and then go to a lecture hall. There I discovered something strange and wonderful. Complex Analysis is a far easier topic than Real Analysis. Both are courses about why calculus works. But why calculus for complex-valued numbers works is a much easier problem than why calculus for real-valued numbers works. It’s dazzling. Part of this is that Complex Analysis, yes, builds on Real Analysis. So Complex can take for granted some things that Real has to prove. I didn’t mind. Given the way I crashed through Intro to Real Analysis I was glad for a subject that was, relatively, a breeze.

As we worked through Complex Variables and Applications so many things, so very many things, got to be easy. The basic unit of complex analysis, at least as we young majors learned it, was in contour integrals. These are integrals whose value depends on the values of a function on a closed loop. The loop is in the complex plane. The complex plane is, well, your ordinary plane. But we say the x-coordinate and the y-coordinate are parts of the same complex-valued number. The x-coordinate is the real-valued part. The y-coordinate is the imaginary-valued part. And we call that summation ‘z’. In complex-valued functions ‘z’ serves the role that ‘x’ does in normal mathematics.

So a closed loop is exactly what you think. Take a rubber band and twist it up and drop it on the table. That’s a closed loop. Suppose you want to integrate a function, ‘f(z)’. If you can always take its derivative on this loop and on the interior of that loop, then its contour integral is … zero. No matter what the function is. As long as it’s “analytic”, as the terminology has it. Yeah, we were all stunned into silence too. (Granted, mathematics classes are usually quiet, since it’s hard to get a good discussion going. Plus many of us were in post-dinner digestive lulls.)

Integrating regular old functions of real-valued numbers is this tedious process. There’s sooooo many rules and possibilities and special cases to consider. There’s sooooo many tricks that get you the integrals of some functions. And then here, with complex-valued integrals for analytic functions, you know the answer before you even look at the function.

As you might imagine, since this is only page 113 of a 341-page book there’s more to it. Most functions that anyone cares about aren’t analytic. At least they’re not analytic everywhere inside regions that might be interesting. There’s usually some points where an interesting function ‘f(z)’ is undefined. We call these “singularities”. Yes, like starships are always running into. Only we rarely get propelled into other universes or other times or turned into ghosts or stuff like that.

So much of the rest of the course turns into ways to avoid singularities. Sometimes you can spackel them over. This is when the function happens not to be defined somewhere, but you can see what it ought to be. Sometimes you have to do something more. This turns into a search for “removable” singularities. And this does something so brilliant it looks illicit. You modify your closed loop, so that it comes up very close, as close as possible, to the singularity, but studiously avoids it. Follow this game of I’m-not-touching-you right and you can turn your integral into two parts. One is the part that’s equal to zero. The other is the part that’s a constant times whatever the function is at the singularity you’re removing. And that ought to be easy to find the value for. (Being able to find a function’s value doesn’t mean you can find its derivative.)

Those tricks were hard to master. Not because they were hard. Because they were easy, in a context where we expected hard. But after that we got into how to move singularities. That is, how to do a change of variables that moved the singularities to where they’re more convenient for some reason. How could this be more convenient? Because of chapter five, series. In regular old calculus we learn how to approximate well-behaved functions with polynomials. In complex-variable calculus, we learn the same thing all over again. They’re polynomials of complex-valued variables, but it’s the same sort of thing. And not just polynomials, but things that look like polynomials except they’re powers of \frac{1}{z} instead. These open up new ways to approximate functions, and to remove singularities from functions.

And then we get into transformations. These are about turning a problem that’s hard into one that’s easy. Or at least different. They’re a change of variable, yes. But they also change what exactly the function is. This reshuffles the problem. Makes for a change in singularities. Could make ones that are easier to work with.

One of the useful, and so common, transforms is called the Laplace-Stieltjes Transform. (“Laplace” is said like you might guess. “Stieltjes” is said, or at least we were taught to say it, like “Stilton cheese” without the “ton”.) And it tends to create functions that look like a series, the sum of a bunch of terms. Infinitely many terms. Each of those terms looks like a number times another number raised to some constant times ‘z’. As the course came to its conclusion, we were all prepared to think about these infinite series. Where singularities might be. Which of them might be removable.

These functions, these results of the Laplace-Stieltjes Transform, we collectively call ‘zeta functions’. There are infinitely many of them. Some of them are relatively tame. Some of them are exotic. One of them is world-famous. Professor Walsh — I don’t mean to name-drop, but I discovered the syllabus for the course tucked in the back of my textbook and I’m delighted to rediscover it — talked about it.

