Case Study: Primes and the Riemann Hypothesis

After watching Barry Mazur’s video lecture on “A Lecture on Primes and the Riemann Hypothesis”, I got inspired to talk about what’s up with Mathematicians and their unruly obsession with prime numbers.

In popular media, such as art, literature and even movies, primes are indicative for something mysterious, a hidden language or code in its fervent use in cryptography and code breaking. It appears in the film Contact and Cube where primes are used in encrypted messages and in the encryption which keeps our credit card details safe from outside meddlers. What type of interest would possibly be worth Mathematician’s time to show to these types of numbers?


Beginner

Prime numbers are often called ‘the building blocks of numbers’. This singular statement is what captures the interest of Mathematicians around the world, and holds much more than meets the eye. As a comparison, we can look to other fields to justify our excitement with their own study of their building blocks: Chemistry has the elements in the Periodic Table and Physics has the Standard Model. While in these fields, their study is justified by the mere fact that they exist, within Mathematics this is less obvious and must be justified in some way. How do we know that such building blocks exist? More importantly, why do we give this title to the primes? In Mathematics this question is quite nicely addressed by quite an important result which gives justice to this excitement. It is rightly given a very fancy and a very luxurious name – ‘The Fundamental Theorem of Arithmetic’. It goes as follows:

Any integer greater than 1 is either prime or a unique product of primes

Let’s break this statement down for those who are a bit unfamiliar with the terms used. It is well known that prime numbers are those numbers that are only divisible by 1 and themselves. Another way to know these numbers as prime in a more applicable manner, is that when you try to break it down into its factors i.e. numbers that divide it, you can’t break it down into smaller parts.

As an example, take the number 12 . If we break it down into its constituent parts, we can see that it is made up of 4 \times 3 or 2 \times 6 (we shall be ignoring 1 \times 12 ). Can this be broken down further? Yes indeed.

prime-factors.png

Notice that after breaking down the number into its factors for as much as we can, we always end up with a product of prime numbers. Try and convince yourself why this is the case.
More importantly, notice that the answer we get is the same. No matter how we decide to factor the number. We can try this with other numbers to experiment! 15 can be broken down maximally to obtain 5 \times 3 and 18 can be broken down maximally to 2 \times 3 \times 3 . This seems to indicate that EVERY possible number can be written down as a one and only one product of primes and The Fundamental Theorem of Arithmetic gives us that justification that this is indeed the case.

What does this mean with regards to the importance of prime numbers? Well, as previously said, you can now see primes through the lens of being the building blocks of the numbers. In other words, every number has a unique ‘ID’ which is exactly its prime factorization.

numbers.png

This gives critical information on the study of the structure of numbers themselves. As instead of studying particular attributes of numbers directly (which is often quite difficult) we can shift the problem to the study of the structure of their primes. This problem is quite a readily used fact in number theory, and so we must know a great deal of information about the structure of primes in general to get a better idea of these other problems.

This in itself, is not an easy feat either, but certainly opens up a route to attack many other problems which would otherwise be virtually impossible to solve practically.


Intermediate

Before moving on through the history of the importance of prime numbers, we now stumble across this quite famous infinite series called ‘The Harmonic Series’.

\sum^{\infty}_{n=1} \frac{1}{n} = 1+\frac{1}{2}+\frac{1}{3}+...

This series infamously diverges, slowly, but surely to infinity. Even though you are adding consecutively smaller and smaller terms, which is a wonder onto itself! This result has been known for quite a long time, since the 14th century, but somewhere around the mid 17th century, a slight variation of this problem proved to be quite a tough nut to crack.

This variation involved squaring each individual term in the Harmonic Series above to obtain:

\sum^{\infty}_{n=1} \frac{1}{n^2} = 1+\frac{1}{2^2}+\frac{1}{3^2}+...

This infamous problem that not even the great Bernoulli family of mathematicians could solve, would later be called ‘The Basel Problem’. People knew that this infinite sum converged to a particular value but were clueless as to what that value actually was.

Leonhard_Euler

Fast forward to 1734, Leonhard Euler, one of the greatest mathematicians of the 18th century managed to solve the problem. He was able to successfully determine that the value this infinite sum converged to was \frac{\pi^2}{6} . For first time viewers, this might quite an extraordinary result given the sudden appearance of the circle constant \pi .

Leonhard Euler continued to study infinite series of this type, but in a more general sense. He was studying series of the form:

\sum^{\infty}_{n=1} \frac{1}{n^s} = 1+\frac{1}{2^s}+\frac{1}{3^s}+...

where s is some whole number greater than 1.
We shall be looking at his quite clever solution to write this generalized infinite sum, in terms of an infinite product…of infinite series. Now…this might immediately might sound dubious for some, and plain gibberish for others. But Euler wasn’t exactly known for keeping to the rules, he was somewhat of a mathematical outlaw within the community, this . Working outside what was commonly accepted to be ‘right’ to stumble on some quite ‘interesting’ results.

