Science Fiction movies are always centered around the possibility of time travel. Imagine traveling into the future or the past to find out some information that you desperately wanted to know. Of course, scientists are often interested in questions about their particular areas of research. In pure mathematics the greatest unsolved problem has to do with a particular function called the Riemann zeta function. The conjecture is that all its solutions lie on a certain axis on the complex plane, a 2-dimensional plane essentially. Here is the original paper by Riemann published in 1859.
Now, you’re probably thinking what in the world is wrong with these folks. Why does this matter? Well, two centuries of research has shown that this pesky function is everywhere. In particular it is closely related to the distribution of prime numbers. Ok, you forgot high school math. A prime number is any number only divisible by 1 and itself. So, a standard programming exercise is to efficiently generate all prime numbers. Of course it starts with 1. 2 is also prime, the only even prime number. Then we have 3. 4 is divisible by 2, so we skip it. 5 is prime. We skip 6. 7 is prime. 8 is not prime. 9 is not. 10 is not. 11 is prime. 12 is not. 13 is prime. 14, 15,16 are not. 17 is prime. And so on. What’s the pattern? If you figure that out, it’s worth billions of dollars! Why?
Well, every time you buy something online, there’s a cryptography protocol that hides your transaction by applying a code so eavesdropping your financial data is not easy. The secrecy protocol is generated using prime numbers. If you can factor a number efficiently using prime numbers, you can crack a lot of secrets. Well, no one has yet figured out the pattern of prime numbers. We know the general distribution, roughly how many there are between two numbers, and a lot more, but no solution to the Riemann zeta function. A famous mathematician once said that when he finally got to meet God, he’d ask him first if the Riemann hypothesis was true.
Back to music. Time travel in music is easy. We can just listen to a recording of some very old music. I’m listening to Gothic music of the 12th and 13th century. Compared to the above Folia recording is interesting. In the 12th and 13th century, they hadn’t figured out what melody was. Like the ancients didn't know the planets revolved around the sun, not the other way around. We are listening to a great old recording by David Munro and the Early Music Consort of London. It’s eerie listening to this because it’s medieval pre-music. These guys hadn’t figured out how to create a tune. So, they hummed and hawed around a bunch of notes, but it just doesn’t click. They hadn’t yet figured out Folia. It’s like the mathematicians don’t know how primes are distributed. Perhaps in 2300 any kid will have a super cool thinking machine implant that can solve this or any other math problem. Word on the AI street is that some hush hush projects in big Bay Area companies are working on getting AI to solve the Riemann hypothesis. What would Riemann have thought of all this?
To me, music and math have been my constant companions for 40+ years. I got into AI reading a book on math, music, and art, and AI. It’s called Gödel, Escher, Bach: An eternal Golden Thread. It won the Pullitzer price for non-fiction. It’s 800 pages, but a mind-blowing book. Check it out!