The Riemann-Davis hypothesis

Few musicians can claim a legacy the size and impact of Miles Dewey Davis III’s (26 May 1926 – 28 September 1991): from The Birth of the Cool (1957) through Kind of Blue (1959) all the way to Bitches Brew (1970) and beyond Miles Davis was not only influencial, he made at every stage of his career (cue Academic Tone) “seminal contributions to the field”; whatever stage of Miles’ career was marked by two things: organisational genius and contagious creativity. Take Kind of Blue for instance. It is known that the session musicians were only given rhythmic lines and general indications on key and tonality for each track they recorded; they had to improvise along those lines and they recorded the album in a two-day session. The result: one of the most critically acclaimed and comercially succesful albums of all time.

Miles Davis made great music and in passing influenced almost every musical genre; he’s credited as one of the creators of “jazz fusion”; he could bring together percussionists of afro-latin rhythms, classically trained pianists and jazz musicians to create music that otherwise-I venture to say-would’ve never existed:

Whatever you may think of the above piece, where do you put it? It’s not “jazz-jazz”, is it? It’s not exactly “rock”; it’s not quite “funk”… yet jazz fans can make sense of it, rock fans can relate to it and funky people might just ask for a little more up-beat to be content. Incidentally, the track above belongs to the recording sessions of what became Jack Johnson (1971) which Miles described as “the best rock band you’ve ever heard”… and in my opinion, he was right; Ali didn’t make it to the official release of the album, though.

Now take this one, called Ascenseur pour l’echefaud (Lift to the Gallows):

It’s the soundtrack of a film of the same title (1957). Background story: Miles was introduced to the director of the film and he invited him to record the soundtrack for it; Miles agreed and less than a month later he came to the studio with his band, gave them a few general directives and harmonic sketches, heard the synopsis of the film and while an edited version of it was projecting, they started recording. The music is jazz… but how do you account for that?

Allow me to repeat myself: how do you account for a bandleader that can inspire his bandmates to improvise a whole (and beautiful, in my opinion) album while the film for which it’ll be the soundtrack is playing right in front of them? It takes a special kind of genius to do that.

Trading God for numbers

Miles’ recording career as a bandleader started in 1951. A hundred years earlier, many miles away from the U.S., in Göttingen, Hannover (in what later became Germany) Georg Friedrich Bernhard Riemann (17 September 1826 – 20 July 1866) submitted his Dissertation Grundlagen für eine allgemeine Theorie der Functionen einer veränderlichen complexen Grösse (more or less: “foundations for a general theory of functions of a complex variable”). Granted, quite far removed from funky-jazz-rock music. Nonetheless, Riemann and Miles had two things in common: genius and virtually ever-lasting legacy.

B. Riemann, source unknown.
B. Riemann, image source unknown.

Riemann’s life falls short for a few months to Miles’ recording career (well, not exactly true: Miles took a 4-year hiatus in the late 70s… but that’s rather unimportant here). Tuberculosis killed him at the age of 39. He was the son of a Lutheran pastor in Breselenz, Hannover. He moved to Göttingen with the intention of studying theology at the university there; however in that university lied the most influential mathematician of his time (and one of the most influential of all times): Carl Friedrich Gauß (anglicised “Gauss”). Riemann, having attended some mathematics lectures (in all probability, some of them given by Gauß), asked for permission of his father to switch from theology to mathematics. Perhaps theology departments around the world don’t realise the genius they lost, but we mathematicians are grateful for the change. Then in the spring of 1847 Riemann moved to Berlin; that was a Mecca for mathematical research in those days (later Göttingen became it, thanks in no small part to the influence of Riemann’s work) and he made the most of it. Being naturally disposed to mathematics he was taken under the wing of Gustav Dirichlet, one of the leading mathematicians of the time. He had a very fertile visual imagination and immediately put it to use to study functions of a complex variable. If you ever took a calculus course imagine it using complex numbers. Complex calculus is in a sense easier that the usual calculus because for a function to be “well-behaved” in the usual (termed “real”) calculus one has to ask for a lot of conditions on the function, whereas minimal conditions are required for a complex function to be “nice”. All this is rather unimportant here, but you can find out more if you’re interested; that’s why I mention it.