That world-famous one is, of course, the Riemann Zeta function. Yes, that same Riemann who keeps turning up, over and over again. It looks simple enough. Almost tame. Take the counting numbers, 1, 2, 3, and so on. Take your ‘z’. Raise each of the counting numbers to that ‘z’. Take the reciprocals of all those numbers. Add them up. What do you get?

A mass of fascinating results, for one. Functions you wouldn’t expect are concealed in there. There’s strips where the real part is zero. There’s strips where the imaginary part is zero. There’s points where both the real and imaginary parts are zero. We know infinitely many of them. If ‘z’ is -2, for example, the sum is zero. Also if ‘z’ is -4. -6. -8. And so on. These are easy to show, and so are dubbed ‘trivial’ zeroes. To say some are ‘trivial’ is to say that there are others that are not trivial. Where are they?

Professor Walsh explained. We know of many of them. The nontrivial zeroes we know of all share something in common. They have a real part that’s equal to 1/2. There’s a zero that’s at about the number \frac{1}{2} - \imath 14.13 . Also at \frac{1}{2} + \imath 14.13 . There’s one at about \frac{1}{2} - \imath 21.02 . Also about \frac{1}{2} + \imath 21.02 . (There’s a symmetry, you maybe guessed.) Every nontrivial zero we’ve found has a real component that’s got the same real-valued part. But we don’t know that they all do. Nobody does. It is the Riemann Hypothesis, the great unsolved problem of mathematics. Much more important than that Fermat’s Last Theorem, which back then was still merely a conjecture.

What a prospect! What a promise! What a way to set us up for the final exam in a couple of weeks.

I had an inspiration, a kind of scheme of showing that a nontrivial zero couldn’t be within a given circular contour. Make the size of this circle grow. Move its center farther away from the z-coordinate \frac{1}{2} + \imath 0 to match. Show there’s still no nontrivial zeroes inside. And therefore, logically, since I would have shown nontrivial zeroes couldn’t be anywhere but on this special line, and we know nontrivial zeroes exist … I leapt enthusiastically into this project. A little less enthusiastically the next day. Less so the day after. And on. After maybe a week I went a day without working on it. But came back, now and then, prodding at my brilliant would-be proof.

The Riemann Zeta function was not on the final exam, which I’ve discovered was also tucked into the back of my textbook. It asked more things like finding all the singular points and classifying what kinds of singularities they were for functions like e^{-\frac{1}{z}} instead. If the syllabus is accurate, we got as far as page 218. And I’m surprised to see the professor put his e-mail address on the syllabus. It was merely “bwalsh@math”, but understand, the Internet was a smaller place back then.

I finished the course with an A-, but without answering any of the great unsolved problems of mathematics.

Reading the Comics, September 24, 2015: Yes, I Do So Edition

Yes, in this roundup of mathematically-themed comic strips I talk seriously about the educational techniques of the fictional Great Smokey Mountains community where the comic strip Barney Google and Snuffy Smith takes place. I accept the implications of this.

John Rose’s Barney Google And Snuffy Smith for the 23rd of September is your standard snarky-response joke. I’m a bit surprised to see that at whatever class level Jughaid’s in they’re using “x” to stand in for the not-yet-known number. I thought empty boxes or question marks were more common. But I also think Miz Prunelly’s not working most effectively by getting angry at Jughaid for not knowing what x is.

Miz Prunelly asks Jughaid what the 'x' in the equation 3 + x = 8 stands for. He insists he doesn't know. She says she'll only ask him one more time. He says that's a relief as he still doesn't know.
John Rose’s Barney Google And Snuffy Smith for the 23rd of September, 2015.

I would suggest trying this: can Jughaid find some possible values of x that are definitely too small? And some possible values that are certainly too big? Then what kinds of numbers are both not-too-small and not-too-big? One standard mathematician’s trick for finding an unknown quantity is to show that it can’t be smaller than some number, giving us a lower bound. And then show it can’t be larger than some number, giving us an upper bound. If the lower bound and the upper bound are the same number, we’re done. If they’re not the same number we might have to go looking, but at least we’ve got a better idea what a correct answer should look like. If the lower bound is a larger number than the upper bound, we have to go back and check whether there actually is an answer, or if we started off in the wrong direction.