We shall be showing the result for when s=1 in the original problem, but this is merely to understand the idea behind it. The idea works for a general s .

The idea begins by considering the following infinite sum.

S_p=\sum^{\infty}_{n=0} \frac{1}{p^n} =  1 + \frac{1}{p} + \frac{1}{p^2}+\dots

where p is some prime number. To give some context, we have that for p=2 and p=5 for example,

S_2=\sum^{\infty}_{n=0} \frac{1}{2^n} =  1 + \frac{1}{2} + \frac{1}{2^2}+\dots

S_5=\sum^{\infty}_{n=0} \frac{1}{5^n} =  1 + \frac{1}{5} + \frac{1}{5^2}+\dots

Euler’s key insight was in the following….consider S_2S_3S_5S_7S_{11}\dots what would this look like?

(1 + \frac{1}{2} + \frac{1}{2^2}+\dots)(1 + \frac{1}{3} + \frac{1}{3^2}+\dots)(1 + \frac{1}{5} + \frac{1}{5^2}+\dots)\dots

Let us begin opening up the brackets as if we were opening up a finite number of brackets.

  1. (1)(1)(1)(1)\dots = 1
  2. (\frac{1}{2})(1)(1)(1)\dots = \frac{1}{2}
  3. (1)(\frac{1}{3} )(1)(1)\dots = \frac{1}{3}
  4. (\frac{1}{2^2})(1)(1)(1)\dots= \frac{1}{2^2}=\frac{1}{4}
  5. (1)(1)(\frac{1}{5})(1)\dots = \frac{1}{5}
  6. (\frac{1}{2})(\frac{1}{3})(1)(1)\dots = \frac{1}{2}.\frac{1}{3}=\frac{1}{6}
  7. \dots 

Do you realize what the trick is? It is to use the Fundamental Theorem of Arithmetic i.e. that any number has a unique way to be broken down into a product of primes, to generate our original infinite sum that Euler was studying! In other words,

\sum^{\infty}_{n=1} \frac{1}{n} = S_2.S_3.S_5\dots= \prod^{ }_p \frac{p}{p-1}

Apart from this result implying that there necessarily must be an infinite number of prime numbers since the Harmonic Series diverges…this result also has an analogous version for a general s which is:

\sum^{\infty}_{n=1} \frac{1}{n^s} = \prod^{ }_p \frac{p^s}{p^s-1}


Expert

The progress up to why Mathematicians find prime numbers interesting should already be well explained up till now. The idea is the following: studying the structure of the building blocks of numbers, will give you an idea on particular structures of the numbers that you would like to inquire about. As mentioned previously, this venture is halted by the brick wall which comes with studying prime numbers. This is because prime numbers appear to have no recognizable pattern behind them. As an example, there are some very basic but yet ancient questions about the primes which have yet to be solved.

The first of such problems is the Twin Prime Conjecture which asks whether or not there are infinitely many twin primes i.e. primes that differ by 2.  As an example, 3 and 5, 5 and 7, 11 and 13 are twin primes, but as the numbers get bigger, there is a general intuition that the primes are more likely to have bigger and bigger gaps. This doesn’t appear to inhibit twin prime formation though as we can find cases such as 1,034,951 and 1,034,953 which is an example of twin primes in the millions. Here is a list enumerating the first 100,000 twin primes! Are there infinitely many such twin primes? This still hasn’t been proved.

Another famous unanswered problem of primes is the Goldbach Conjecture which asks whether or not any even number greater than 2 can be written as the sum of 2 primes. We can be satisfied that it works for 20=17+3 or 100=11+89 but this very old conjecture is still as of yet unproved.

BernhardRiemannAWeger

The journey of how things further progressed from Euler continues to Bernhard Riemann who continued from Euler’s work on the infinite sum \sum^{\infty}_{n=1} \frac{1}{n^s} which was previously only defined for s\geq 1 and generalizing it further to allow s to take complex values of the form s = \sigma + it . This gave rise to Riemann’s own conjecture, the famous Riemann Hypothesis which is in fact one of the 7 Millenium Prizes and the single most important theorem in mathematics which is as of yet, unsolved.

I would normally explain what exactly is the Riemann Hypothesis and how it relates to the study of prime numbers as we have been doing, but I cannot possibly improve on YouTube’s favourite mathematician James Grime (singingbanana) who gave a truly brilliant summary of the entire history behind the Riemann Hypothesis, its influences and repercussions.


That should conclude my case study on the Primes and the Riemann Hypothesis. I hope that you enjoyed reading and got something out of it that you might not have before, irrespective of your mathematical level. I have also included some other great resources about prime numbers and more details on the Riemann Hypothesis in the links below!

Thanks for reading!

-Sinthorel

Resources:

Prime Factorization and why 1 is not prime:

Another way how to obtain Euler’s general result:

A great blog explaining the Riemann Hypothesis and Euler’s result in more detail:
http://www.riemannhypothesis.info/category/euler-product/

 

Advertisements

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s