In his short academic career Riemann made “seminal contributions” to almost every field of the mathematics of his day. The ideas in his dissertation later became the theory of Riemann surfaces which are still studied today; he studied problems in electrodynamics and thermodynamics that led him to new mathematics, especially in the theory of conformal maps; he single-handedly created Riemannian geometry, which is the study of curvature in spaces of arbitrary dimension, ideas that became later fundamental for grounding the theory of general relativity on sound mathematics; and he lend his name to one of the most famous open problems in contemporary mathematics: the Riemann hypothesis. I want to give you a glimpse of the latter contribution.

Building the building blocks

Positive integers, or whole numbers, are familiar from everyday experience; we count with them (Spanish pun very much intended). Negative integers arise when substracting a whole number from one which is stricty lesser; we use them to represent debts. For instance, if I earned $10 and owe you $15, if I give you my $10 I still owe you $5, so I end up with -5 “coins”. Among integers there are many special families. One is the sequence of even numbers: 2, 4, 6, 8… It’s easy to generate that sequence: you take any integer and multiply it by two. Likewise, odd numbers are generated by taking any whole number, multiplying it by two and substracting 1 from the result: 1->1, 2->3, 3->5, etcetera.

Perhaps the most interesting family of integers is that of prime numbers: those that cannot be divided by any other number, save 1. For instance: 17 is prime because dividing it by any other integer doesn’t return an integer as a result, whereas 14 is not prime because 14/2=7; neither is 45 because 45/5=9. A number that is not prime is called composite; when dividing a composite number by a prime number there are two possibilities: either the result is a prime number or another composite. If the latter is the case, we can divide it further by the same or another prime number, and the same two possibilities arise… and so on. For example, 210 is composite and dividing it by 2 gives 105; dividing the latter by 3 gives 35; dividing by 5 gives 7 and 7 is prime. Put another way (and this is the crucial idea) 210=2·3·5·7 (the dot denotes multiplication). It turns out that this can be done for any number; this is the so-called fundamental theorem of arithmetic: every (positive) integer (excepting 1) is a product of primes (repetitions are allowed; for example, factor 72 into primes). In this sense, prime numbers can be considered as the building blocks for all other numbers.

To the point (finally!): whereas even and odd numbers are easy to generate, there is no straightforward way to generate prime numbers. What this means is that there is so far no “recipe” to generate prime numbers in the same way there is one to generate even numbers (take a number and then multiply it by 2).

In 1859 Riemann submitted a work to the Berlin Academy of Sciences, titled Über die anzahl der Primzahlen unter einer gegegen Grösse, which translates (roughly) into “On the quantity of prime numbers less than a given quantity”. Using ideas from complex analysis (in which he was an expert) he proposed a way for measuring how many prime numbers are less than a given number x; this quantity is usually denoted by \pi(x). The Riemann hypothesis states that this quantity is controlled by a complex function, called the Riemann-zeta function, in such a way that the zeroes of this function (that is, the points at which the function returns the value zero) predict very accurately the value of \pi(x). Rephrasing this: when x is really big the difference between the number \pi(x) and the value predicted by the Riemann hypothesis becomes very small. This is not as straightforward as multiplying a given integer by 2, but it would help us to know just how many primes are there below a given number. This is an unsolved (and VERY difficult) problem in mathematics. For more information you can go here.

Abstract legacies

Of course there is no direct relation between the life and works of Miles Davis and those of Bernhard Riemann. One can draw an analogy though: both men were unrelenting in their pursuit of the discipline that gave their lives meaning. Miles said that he didn’t like dwelling in the past, regarding his past work as “overheated turkey”; he strived for experimentation, for new ways of making music he could enjoy making. Riemann traded his passion for God–though, allegedely, remained very religious throughout his life–for a new passion: mathematics. In doing so he became one of the most important mathematicians of history; he never gave up his willing of understanding the ideas behind the problems of the mathematics of his time. In his pursuit of understanding he contributed problems and theories that keep mathematicians busy to this day. Certainly, the legacies of both men are ever-lasting, though very abstract (music is abstract in the sense that, when it is not being played, it is not clear what it is). That is, in my opinion, the sine qua non characteristic of genius: exploring a discipline so thoroughly that the exploration gives rise to innovation. My hypothesis, the Riemann-Davis Hypothesis, is that genius, though scarce, arises when someone is disciplined and curious enough to transform a field anew, disregarding current opinions or practical difficulties, proposing new ideas and developing them to their utmost consequences.

Genius is a word that is gratuitously used to express admiration or preference for someone’s work. Of course it is a matter of opinion, but whether or not you like or understand the work of Miles or of Riemann I find it very hard to deny that they are works of genius. I find them both inspiring and beautiful. I hope I’ve managed to convince you they are.