Scott Adams’s Dilbert Classics for the 23rd of September (a rerun from the 30th of July, 1992) mentions “conversational geometry”. It’s built on a bit of geometry that somehow escaped into being a common allusion, and that occasionally riles up grammar nerds. The problem is trying to use “turned around 360 degrees” for “turned completely around”. 360 degrees is certainly turning something all the way around, but it leaves the thing back where it started, apparently unchanged. (Well, there are some oddball structures where you can rotate something 360 degrees and have it not back the way it started, but those only occur in abstract mathematical constructions and in some — not all! — subatomic particles. Yes, it’s weird. It’s like that.)

The grammar nerd will insist that what’s meant is to turn something 180 degrees, reversing its direction. Or maybe changed 90 degrees, looking perpendicular to whatever the original situation was. Personally I can’t get upset about a shorthand English phrase not making literal sense, because there are only about six shorthand English phrases that make even the slightest literal sense, and four of those are tapas orders. Eventually you have to stop with the rage and just say something already. And rotating 360 degrees is a different process from rotating not at all. You move, you break your focus, you break your attention. Even if you face the same things again you face them having refreshed your perceptions. You might now see something you had not before.

John Zakour and Scott Roberts’s Maria’s Day for the 23rd of September asserts that mathematics is important so that one can check one’s accountants. This is true, although it’s hardly everything mathematics is enjoyable for. And while I don’t often get to call attention to comic strip artwork, do look at the different papers; there’s some fun there.

Pab Sungenis’s New Adventures of Queen Victoria for the 24th of September — and the days around it — have seen Victoria and Nikola Tesla facing the end result of too much holiday creep: a holiday singularity. By a singularity a mathematician means a point where stuff gets weird: where a function isn’t defined, where a surface breaks off, where several independent solutions suddenly stop being independent, that sort of thing. It’ll often correspond with some measure becoming infinitely large (as a positive or a negative number), though I don’t think it’s safe to say that always happens.

We generally can’t say what’s happening at a singularity. But the existence of a singularity, and what it behaves like, can tell us something about what’s happening away from the singularity. It can happen, for example, that a singularity is removable. That is, if a function is undefined for some values, perhaps we can come up with a logically compelling definition for what it might do at those values. If you can remove a singularity then we call this a “removable singularity”. This serves to show you don’t necessarily need grad school to understand everything mathematicians are saying. Sometimes a singularity can’t be removed, and those are known as “nonremovable singularities” or “essential singularities” or sometimes some other nastier names.

Usually, if one has a singularity in a mathematical construct, then information about one side of the singularity isn’t enough to extrapolate what might be on the other side. This makes the literary use of a “singularity” as “something magical that does whatever the plot requires” justified enough. Tesla here is clearly using the idea of reaching an infinitely vast, or an infinitely dense, holiday concentration as a singularity. I grant that would be singular enough. The strip does make me think of a fun sequence in Walt Kelly’s Pogo where one year the Bun Rabbit decided to get all the holiday-celebrating done first thing in the year, to clear out the rest. He went about banging the drum and listing every holiday ever, which is what made me aware of the New Jersey Big Sea Day.

Shaenon K Garrity and Jeffrey C Wells’s Skin Horse for the 24th of September includes a sequence identified as the “Catalan Series”. I’d have said “sequence” myself. The Catalan sequence describes (among other things) how many ways you can break down a regular polygon into a particular number of triangles. A square can be broken down into two triangles just two ways (if orientation counts, which for this problem, it does). A pentagon can be broken down into three triangles in five ways. A hexagon can be broken down into four triangles in fourteen ways, and so on. (The key is you break the polygon into a number of triangles that’s two less than the number of sides. So if you had a 9-sided polygon, you’d break it up into 7 triangles. If you had a 20-sided polygon, you’d break it up into 18 triangles.) The sequence describes more stuff than that, but this is an easy-to-understand application. As the name of the sequence suggests, it comes to us from the Belgian-French mathematician Eugène Charles Catalan (1814 – 1894).

Catalan’s name also might be faintly familiar for a conjecture he posed in 1844, which was finally proven true in 2002 by Preda Mihăilescu. His conjecture is based on observing that the number 2 raised to the third power is 8, while the number 3 raised to the second power is 9, quite close together. Catalan conjectured this was the only case of consecutive powers. That is, there’s nothing like 15 to the twentieth power being one less than 12 to the twenty-fourth power or anything like that. I’m afraid I don’t know enough of this kind of mathematics, known as number theory, to say whether that’s of use for anything more than settling curiosity on the point.