Progress On Riemann?

The Riemann Hypothesis is one of the greatest unsolved math problems in the world today – probably. If not, it’s easily one of the most famous.

It centers on the Riemann Zeta Function, given by

The function being defined over the complex plane. The Hypothesis is, in its most basic form, that if , then s = 1/2 + t*i, for some real number t. That any zero of the zeta function has a real part of one half.

Why this is interesting though is the connection with prime numbers. There’s a function called the prime counting function, pi(x), and it basically gives the number of prime numbers between 0 and n. As the distribution of primes is so bizarre, being able to calculate pi(x) correctly, without just counting all the prime numbers from 1 to x, would be a very neat trick. We have some very nice approximations for pi(x), involving integrals of relatively simple functions, but the question is the, how good are these approximations?

So enters the Riemann Hypothesis. Basically, it’s been shown that the Riemann Hypothesis, as stated above, is equivalent to saying, for our best approximation of pi(x), pi(x) differs from this approximation by no more than some constant times , for x sufficiently large.

Which is a very interesting result.

But a proof was proposed a few months ago, and it seems really quite promising, and at the very least opens up some new avenues for further inquiry. As usual for such things, I don’t have the background to explain fully, but I think I can break down a couple of the key points to an understandable level. Maybe Mark can pick this up?

Prime Polynomials

Primes are of interest. Indeed, I would go so far as to say they are the stereotypical Object of Interest people think of when they think about mathematics.

When writing posts like these, I’m never really sure how basic to go. I’ve been tempted at times to define what functions are, for example. One of my goals in writing about math is to make it as accessible as possible, and I’m pretty sure I fail a lot at that, but then again I don’t want to go so basic I can’t talk about anything of interest.

The point is, it will satisfy me enough to say, primes are integer numbers that are only divisible by themselves and 1. 2, 3, 5, 7, 101 are classic examples, while 10 is very much not prime.

Primes are of interest for many reasons, but they tickle the fancy of many due to their rather inexplicable nature. For example, the distribution of primes, where they fall on the number line, while being highly structured, seems largely without form or pattern. We can’t really predict what numbers are prime, where they are, or how many there are in a certain range.

Something that would be nice would be a function that generates just prime numbers. Now, given a set of prime numbers, like 2, 3, 5, 7, you can construct, for example, a polynomial f(x) such that f(1) = 2, f(2) = 3, f(3) = 5, f(4) = 7, and so on. An excellent example of such a thing is f(n) = n² + n + 41, which produces primes for n = 0 to n = 40. Do you see why f(41) is not prime, though?

So the question is, in general, can you construct a polynomial that only gives prime numbers on integer inputs?

Too Many Words On Divisibility

In this one post I’m working on, of primes and polynomials, there’s this one part of the proof which requires a small lemma. And this has caused no end of vexation for me.

The problem is that the lemma is not directly related to what I’m trying to prove, and is not trivial enough to simply state. So it requires a proof, but were I to prove it, it would be this block of text in the body of my main proof completely unrelated to the point and topic I wanted to focus on. When I write these posts, I really try to keep everything focused and to the point to minimize the chance of losing anyone along the way. Moreover, while it’s a lemma about divisibility and remainders, a subject most easily communicated in the form of modular arithmetic, I don’t honestly know how many people who read this have exposure to that subject, so how much would I have to say to build up to the point of proving this lemma in that form? All this goes to say, how can I make the whole proof most accessible?

It’s a style and communication thing.

So ultimately, I’ve decided to let this one lemma, however simple, to have its own post. But, as a result, I will end up shedding far too many words on far too small a result.

The desired result is this, that for a prime p and two integers m and n, (1 + m*p)ⁿ gives a remainder of 1 when divided by p.

If this seems trivial to you, move on and be merry. If not, check below the fold. It’s a good divisibility proof, with some basic induction to boot.

Mathematicians Are Best!

“There are three men on a train. One of them is an economist and one of them is a logician and one of them is a mathematician. And they have just crossed the border into Scotland (I don’t know why they are going to Scotland) and they see a brown cow standing in a field from the window of the train (and the cow is standing parallel to the train). And the economist says, ‘Look, the cows in Scotland are brown.’ And the logician says, ‘No. There are cows in Scotland of which at least one is brown.’ And the mathematician says, ‘No. There is at least one cow in Scotland, of which one side appears to be brown.’ And this is funny because economists are not real scientists and because logicians think more clearly, but mathematicians are best.”

The Curious Incident of the Dog in the Night-Time (2003)

… I’m Really Rather Tired

And I can’t do this any more.

This is an issue I’ve been wrestling with for a while now. When I first went off to college, I was full of excitement and verve. I could see the universe, in all its entirety, a vast mathematical web stretching out to infinity, waiting to be unwound, solved, and expounded upon. Mathematics seemed the key to everything – and I was to be the one to solve it.

But then it began – it’s hard to know how it started. A creeping depression that … I was wrong. Something I certainly wasn’t used to. Was it Godel’s Incompleteness Theorem? A statement of fact that not everything true was provable? Could it be that some problems just couldn’t be solved? Was that any consolation for the growing certainty that I was not going to be the one to solve them? Was it the fact that so much of mathematics – set theory, for example – was, in point of fact, boring?

Theorems lost their meaning as they became little more than recombinations of axioms that have little to no bearing or significance. Weeks of proving result after result that we trivially know to be true, while the questions of interest slip further into the realm of unknowable. You end up asking yourself, what is it you’re really after? You thought you enjoyed math, but that was before, evidently, you knew what math really was.

Despair is the only word to describe it. The hopeless realization that the path you’ve set yourself upon leads to a castle, yes, but one built on a foundation of groundless assumptions. Does anyone really know what Knot Theory is good for? You find yourself trapped in a world of theory where anything can be what you define it to be. And never knowing what it really is.

I’m tired. Really tired. And I’ve had enough.

I’m giving up maths. I apologize to my few, interested readers, who have followed me all this time, but I really cannot carry myself any further, having lost all interest in what I thought I held dear. I will be dropping out of the Mellon College of Science and entering the College of Humanities and Social Sciences to pursue a degree in philosophy, and a minor in creative writing.

Thank you for your time and attention – but honestly, let’s all move on to something -real-.